Weakly nonlinear models for hydroelastic water waves

WEAKL Y NONLINEAR MODELS F OR HYDR OELASTIC W A TER W A VES DIEGO ALONSO-OR ´ AN, RAF AEL GRANERO-BELINCH ´ ON, AND JULIANA S. ZIEBELL Abstract. In this w ork, w e deriv e reduced in terface models for hydroelastic water wa v es coupled to a nonlinear visco elastic plate. In a weakly nonlinear small-steepness regime w e obtain bidirectional nonlocal evolution equations capturing the interface dynamics up to quadratic order, and w e also derive tw o unidirectional models describing one-way propagation while retaining the leading disper- sive and dissipative eﬀects induced by the plate. Remark ably , one of the bidirectional mo del has a doubly nonlinear structure in the sense that there there is a nonlinear elliptic op erator acting on the acceleration of the interface. W e pro ve lo cal well-posedness for the bidirectional model for small data via a two-parameter regularization and nested ﬁxed p oints. F or the unidirectional mo dels, we obtain local well-posedness for arbitrary data and global well-posedness for small data. Contents 1. In tro duction 2 2. Setting of the problem 7 3. F ormal asymptotic expansion 14 4. W ell-p osedness result for the bidirectional model 22 5. W ell-p osedness results for the unidirectional models 33 App endix A. Geometric form ulas for graph surfaces 43 App endix B. Nondimensionalisation of the Fluid–Structure System 44 Ac knowledgemen ts 48 References 48 Dep ar t amento de An ´ alisis Ma tem ´ atico and Instituto de Ma tem ´ aticas y Aplicaciones (IMAULL), Univer- sidad de La Laguna, C/Astrof ´ ısico Francisco S ´ anchez s/n, 38271, La Laguna, Sp ain. d alonsoo@ull.edu.es Dep ar t amento de Ma tem ´ aticas, Est ad ´ ıstica y Comput aci ´ on, Universid ad de Cant abria, A vda. Los Castros s/n, Sant ander, Sp ain. r af ael.granero@unican.es Dep ar t amento de Ma tem ´ atica Pura e Aplicada Universidade Federal do Rio Grande do Sul Por to Ale- gre, RS 91509, Brazil j ulianaziebell@ufrgs.br 2010 Mathematics Subject Classiﬁc ation. 35Q35, 76B15. Key words and phr ases. hydroelastic wav es; ﬂuid–structure in teraction; asymptotic models; well-posedness. 1 2 D. ALONSO-OR ´ AN, R. GRANERO-BELINCH ´ ON, AND J. S. ZIEBELL 1. Introduction The in teraction betw een incompressible ﬂuids and deformable elastic structures stands as one of the most challenging areas in mo dern mathematical analysis of partial diﬀerential equations. The past t wo decades ha v e seen a substantial expansion of the ﬁeld, driven by applications ranging from na v al h ydro dynamics [ 3 ] and aero elasticity to the mathematics of o ceanic w a v es interacting with sea ice or ﬂoating plates, [ 28 ]. At a mathematical lev el, these problems combine free-boundary ﬂuid dynamics with the nonlinear geometry and high-order mechanics of thin structures, leading to systems in which h yp erb olic (inertia), disp ersiv e (b ending), and sometimes dissipativ e (visco elastic) eﬀects in teract in a genuinely nonlo cal w a y . In this context, one of the most prominent and ph ysically relev an t settings is h ydro elasticit y , where surface wa v es propagate beneath a deformable medium such as sea ice or an elastic plate, and where geometric shell mo dels pro vide a natural framework to describe the elastic forces. In con trast with clas- sical water-w a v e theory , in which the interface is massless and ob eys the Bernoulli dynamic condition, the h ydro elastic interface carries mass, elasticit y , and, in man y situations, damping. A fundamen tal reference in this direction is the work of Plotniko v and T oland [ 27 ], who derived a general geomet- rically nonlinear model for an elastic shell of Cosserat t yp e in teracting with an irrotational Euler ﬂuid. Existing studies of h ydroelastic w a ves are primarily dev oted to trav eling-w a v e solution, either establishing their existence or computing them, most notably in the massless case [ 29 ] and, when the in terface mass is included, in [ 9 , 30 ]. In tw o spatial dimensions the situation has also b een extensiv ely studied, most notably in the vortex-sheet formulation introduced by Ambrose and Liu [ 7 ]. There, the in terface is assumed to b e a one-dimensional curve with elastic prop erties, and the ﬂuid velocity is describ ed by the Birkhoﬀ–Rott integral. Despite the apparent simplicit y of the geometry , the cou- pling b et w een elasticity and the Kelvin–Helmholtz instabilit y of the vortex sheet leads to a highly delicate system. The authors obtained lo cal w ell-p osedness in Sobolev spaces and studied regularity mec hanisms arising from the elastic terms. Bey ond hydroelasticity , sev eral lines of work hav e addressed ﬂuid-structure in teraction problems in- v olving three-dimensional incompressible ﬂuids and deformable solids. One classical approach, initi- ated b y Coutand and Shk oller [ 15 ], emplo ys a fully Lagrangian framework for the elastodynamics of the solid and transforms the ﬂuid equations to the material co ordinates of the structure. This metho d has prov en particularly eﬀectiv e for ﬂuid–solid systems in which the structure retains a volumetric geometry . More recently , Kuk avica, T uﬀaha, and Ziane [ 22 , 23 ] dev eloped a sharp regularit y theory for incompressible ﬂuids interacting with elastic solids, establishing lo cal w ell-posedness under minimal Sob olev assumptions. Their w ork highligh ts the need for precise control of the deformation of the elastic region, the mapping of the ﬂuid equations to a moving coordinate system, and the geometric terms app earing in the transformed system. In the con text of thin structures, mo dels based on the Koiter shell energy hav e emerged as a natural limit of three-dimensional elasticit y . The work of Muha and Cani´ c [ 26 ] represen ts a landmark result WEAKL Y NONLINEAR MODELS F OR HYDR OELASTIC W A TER W A VES 3 in this direction: they establish the existence of weak solutions for the coupling b etw een a viscous incompressible ﬂuid and a Koiter shell, using an op erator-splitting approac h together with Aleksandro v transformations of the moving domain. Their setting assumes a cylindrical structure and exploits symmetries to obtain compactness and energy b ounds that are otherwise diﬃcult to establish. See also [ 24 ] where the existence theory has also b een extended to thermally coupled ﬂuid–structure in teraction models. A further developmen t is due to Balakrishna, Kuk avica, Muha, and T uﬀaha [ 10 ], who studied the in teraction b etw een the three-dimensional Euler equations and a nonlinear tw o-dimensional Koiter plate represen ted as a graph. Their mo del incorporates the full geometric nonlinearity of the Koiter energy , the inertia of the plate, and Kelvin–V oigt damping, and it is p osed on a time-dependent ﬂuid domain whose upp er boundary is the unkno wn graph. Related in viscid free-b oundary ﬂuid– structure mo dels, including v ariants with alternativ e structural la ws, diﬀeren t solution concepts, and global-in-time results in weak frameworks hav e also b een in v estigated in [ 1 , 2 , 8 , 20 , 21 , 31 ]. 1.1. Con tributions and main results. This work addresses a three-dimensional in viscid ﬂuid– structure in teraction problem in whic h an incompressible Euler ﬂow evolv es beneath a nonlinear visco elastic plate represen ted as a graph. Starting from the geometric Euler–plate system ( 2.14 ), w e pursue a tw ofold program. On the one hand, w e deriv e reduced in terface equations in a weakly nonlinear, small-steepness regime, obtaining asymptotic mo dels that capture the leading hydroelastic dynamics. On the other hand, we provide a well-posedness theory for these reduced equations. A similar program has b een implemented for ﬂuid-solid in teraction problems in the case of Stokes law [ 11 , 12 ] and Darcy la w [ 6 ]. Assuming irrotationality , we reduce the Euler–plate system to a b oundary formulation and p erform a w eakly nonlinear small-steepness expansion in ε “ H { L . T runcating at quadratic accuracy yields tw o closed bidirectional in terface equation for the renormalized displacement f “ η p 0 q ` εη p 1 q , ` I ` ΥΛ ˘ f tt ` δ Λ 3 f t ` ´ Λ ` β 4 Λ 5 ¯ f “ ε N r f s , (1.1) where the quadratic forcing N r f s dep ends on the bidirectional mo del under consideration. On the one hand, the ﬁrst bidirectional mo del has a nonlinearit y given b y ( 3.24 ). This ﬁrst bidirectional mo del is doubly nonlinear in the sense that there is a nonlinear elliptic operator acting on the acceleration f tt of the mem brane. On the other hand, a second bidirectional model (with the same accuracy) has a nonlinearity given by explicitly in ( 3.31 ). Moreov er, by restricting to one horizon tal dimension and in tro ducing the c haracteristic v ariables ξ “ x ´ t, τ “ εt, f p x, t q “ F p ξ , τ q , 4 D. ALONSO-OR ´ AN, R. GRANERO-BELINCH ´ ON, AND J. S. ZIEBELL w e obtain t w o unidirectional slo w-mo dulation mo dels. The ﬁrst one, asso ciated with the quadratic- precision bidirectional equation ( 3.24 ), can b e written as F τ “ 1 ε ` a ` b H ˘ ˆ p I ` ΥΛ q F ξ ` β 4 Λ F ξξ ξ ` δ Λ F ξξ ` H F ˙ ´ ` a ` b H ˘ p 2 F ξ Λ F ´ J Λ , F K F ξ q , (1.2) where a “ a p Λ , B ξ q and b “ b p Λ , B ξ q are the real F ourier multipliers deﬁned in ( 3.42 ) (all multipliers understo od with resp ect to ξ ). The second one, asso ciated with the alternativ e quadratic closure ( 3.30 ), takes the form F τ “ 1 ε ´ α ` γ H ¯” p I ` ΥΛ q F ξξ ` β 4 Λ F ξξ ξ ξ ` δ Λ F ξξ ξ ` Λ F ı ´ ´ α ` γ H ¯ « 1 2 Λ ´ F 2 ξ ´ p Λ F q 2 ¯ ´ J Λ , F ξ K F ξ ` F ξξ Λ F ` J Λ , F K T F ´ F ξ HT F ` β 4 ´ J Λ , F K T F ξξ ξ ξ ´ F ξ Λ T F ξξ ξ ¯ ` δ ´ J Λ , F K T F ξξ ξ ´ F ξ Λ T F ξξ ¯ ﬀ , (1.3) where α “ α p Λ , B ξ q and γ “ γ p Λ , B ξ q are the real F ourier m ultipliers in tro duced in ( 3.47 ), and T “ p I ` ΥΛ q ´ 1 Λ is deﬁned in ( 3.27 ). The second purpose of this paper is to establish a w ell-p osedness theory for the reduced mo dels deriv ed ab o v e. More precisely: ‚ Bidir e ctional lo c al wel l-p ose dness for smal l data. W e prov e lo cal existence and uniqueness for the bidirectional interface equation ( 3.24 ) for mean-zero initial data f 0 P H 3 0 p T 2 q , f t p¨ , 0 q P H 1 0 p T 2 q , } f 0 } H 3 suﬃcien tly small , see Theorem 4.1 . A key diﬃculty in the bidirectional mo del ( 3.24 ) is that, once written as the ﬁrst-order system ( 4.1 ), the coupling in v olv es the unkno wn time deriv ative through N ` f , B t v ˘ , and the problem is not a standard semilinear ev olution and in fact it has a doubly nonlinear structure akin to v t ` N 1 p v t q “ N 2 p v q , with N 1 b eing a nonlinear elliptic op erator. Our approac h is therefore based on a t w o-parameter regularization and a nested ﬁxed-point scheme; see Steps 1–5 in the proof of Theorem 4.1 . ‚ Unidir e ctional lo c al wel l-p ose dness and smal l-data glob al de c ay (ﬁrst r e duction). F or the unidi- rectional slow-modulation mo del ( 3.43 ) (equiv alen tly ( 1.2 )), w e establish lo cal w ell-p osedness for arbitrary mean-zero data F 0 P H 2 p T q ; see Theorem 5.1 . The argument relies on a priori energy estimates in the symmetrized form ulation together with a classical F riedric hs molliﬁcation scheme. Moreo ver, for suﬃciently small H 2 data w e prov e global-in-time existence and exp onential decay of the natural energy; see Theorem 5.2 . ‚ Glob al smal l-data wel l-p ose dness (se c ond r e duction). F or the more singular unidirectional mo del ( 3.48 ) (equiv alently ( 1.3 )), we obtain global well-posedness for suﬃciently small mean-zero data F 0 P H 3 p T q ; see Theorem 5.3 . The proof is based on an energy method with a bo otstrap/absorption WEAKL Y NONLINEAR MODELS F OR HYDR OELASTIC W A TER W A VES 5 step which allows the highest-order nonlinear terms to b e controlled by the linear dissipation as long as the solution remains small. 1.2. Notation and preliminary estimates. In what follows we collect the notation and the analytic to ols that will b e used throughout the paper, in particular basic prop erties of the F ourier m ultipliers in volv ed and the Sobolev/pro duct estimates needed in the w ell-posedness analysis. Notation and functional setting. W e w ork with horizon tal v ariables x “ p x 1 , x 2 q P T 2 , v ertical v ariable x 3 P p´8 , 0 q , and time t ě 0. W e write ∇ “ pB x 1 , B x 2 , B x 3 q , ∆ “ B 2 x 1 ` B 2 x 2 ` B 2 x 3 , ∇ x “ pB x 1 , B x 2 q , ∆ x “ B 2 x 1 ` B 2 x 2 , and B t for the time deriv ativ e. Throughout the pap er the notation A ≲ B means A ď C B for a constan t C ą 0 which may dep end on ﬁxed parameters (e.g. Υ , δ , β ) but not on the ev olving functions. F or d P t 1 , 2 u and f : T d Ñ C w e write the F ourier series f p x q “ ÿ k P Z d p f p k q e ik ¨ x , p f p k q “ 1 p 2 π q d ż T d f p x q e ´ ik ¨ x dx, | k | “ p k 2 1 ` ¨ ¨ ¨ ` k 2 d q 1 { 2 . F or s P R w e deﬁne } f } 2 H s p T d q : “ ÿ k P Z d p 1 ` | k | 2 q s | p f p k q| 2 , x f y : “ 1 p 2 π q d ż T d f , H s 0 p T d q : “ t f P H s p T d q : x f y “ 0 u . On mean-zero functions we also use the homogeneous seminorm } f } 2 9 H s p T d q : “ ÿ k ‰ 0 | k | 2 s | p f p k q| 2 , and interpret Λ ´ 1 mo de-b y-mo de on k ‰ 0. By P oincar´ e, for s ě 0 and f P H s 0 p T d q , } f } H s p T d q „ } f } 9 H s p T d q . (1.4) Basic estimates and ine qualities. On T 2 w e denote Λ : “ p´ ∆ x q 1 { 2 , that is x Λ f p k q “ | k | p f p k q , and we use the Riesz transforms y R j f p k q “ ´ i k j | k | p f p k q , k ‰ 0 , j “ 1 , 2 , R f “ p R 1 f , R 2 f q . They are b ounded on Sobolev spaces: for all s P R , } R j f } H s p T 2 q ≲ } f } H s p T 2 q , } R j f } 9 H s p T 2 q ≲ } f } 9 H s p T 2 q . (1.5) W e also use the Sob olev em b edding H 2 p T 2 q ã Ñ W 1 , 8 p T 2 q . (1.6) On H s 0 p T 2 q the resolv ent p I ` ΥΛ q is in vertible mode-by-mode with symbol p 1 ` Υ | k |q ´ 1 for k ‰ 0, and in particular }p I ` ΥΛ q ´ 1 G } H s p T 2 q ď } G } H s p T 2 q , G P H s 0 p T 2 q . (1.7) 6 D. ALONSO-OR ´ AN, R. GRANERO-BELINCH ´ ON, AND J. S. ZIEBELL When restricting to one horizon tal dimension w e work on T with v ariable ξ . F ourier multipliers Λ “ | D ξ | and Λ ´ 1 are understo o d with resp ect to ξ . W e use the Hilb ert transform H , x H g p k q “ ´ i sgn p k q p g p k q , k P Z , so that H ˚ “ ´ H on L 2 p T q and H 2 “ ´ I on mean-zero functions. On nonzero mo des we ha v e the iden tities Λ “ H B ξ , B ξ Λ ´ 1 “ ´ H , B ξ H “ H B ξ , (1.8) and, for mean-zero f and an y s P R , } H f } 9 H s p T q “ } f } 9 H s p T q . (1.9) W e also use T ricomi’s identities: for suﬃcien tly regular real-v alued f , g , H ` f H g ` g H f ˘ “ p H f qp H g q ´ f g, H ` f H f ˘ “ 1 2 ` p H f q 2 ´ f 2 ˘ . (1.10) Next, we gather products, commutators and auxiliary m ultipliers estimates in one dimension. F or a linear op erator A and a function f we write J A, f K g : “ A p f g q ´ f Ag . W e will rep eatedly use that for s ą 1 2 and f P H s 0 p T q , } f } L 8 p T q ≲ s } f } 9 H s p T q , (1.11) so in particular H s p T q is a Banach algebra for s ą 1 2 , with } f g } H s p T q ≲ s } f } H s p T q } g } H s p T q , } f g } 9 H s p T q ≲ s } f } L 8 } g } 9 H s ` } g } L 8 } f } 9 H s . (1.12) W e also use the Gagliardo–Nirenberg inequalit y } f } 2 9 W 1 , 4 p T q “ } f ξ } 2 L 4 p T q ≲ } f } L 8 p T q } f } 9 H 2 p T q . (1.13) Giv en Υ ą 0, we deﬁne the F ourier m ultiplier T : “ p I ` ΥΛ q ´ 1 Λ , x T f p k q “ | k | 1 ` Υ | k | p f p k q , k ‰ 0 , (1.14) so that for all s P R and mean-zero f , } T f } 9 H s p T q ď } f } 9 H s p T q , } Λ T f } 9 H s p T q ≲ Υ } f } 9 H s ` 1 p T q . (1.15) Finally , for ν P p 0 , 1 s consider J ν the heat k ernel on T d ( d “ 1 , 2) deﬁned b y y J ν f p k q : “ e ´ ν | k | 2 p f p k q , k P Z d . Then J ν is self-adjoint on L 2 , it commutes with B ξ and Λ (and with H when d “ 1), and J ν f Ñ f in H s as ν Ñ 0 for all s P R . Moreov er, for every s P R and m ě 0, } J ν f } H s ≲ } f } H s , } J ν f } H s ` m ≲ ν ´ m } f } H s . (1.16) T o close this preliminary section, w e record a standard represen tation form ula for the P oisson problem: WEAKL Y NONLINEAR MODELS F OR HYDR OELASTIC W A TER W A VES 7 Lemma 1.1 (Poisson problem with Dirichlet data) . Let b “ b p x, x 3 q and g “ g p x q be smooth, 2 π –p erio dic in x P T 2 , and assume that b p¨ , x 3 q decays suﬃciently fast as x 3 Ñ ´8 . Consider the problem $ ’ ’ ’ & ’ ’ ’ % ∆ u “ b in Ω : “ T 2 ˆ p´8 , 0 q , u “ g on Γ : “ T 2 ˆ t 0 u , u p¨ , x 3 q Ñ 0 as x 3 Ñ ´8 . (1.17) Then ( 1.17 ) admits a unique solution u , and for every nonzero F ourier mo de k P Z 2 zt 0 u one has B x 3 p u p k , 0 q “ | k | p g p k q ` ż 0 ´8 e | k | y 3 p b p k , y 3 q dy 3 . (1.18) Equiv alently , in operator form on mean–zero functions, B x 3 u p¨ , 0 q “ Λ g ` ż 0 ´8 e y 3 Λ b p¨ , y 3 q dy 3 . (1.19) Pr o of. T aking the F ourier series in x reduces ( 1.17 ) to pB 2 x 3 ´ | k | 2 q p u p k , x 3 q “ p b p k , x 3 q , p u p k , 0 q “ p g p k q , p u p k , x 3 q Ñ 0 as x 3 Ñ ´8 , whic h is solved explicitly by v ariation of constan ts. Diﬀeren tiating at x 3 “ 0 giv es ( 1.18 ). A detailed deriv ation can b e found for instance in [ 16 , 17 , Lemma A.1]. □ 1.3. Organization of the man uscript. The manuscript is organized as follows. In Section 2 we in tro duce the full three-dimensional Euler–plate system, discuss the geometric bending energy and the asso ciated structural forces, and rewrite the problem in p otential v ariables. W e then nondimensionalize the equations and derive an Arbitrary Lagrangian–Eulerian (ALE) formulation on a ﬁxed reference domain. In Section 3 we p erform a small-steepness asymptotic expansion of the ALE system and obtain, up to quadratic order, t w o bidirectional nonlocal in terface models for the surface displacemen t. In Section 4 w e deriv e tw o one-dimensional unidirectional reductions capturing one-wa y propagation while retaining the leading disp ersive and dissipativ e eﬀects. In Section 4 we prov e local w ell-p osedness for the ﬁrst of the bidirectional mo del for small H 3 initial data. In Section 5 we establish well-pose dness results for the unidirectional mo dels, including lo cal well-posedness for arbitrary data and global well- p osedness for small data. Finally , App endix A collects geometric identities for graph surfaces, and App endix B con tains the details of the non-dimensionalisation. 2. Setting of the pr oblem 2.1. The ﬂuid structure in teraction system. In this section we presen t a complete formulation of the ﬂuid–structure in teraction system under consideration. The mo del consists of a three–dimensional incompressible, inviscid ﬂuid ev olving b elo w a deformable elastic surface. The ﬂuid o ccupies the time- dep enden t domain Ω p t q “ tp x 1 , x 2 , x 3 q P R 3 : ´ Lπ ă x 1 , x 2 ă Lπ , x 3 ă η p t, x 1 , x 2 q , t P r 0 , T su , (2.1) 8 D. ALONSO-OR ´ AN, R. GRANERO-BELINCH ´ ON, AND J. S. ZIEBELL where L ą 0 and the elastic free–b oundary surface o ccupied b y the plate is represented as the graph Γ p t q “ tp x 1 , x 2 , η p x 1 , x 2 , t qq : p x 1 , x 2 q P L S 1 ˆ L S 1 , t P r 0 , T su , (2.2) The ﬂuid is assumed to be incompressible, inviscid and of constan t density ρ f ą 0. Therefore, v elo city u “ p u 1 , u 2 , u 3 q and pressure p in the ﬂuid region satisfy the incompressible Euler equations ρ f ` B t u ` p u ¨ ∇ q u ˘ “ ´ ∇ p ´ ρ f g e 3 in Ω p t q , (2.3a) ∇ ¨ u “ 0 in Ω p t q . (2.3b) where g ą 0 denotes gravit y and e 3 “ p 0 , 0 , 1 q . Since the domain extends inﬁnitely deep, w e imp ose the decay condition u p t, x 1 , x 2 , x 3 q Ñ 0 , p p t, x 1 , x 2 , x 3 q Ñ p 8 p t q , x 3 Ñ ´8 . (2.4) Moreo ver, the ﬂuid is assumed to b e p erio dic in the horizontal directions, i.e., for k “ p k 1 , k 2 q P Z 2 w e impose u p t, x 1 ` 2 π Lk 1 , x 2 ` 2 π Lk 2 , x 3 q “ u p t, x 1 , x 2 , x 3 q , t P r 0 , T s , (2.5) and analogously for the pressure p p t, x 1 ` 2 π Lk 1 , x 2 ` 2 π Lk 2 , x 3 q “ p p t, x 1 , x 2 , x 3 q . (2.6) 2.2. Boundary conditions, geometric b ending energy and structural forces. The motion of the free b oundary Γ p t q is gov erned by tw o conditions: a kinematic condition expressing that the in terface is transp orted by the ﬂuid, and a dynamic condition expressing balance of normal stresses. The kinematic b oundary c ondition. Since the free surface consists of material particles of the ﬂuid, its normal velocity must coincide with that of the ﬂuid. Noticing that the up ward unit normal to Γ p t q is giv en b y n p x, t q “ p´ ∇ x η p x, t q , 1 q T a 1 ` | ∇ x η p x, t q| 2 , x “ p x 1 , x 2 q , w e ha ve that B t η “ u ¨ a 1 ` | ∇ x η | 2 n, on Γ p t q (2.7) Ge ometric b ending ener gy, inertia, and damping. Before form ulating the dynamic boundary condition, w e describ e the forces acting on the elastic plate. These consist of (i) a nonlinear bending force obtained as the ﬁrst v ariation of a curv ature-dep endent energy , (ii) an inertial response arising from the mass densit y of the plate, and (iii) a dissipative con tribution mo delling structural damping. T ogether these terms form the structural side of the evolution equation on the free b oundary . F ollowing the hydroelastic shell model of Plotniko v and T oland [ 27 ], we assume that the restoring elastic force deriv es from a curv ature-based b ending energy of the form E b r η s “ ż L S 1 ˆ L S 1 W b ` H p η q ˘ a 1 ` | ∇ x η | 2 dx, (2.8) WEAKL Y NONLINEAR MODELS F OR HYDR OELASTIC W A TER W A VES 9 where H p η q denotes the mean curv ature of the graph Γ p t q and W b is a prescrib ed b ending density . In the context of hydroelasticity , membrane strains are typically negligible compared to b ending, and the energy dep ends solely on curv ature. A widely used and physically relev ant choice is W b p H q “ B 2 H 2 , whic h leads to the Willmore-t yp e energy E b r η s “ ż L S 1 ˆ L S 1 B 2 H p η q 2 a 1 ` | ∇ x η | 2 dx. (2.9) The elastic force asso ciated with this energy is obtained from its ﬁrst v ariation. As sho wn in [ 27 ], one obtains the fourth-order nonlinear geometric op erator E p η q “ B 2 ∆ Γ H p η q ` B H 3 p η q ´ B H p η q K p η q , (2.10) where ∆ Γ is the Laplace–Beltrami op erator and K is the Gauss curv ature of the evolving surface. The op erator retains the full geometric nonlinearity of the plate and dep ends explicitly on η , ∇ x η , and ∇ 2 x η . F or conv enience, explicit expressions for H , K , ∆ Γ , and the metric co eﬃcients in terms of η are collected in Appendix A . In addition to curv ature-driven elasticity , the plate p ossesses inertia: with mass densit y ρ s h per unit area, vertical acceleration produces the inertial force ρ s h B tt η . (2.11) Here ρ s denotes the volumetric densit y of the solid material and h ą 0 is the (constant) thic kness of the plate, so that ρ s h is the mass p er unit area of the structure. This hyperb olic term reﬂects the kinetic resp onse of the elastic sheet and pla ys a central role in the time-dep endent hydroelastic dynamics. Finally , structural or visco elastic damping may b e present. A standard and analytically con venien t c hoice is Kelvin–V oigt damping, mo delled by D p η t q “ ´ γ B t ∆ x η , γ ě 0 , (2.12) corresp onding to a Rayleigh dissipation p otential R p η q “ γ 2 ş |B t ∇ x η | 2 dx . This term introduces high- order dissipation compatible with the geometric structure of the bending op erator and app ears natu- rally in Koiter-t yp e plate mo dels [ 10 ]. These three con tributions com bine to giv e the complete structural response of the plate: the nonlinear b ending op erator E p η q in ( 2.10 ), the inertial force ρ s h η tt in ( 2.11 ), and the damping term D p η t q in ( 2.12 ). T ogether they form the left-hand side of the ev olution equation on the free boundary , and their balance with the normal ﬂuid traction yields the dynamic b oundary condition on Γ p t q considered in the next subsection. 10 D. ALONSO-OR ´ AN, R. GRANERO-BELINCH ´ ON, AND J. S. ZIEBELL Dynamic b oundary c ondition: b alanc e of normal for c es. With the elastic, inertial, and damping forces iden tiﬁed, w e no w state the dynamic boundary condition on the free surface. F or an in viscid, incom- pressible ﬂuid, the Cauch y stress tensor is σ f “ ´ pI . The traction exerted by the ﬂuid across the in terface with out ward unit normal n is therefore σ f n “ ´ p n . In the plate equation, ho w ever, the relev ant quan tit y is the normal force exerted b y the ﬂuid on the structure. By Newton’s third la w, this force is the opp osite of the traction exerted by the structure on the ﬂuid. Consequen tly , the normal ﬂuid force acting on the plate is T ﬂuid “ ´p σ f n q ¨ n “ p, on Γ p t q . Balancing this ﬂuid force with the inertial, elastic, and damping contributions of the plate yields Newton’s second la w on the moving b oundary: ρ s h B tt η ` E p η q ´ γ B t ∆ x η “ p on Γ p t q . (2.13) Equation ( 2.13 ) constitutes the fully nonlinear dynamic boundary condition gov erning the evolution of the elastic surface. 2.3. The full Euler–plate system and dimensionless system. Equation ( 2.13 ), together with the kinematic condition ( 2.7 ) and the Euler equations ( 2.3 ), completes the formulation of the ﬂuid– structure interaction problem. F or con v enience, w e collect the full system below: ρ f pB t u ` p u ¨ ∇ q u q “ ´ ∇ p ´ ρ f g e 3 in Ω p t q , (2.14a) ∇ ¨ u “ 0 in Ω p t q , (2.14b) B t η “ u ¨ a 1 ` | ∇ x η | 2 n on Γ p t q , (2.14c) ρ s h B tt η ` E p η q ´ γ B t ∆ x η “ p on Γ p t q . (2.14d) where E p η q is giv en in ( 2.10 ). Assuming that the ﬂow is irrotational, we in tro duce a potential ϕ such that u “ ∇ ϕ and denote b y ψ p x 1 , x 2 , t q “ ϕ p t, x 1 , x 2 , η p x 1 , x 2 , t qq the surface p otential. T aking the trace of the Bernoulli relation p “ ´ ρ f ` B t ϕ ` 1 2 | ∇ ϕ | 2 ` g x 3 ˘ in Ω p t q , yields p “ ´ ρ f ´ B t ϕ ` 1 2 | ∇ ϕ | 2 ` g η ¯ on Γ p t q . Moreo ver, using the c hain rule, w e can write B t ψ “ B t ϕ ` B x 3 ϕ B t η , on Γ p t q . and using the kinematic boundary condition in ( 2.14c ), B t ψ “ B t ϕ ` B x 3 ϕ ´ ∇ ϕ ¨ a 1 ` | ∇ x η | 2 n ¯ , on Γ p t q . WEAKL Y NONLINEAR MODELS F OR HYDR OELASTIC W A TER W A VES 11 Th us, the ﬂuid–structure interaction system admits the follo wing p otential formulation ∆ ϕ “ 0 in Ω p t q , (2.15a) ϕ “ ψ on Γ p t q , (2.15b) B t η “ ∇ ϕ ¨ a 1 ` | ∇ x η | 2 n on Γ p t q , (2.15c) B t ψ “ ´ ρ s h ρ f B 2 t η ´ 1 ρ f E p η q ` γ ρ f B t ∆ x η ` B x 3 ϕ ` ∇ ϕ ¨ a 1 ` | ∇ x η | 2 n ˘ ´ 1 2 | ∇ ϕ | 2 ´ g η on Γ p t q . (2.15d) where we recall that E p η q is giv en in ( 2.10 ). Remark 2.1 (General b ending energies) . The structure of the mo del remains unchanged for general b ending energies of the form E b r η s “ ż L S 1 ˆ L S 1 W b p H p η qq , a 1 ` | ∇ x η | 2 dx, with W b smo oth. The corresp onding elastic operator is E p η q “ 1 2 ∆ Γ W 1 b p H q ` 2 H p H W 1 b p H q ´ W b p H qq ´ K W 1 b p H q . The c hoice W b p H q “ B 2 H 2 is made for physical and analytic clarity , but more general mo dels arise in Cosserat shell theory . Remark 2.2 (Dep endence on the Gauss curv ature) . Bending energies dep ending on both H and K can also b e considered: E b r η s “ ż L S 1 ˆ L S 1 W p H , K q a 1 ` | ∇ x η | 2 dx. Suc h energies lead to additional terms inv olving the v ariation of K and may require higher regularity . These app ear in the literature on elastic shells and biomembranes, though we do not pursue them here. Dimensionless quantities and system. In order to deriv e the asymptotic mo dels, w e provide the non- dimensionalization of the system ( 2.15a )-( 2.15d ). Let L denote the typical horizontal wa velength and H the typical vertical amplitude of the elastic sheet. W e in tro duce the dimensionless spatial and temp oral v ariables x “ L ˜ x, x 3 “ L ˜ x 3 , t “ d L g ˜ t, and the non-dimensional surface displacemen t, p otential and surface p otential η p x, t q “ H ˜ η p ˜ x, ˜ t q , ϕ p x, x 3 , t q “ H a g L ˜ ϕ p ˜ x, ˜ x 3 , ˜ t q , ψ p x, t q “ H a g L ˜ ψ p ˜ x, ˜ t q . Accordingly , w e deﬁne the non-dimensionalized ﬂuid domain r Ω p t q “ tp ˜ x 1 , ˜ x 2 , ˜ x 3 q P R 3 : ´ π ă ˜ x 1 , ˜ x 2 ă π , ˜ x 3 ă ε ˜ η p ˜ x, ˜ t q , t P r 0 , T su , 12 D. ALONSO-OR ´ AN, R. GRANERO-BELINCH ´ ON, AND J. S. ZIEBELL and free surface ˜ Γ p t q “ tp ˜ x 1 , ˜ x 2 , ε η p ˜ x, ˜ t qq : ´ π ă ˜ x 1 , ˜ x 2 ă π , t P r 0 , T su . Here we introduce the nondimensional parameters Υ : “ ρ s h ρ f L , β : “ H B ρ f g L 4 , δ : “ γ ρ f a g L 3 , ε : “ H L . The non-dimensionalisation of the potential formulation ( 2.15a )–( 2.15d ) is carried out in detail in App endix B . In particular, after in tro ducing the dimensionless v ariables and parameters and dropping tildes for notational simplicity , the ﬂuid–structure system takes the form ∆ ϕ “ 0 in Ω p t q , (2.16) ϕ “ ψ on Γ p t q , (2.17) B t η “ ∇ ϕ ¨ p´ ε ∇ x η , 1 q on Γ p t q , (2.18) B t ψ “ ´ Υ B tt η ´ β E p η ; ε q ` δ B t ∆ x η ` ε B x 3 ϕ p ∇ ϕ ¨ p´ ε ∇ x η , 1 qq ´ ε 2 | ∇ ϕ | 2 ´ η , on Γ p t q . (2.19) The dimensionless op erator E p η ; ε q is given explicitly b y E p η ; ε q “ 1 2 L Γ p η ; ε q ` ε 2 ´ H p η ; ε q 3 ´ H p η ; ε q K p η ; ε q ¯ , where L Γ p η ; ε q “ 1 ? α B x i ` ? α α ij B x j H p η ; ε q ˘ , and H p η ; ε q “ 1 2 ? α ´ α 11 η x 1 x 1 ` 2 α 12 η x 1 x 2 ` α 22 η x 2 x 2 ¯ , K p η ; ε q “ η x 1 x 1 η x 2 x 2 ´ η 2 x 1 x 2 α 2 . α “ 1 ` ε 2 | ∇ x η | 2 , α 11 “ 1 ` ε 2 η 2 x 2 α , α 22 “ 1 ` ε 2 η 2 x 1 α , α 12 “ ´ ε 2 η x 1 η x 2 α . 2.4. The Arbitrary Lagrangian-Eulerian (ALE) formulation. In this section, our goal is to transform the free-b oundary problem in to a ﬁxed reference domain Ω “ T 2 ˆ p´8 , 0 q , Γ “ T 2 ˆ t 0 u . T o this end, w e introduce a time-dep endent diﬀeomorphism Ψ p¨ , t q : Ω Ý Ñ Ω p t q , deﬁned by Ψ p x 1 , x 2 , x 3 , t q “ ` x 1 , x 2 , x 3 ` ε η p x 1 , x 2 , t q ˘ . This mapping straigh tens the mo ving free surface x 3 “ εη p x 1 , x 2 , t q in to the ﬁxed b oundary x 3 “ 0. Giv en a scalar function f deﬁned on Ω p t q , we deﬁne its ALE coun terpart on the reference domain Ω b y composition with Ψ, that is, F “ f ˝ Ψ . WEAKL Y NONLINEAR MODELS F OR HYDR OELASTIC W A TER W A VES 13 In particular, w e introduce the ALE p oten tial Φ “ ϕ ˝ Ψ . W e no w compute the gradient of the diﬀeomorphism Ψ. A direct calculation yields ∇ Ψ “ ˜ 1 0 0 0 1 0 ε B x 1 η p x, t q ε B x 2 η p x, t q 1 ¸ , p ∇ Ψ q ´ 1 “ ˜ 1 0 0 0 1 0 ´ ε B x 1 η p x, t q ´ ε B x 2 η p x, t q 1 ¸ . Adopting the Einstein summation con v ention and deﬁning the ALE pullback F : “ f ˝ Ψ, the c hain rule yields pB x j f q ˝ Ψ “ A k j B x k F , A “ p ∇ Ψ q ´ 1 . Consequen tly , p ∆ f q ˝ Ψ “ A i j B x i ` A k j B x k F ˘ . Using the explicit form of A and expanding up to ﬁrst order in ε , w e obtain p ∆ f q ˝ Ψ “ ∆ F ´ ε ” p η x 1 x 1 ` η x 2 x 2 q B x 3 F ` η x 1 pB x 3 x 1 F ` B x 1 x 3 F q ` η x 2 pB x 3 x 2 F ` B x 2 x 3 F q ı ` O p ε 2 q . Finally , the comp osition of equations ( 2.16 )-( 2.19 ) the system with the diﬀeomorphism reads to ∆Φ ´ ε ” p η x 1 x 1 ` η x 2 x 2 q Φ x 3 ` η x 1 ` Φ x 3 x 1 ` Φ x 1 x 3 ˘ ` η x 2 ` Φ x 3 x 2 ` Φ x 2 x 3 ˘ ı “ O p ε 2 q in Ω , (2.20) Φ “ ψ on Γ , (2.21) B t η “ Φ x 3 ´ ε ` η x 1 Φ x 1 ` η x 2 Φ x 2 ˘ ` O p ε 2 q on Γ , (2.22) B t ψ “ ´ Υ B tt η ´ β 4 ∆ 2 x η ` δ B t ∆ x η ´ η ` ε 2 ´ Φ 2 x 3 ´ | ∇ x Φ | 2 ¯ ` O p ε 2 q on Γ . (2.23) Remark 2.3 (Leading–order reduction of the b ending op erator) . At leading order in the small– steepness parameter ε , the geometric bending operator E p η ; ε q reduces to a linear biharmonic operator. Indeed, α “ 1 ` O p ε 2 q , α ij “ δ ij ` O p ε 2 q , so that H p η ; ε q “ 1 2 ∆ x η ` O p ε 2 q , E p η ; ε q “ 1 4 ∆ 2 x η ` O p ε 2 q . Accordingly , in equation ( 3.14 ) the elastic resp onse of the plate is go v erned by a biharmonic op erator, while nonlinear geometric eﬀects are encapsulated in the O p ε 2 q terms. 14 D. ALONSO-OR ´ AN, R. GRANERO-BELINCH ´ ON, AND J. S. ZIEBELL 3. F ormal asymptotic exp ansion W e no w deriv e a hierarc hy of appro ximate models b y p erforming a formal small–steepness expansion in the parameter ε in the ALE system ( 3.11 )–( 3.14 ). Sp eciﬁcally , we seek Φ, ψ and η as p ow er series in ε of the form Φ p x 1 , x 2 , x 3 , t q “ 8 ÿ n “ 0 ε n Φ p n q p x 1 , x 2 , x 3 , t q , (3.1) ψ p x 1 , x 2 , t q “ 8 ÿ n “ 0 ε n ψ p n q p x 1 , x 2 , t q , η p x 1 , x 2 , t q “ 8 ÿ n “ 0 ε n η p n q p x 1 , x 2 , t q , (3.2) Substituting ( 3.1 )–( 3.2 ) into ( 3.11 )–( 3.14 ) and collecting terms of equal p ow ers of ε yields, at each order n ě 0, a linear b oundary v alue problem for Φ p n q in Ω coupled to ev olution equations for ψ p n q and η p n q on Γ “ t x 3 “ 0 u . In particular, the leading order n “ 0 provides the principal (linear) h ydro elastic mo del, while n “ 1 captures the ﬁrst nonlinear correction. The initial data are expanded consistently as Φ p x, x 3 , 0 q “ 8 ÿ n “ 0 ε n Φ p n q p x, x 3 , 0 q , ψ p x, 0 q “ 8 ÿ n “ 0 ε n ψ p n q p x, 0 q , η p x, 0 q “ 8 ÿ n “ 0 ε n η p n q p x, 0 q , so that Φ p n q p¨ , ¨ , 0 q , ψ p n q p¨ , 0 q and η p n q p¨ , 0 q represen t the n -th order initial proﬁles in the expansion. The c ase n “ 0 . W e substitute the formal expansions ( 3.1 )-( 3.2 ) into the ALE system ( 3.11 )–( 3.14 ) and retaining only the leading-order contributions yields the following system for p Φ p 0 q , ψ p 0 q , η p 0 q q : ∆Φ p 0 q “ 0 in Ω , (3.3) Φ p 0 q “ ψ p 0 q on Γ , (3.4) B t η p 0 q “ B x 3 Φ p 0 q on Γ , (3.5) B t ψ p 0 q “ ´ Υ B tt η p 0 q ´ β 4 ∆ 2 x η p 0 q ` δ B t ∆ x η p 0 q ´ η p 0 q on Γ . (3.6) T aking the F ourier series in x P T 2 , the unique decaying solution of ( 3.3 )–( 3.4 ) is y Φ p 0 q p k , x 3 , t q “ y ψ p 0 q p k , t q e | k | x 3 , k P Z 2 , x 3 ď 0 , and in particular B x 3 Φ p 0 q ˇ ˇ Γ “ Λ ψ p 0 q . Therefore, the kinematic condition ( 3.5 ) b ecomes B t η p 0 q “ Λ ψ p 0 q on Γ . (3.7) Diﬀeren tiating ( 3.7 ) in time and using ( 3.6 ) yields a closed equation for η p 0 q : B tt η p 0 q “ Λ ´ ´ Υ B tt η p 0 q ´ β 4 ∆ 2 x η p 0 q ` δ B t ∆ x η p 0 q ´ η p 0 q ¯ , on Γ . (3.8) WEAKL Y NONLINEAR MODELS F OR HYDR OELASTIC W A TER W A VES 15 Equiv alently , ` I ` ΥΛ ˘ B tt η p 0 q ` β 4 Λ∆ 2 x η p 0 q ´ δ Λ B t ∆ x η p 0 q ` Λ η p 0 q “ 0 . (3.9) Let p η p 0 q p k , t q denote the F ourier co eﬃcient of η p 0 q . F or k ‰ 0, using x Λ f p k q “ | k | p f p k q and z ∆ x f p k q “ ´| k | 2 p f p k q , equation ( 3.8 ) b ecomes p 1 ` Υ | k |q B 2 t p η p 0 q p k , t q ` δ | k | 3 B t p η p 0 q p k , t q ` ´ | k | ` β 4 | k | 5 ¯ p η p 0 q p k , t q “ 0 . (3.10) The c ase n “ 1 . Similarly , in the case n “ 1 the leading-order con tributions yields the following system for p Φ p 1 q , ψ p 0 q , η p 1 q q : ∆Φ p 1 q “ ” p η p 0 q x 1 x 1 ` η p 0 q x 2 x 2 q Φ p 0 q x 3 ` η p 0 q x 1 ` Φ p 0 q x 3 x 1 ` Φ p 0 q x 1 x 3 ˘ ` η p 0 q x 2 ` Φ p 0 q x 3 x 2 ` Φ p 0 q x 2 x 3 ˘ ı in Ω , (3.11) Φ p 1 q “ ψ p 1 q on Γ , (3.12) B t η p 1 q “ Φ p 1 q x 3 ´ ` η p 0 q x 1 Φ p 0 q x 1 ` η p 0 q x 2 Φ p 0 q x 2 ˘ on Γ , (3.13) B t ψ p 1 q “ ´ Υ B tt η p 1 q ´ β 4 ∆ 2 x η p 1 q ` δ B t ∆ x η p 1 q ´ η p 1 q ` 1 2 ´ p Φ p 0 q x 3 q 2 ´ | ∇ x Φ p 0 q | 2 ¯ on Γ . (3.14) A conv enien t form for the bulk source term is to write b p x 1 , x 2 , x 3 , t q “ ∆ x η p 0 q B x 3 Φ p 0 q ` 2 ∇ x η p 0 q ¨ ∇ x B x 3 Φ p 0 q . Therefore, by using Lemma 1.1 w e can compute B x 3 y Φ p 1 q p k , 0 , t q “ | k | y ψ p 1 q p k , t q ` ż 0 ´8 p b p k , y 3 , t q e | k | y 3 dy 3 , k “ p k 1 , k 2 q P Z 2 . (3.15) In order to provide an explicit expression for the previous integral term, w e ﬁrst notice that p b p k , x 3 q “ ÿ m P Z 2 { ∆ x η p 0 q p k ´ m q { B 3 Φ p 0 q p m, x 3 q ` 2 ÿ m P Z 2 { ∇ η p 0 q p k ´ m q ¨ { ∇ B 3 Φ p 0 q p m, x 3 q “ ÿ m P Z 2 ´ ´ | k ´ m | 2 ¯´ | m | p ψ p 0 q p m q e | m | x 3 ¯ p η p 0 q p k ´ m q ` 2 ÿ m P Z 2 ´ i p k ´ m q p η p 0 q p k ´ m q ¯ ¨ ´ im | m | p ψ p 0 q p m q e | m | x 3 ¯ “ ´ ÿ m P Z 2 | m | ` | k | 2 ´ | m | 2 ˘ p η p 0 q p k ´ m q p ψ p 0 q p m q e | m | x 3 . (3.16) Th us, using ( 3.15 ) and ( 3.16 ), w e obtain B x 3 Φ p 1 q “ Λ ψ p 1 q ´ r Λ , η p 0 q z Λ ψ p 0 q , on Γ . (3.17) Com bining ( 3.17 ) and ( 3.13 ), the ev olution equation for η p 1 q b ecomes B t η p 1 q “ Λ ψ p 1 q ´ r Λ , η p 0 q z Λ ψ p 0 q ´ ` η p 0 q x 1 Φ p 0 q x 1 ` η p 0 q x 2 Φ p 0 q x 2 ˘ , on Γ , (3.18) 16 D. ALONSO-OR ´ AN, R. GRANERO-BELINCH ´ ON, AND J. S. ZIEBELL Diﬀeren tiating ( 3.18 ) in time and using ( 3.14 ) w e deduce B tt η p 1 q “ Λ ´ ´ Υ B tt η p 1 q ´ β 4 ∆ 2 x η p 1 q ` δ B t ∆ x η p 1 q ´ η p 1 q ` 1 2 ` p Λ ψ p 0 q q 2 ´ | ∇ x ψ p 0 q | 2 ˘ ¯ (3.19) ´ B t r Λ , η p 0 q z Λ ψ p 0 q ´ B t ´ η p 0 q x 1 Φ p 0 q x 1 ` η p 0 q x 2 Φ p 0 q x 2 ¯ . Rearranging the terms and collecting all contributions inv olving η p 1 q on the left-hand side, we obtain ` I ` ΥΛ ˘ B tt η p 1 q ` β 4 Λ∆ 2 x η p 1 q ´ δ Λ B t ∆ x η p 1 q ` Λ η p 1 q (3.20) “ 1 2 Λ ´ p Λ ψ p 0 q q 2 ´ | ∇ x ψ p 0 q | 2 ¯ ´ B t ´ r Λ , η p 0 q z Λ ψ p 0 q ¯ ´ B t ´ η p 0 q x 1 Φ p 0 q x 1 ` η p 0 q x 2 Φ p 0 q x 2 ¯ . Next, we eliminate ψ p 0 q in fa vour of η p 0 q using the relation ( 3.7 ) On nonzero F ourier mo des, this yields ψ p 0 q “ Λ ´ 1 B t η p 0 q , Λ ψ p 0 q “ B t η p 0 q , ∇ x ψ p 0 q “ ∇ x Λ ´ 1 B t η p 0 q . (3.21) Therefore, ( 3.20 ) can b e rewritten as ` I ` ΥΛ ˘ B tt η p 1 q ` β 4 Λ∆ 2 x η p 1 q ´ δ Λ B t ∆ x η p 1 q ` Λ η p 1 q (3.22) “ 1 2 Λ ´ pB t η p 0 q q 2 ´ ˇ ˇ ∇ x Λ ´ 1 B t η p 0 q ˇ ˇ 2 ¯ ´ B t ´ r Λ , η p 0 q z B t η p 0 q ¯ ´ B t ´ ∇ x η p 0 q ¨ ∇ x Λ ´ 1 B t η p 0 q ¯ . Com bining the leading-order equation for η p 0 q with ε times ( 3.22 ), and recalling that f “ η p 0 q ` εη p 1 q , w e obtain, up to terms of order O p ε 2 q , ` I ` ΥΛ ˘ B tt f ` β 4 Λ∆ 2 x f ´ δ Λ B t ∆ x f ` Λ f (3.23) “ ε « 1 2 Λ ´ pB t η p 0 q q 2 ´ ˇ ˇ ∇ x Λ ´ 1 B t η p 0 q ˇ ˇ 2 ¯ ´ B t ´ r Λ , η p 0 q z B t η p 0 q ¯ ´ B t ´ ∇ x η p 0 q ¨ ∇ x Λ ´ 1 B t η p 0 q ¯ ﬀ . Since εf “ εη p 0 q ` O p ε 2 q , w e may replace η p 0 q b y f in the right-hand side at the cost of an O p ε 2 q error. Hence, neglecting O p ε 2 q terms and recalling that ∆ 2 x “ Λ 4 and R “ ∇ x Λ ´ 1 the equation closes as ` I ` ΥΛ ˘ f tt ` δ Λ 3 f t ` ´ Λ ` β 4 Λ 5 ¯ f “ ε « 1 2 Λ ´ pB t f q 2 ´ ˇ ˇ R B t f ˇ ˇ 2 ¯ ´ B t p J Λ , f K B t f q ´ B t ´ ∇ x f ¨ R B t f ¯ ﬀ . (3.24) WEAKL Y NONLINEAR MODELS F OR HYDR OELASTIC W A TER W A VES 17 In the sequel we derive from ( 3.24 ) a diﬀerent equation of the same order of precision, i.e., neglecting terms of order O p ε 2 q . T o that purp ose, w e start from ( 3.22 ) and expand the time deriv atives in the forcing. Using the product rule, B t ` r Λ , η p 0 q z B t η p 0 q ˘ “ r Λ , B t η p 0 q z B t η p 0 q ` r Λ , η p 0 q z B tt η p 0 q , and B t ´ ∇ x η p 0 q ¨ R B t η p 0 q ¯ “ ∇ x B t η p 0 q ¨ R B t η p 0 q ` ∇ x η p 0 q ¨ R B tt η p 0 q . Hence ( 3.22 ) ma y b e rewritten as ` I ` ΥΛ ˘ B tt η p 1 q ` β 4 Λ∆ 2 x η p 1 q ´ δ Λ B t ∆ x η p 1 q ` Λ η p 1 q (3.25) “ 1 2 Λ ´ pB t η p 0 q q 2 ´ ˇ ˇ R B t η p 0 q ˇ ˇ 2 ¯ ´ r Λ , B t η p 0 q z B t η p 0 q ´ r Λ , η p 0 q z B tt η p 0 q ´ p ∇ x B t η p 0 q q ¨ R B t η p 0 q ´ p ∇ x η p 0 q q ¨ R B tt η p 0 q . W e no w eliminate B tt η p 0 q b y means of the leading-order closure ( 3.9 ). Solving ( 3.9 ) for B tt η p 0 q giv es B tt η p 0 q “ ´p I ` ΥΛ q ´ 1 ˆ β 4 Λ∆ 2 x η p 0 q ´ δ Λ B t ∆ x η p 0 q ` Λ η p 0 q ˙ . (3.26) Moreo ver, in tro ducing the F ourier m ultiplier T : “ p I ` ΥΛ q ´ 1 Λ . (3.27) equation ( 3.26 ) b ecomes B tt η p 0 q “ ´ β 4 T ∆ 2 x η p 0 q ` δ T B t ∆ x η p 0 q ´ T η p 0 q . (3.28) Substituting ( 3.28 ) into ( 3.25 ) yields a forcing dep ending only on η p 0 q and B t η p 0 q : ` I ` ΥΛ ˘ B tt η p 1 q ` β 4 Λ∆ 2 x η p 1 q ´ δ Λ B t ∆ x η p 1 q ` Λ η p 1 q (3.29) “ 1 2 Λ ´ pB t η p 0 q q 2 ´ ˇ ˇ R B t η p 0 q ˇ ˇ 2 ¯ ´ r Λ , B t η p 0 q z B t η p 0 q ´ p ∇ x B t η p 0 q q ¨ R B t η p 0 q ` r Λ , η p 0 q z T η p 0 q ` p ∇ x η p 0 q q ¨ R ` T η p 0 q ˘ ` β 4 ´ r Λ , η p 0 q z T ∆ 2 x η p 0 q ` p ∇ x η p 0 q q ¨ R ` T ∆ 2 x η p 0 q ˘ ¯ ´ δ ´ r Λ , η p 0 q z T B t ∆ x η p 0 q ` p ∇ x η p 0 q q ¨ R ` T B t ∆ x η p 0 q ˘ ¯ . Finally , deﬁning the renormalized v ariable f “ η p 0 q ` ε η p 1 q , 18 D. ALONSO-OR ´ AN, R. GRANERO-BELINCH ´ ON, AND J. S. ZIEBELL adding the n “ 0 equation ( 3.9 ) to ε times ( 3.29 ), and neglecting O p ε 2 q terms, we obtain a closed O p ε q mo del for f of the form ` I ` ΥΛ ˘ f tt ` δ Λ 3 f t ` ´ Λ ` β 4 Λ 5 ¯ f “ ε N r f s , (3.30) where the nonlinear forcing is N r f s “ 1 2 Λ ´ p f t q 2 ´ ˇ ˇ R f t ˇ ˇ 2 ¯ ´ J Λ , f t K f t ´ p ∇ x f t q ¨ R f t (3.31) ` J Λ , f K T f ` p ∇ x f q ¨ R ` T f ˘ ` β 4 ´ J Λ , f K T ∆ 2 x f ` p ∇ x f q ¨ R ` T ∆ 2 x f ˘ ¯ ´ δ ´ J Λ , f K T B t ∆ x f ` p ∇ x f q ¨ R ` T B t ∆ x f ˘ ¯ . 3.1. The unidirectional models. In this section we deriv e unidirectional asymptotic models asso ci- ated with the bidirectional systems obtained ab ov e. T o that purp ose, as in [ 4 , 5 , 14 , 16 , 18 ], we restrict to one horizon tal dimension x P T and introduce the unidirectional v ariables ξ “ x ´ t, τ “ εt, f p x, t q “ F p ξ , τ q . The unidir e ctional mo del for system ( 3.24 ) . By the c hain rule, B t “ ´B ξ ` ε B τ , B x “ B ξ , (3.32) and consequently f t “ ´ F ξ ` εF τ , f tt “ F ξξ ´ 2 εF ξτ ` O p ε 2 q , f xx “ F ξξ , f xxxx “ F ξξ ξ ξ . (3.33) Substituting ( 3.33 ) into ( 3.24 ) and discarding O p ε 2 q terms yields the p ξ , τ q -equation p I ` ΥΛ q ` F ξξ ´ 2 εF ξτ ˘ ` β 4 Λ F ξξ ξ ξ ` δ Λ F ξξ ξ ´ ε δ Λ F τ ξ ξ ` Λ F (3.34) “ ε « 1 2 Λ ´ F 2 ξ ´ ` Λ F ˘ 2 ¯ ´ B ξ ´ J Λ , F K F ξ ¯ ` B ξ ´ F ξ Λ F ¯ ﬀ , where all F ourier m ultipliers are understoo d with resp ect to ξ , and w e used B t B xx f “ ´ F ξξ ξ ` εF τ ξ ξ . In tegrating ( 3.34 ) once in ξ and using that Λ comm utes with B ξ , we obtain (up to an additiv e function of τ which v anishes under the mean-zero conv en tion) p I ` ΥΛ q ` F ξ ´ 2 εF τ ˘ ` β 4 Λ F ξξ ξ ` δ Λ F ξξ ´ ε δ Λ F τ ξ ` H F (3.35) “ ε « 1 2 H ´ F 2 ξ ´ ` Λ F ˘ 2 ¯ ´ J Λ , F K F ξ ` F ξ Λ F ﬀ . Moreo ver, b y means of the T ricomi iden tit y for the Hilb ert transform ( 1.10 ), w e can write 1 2 H ´ F 2 ξ ´ ` Λ F ˘ 2 ¯ “ 1 2 H ´ F 2 ξ ´ ` H F ξ ˘ 2 ¯ “ F ξ H F ξ “ F ξ Λ F . (3.36) WEAKL Y NONLINEAR MODELS F OR HYDR OELASTIC W A TER W A VES 19 Th us, ( 3.35 ) simpliﬁes to p I ` ΥΛ q ` F ξ ´ 2 εF τ ˘ ` β 4 Λ F ξξ ξ ` δ Λ F ξξ ´ ε δ Λ F τ ξ ` H F “ ε ´ ´ J Λ , F K F ξ ` 2 F ξ Λ F ¯ . (3.37) Rewriting ( 3.35 ), moving the τ -terms to the left-hand side, w e obtain the ev olution equation ε ´ 2 p I ` ΥΛ q ` δ Λ B ξ ¯ F τ “ p I ` ΥΛ q F ξ ` β 4 Λ F ξξ ξ ` δ Λ F ξξ ` H F ´ ε ´ 2 F ξ Λ F ´ J Λ , F K F ξ ¯ . (3.38) Starting from ( 3.38 ), we write ε M F τ “ p I ` ΥΛ q F ξ ` β 4 Λ F ξξ ξ ` δ Λ F ξξ ` H F ´ ε « 2 F ξ Λ F ´ J Λ , F K F ξ ﬀ , (3.39) where M : “ 2 p I ` ΥΛ q ` δ Λ B ξ , M ˚ : “ 2 p I ` ΥΛ q ´ δ Λ B ξ . Multiplying ( 3.39 ) on the left b y M ˚ and using M ˚ M “ 4 p I ` ΥΛ q 2 ´ δ 2 Λ 2 B 2 ξ , w e obtain ε p M ˚ M q F τ “ M ˚ ˆ p I ` ΥΛ q F ξ ` β 4 Λ F ξξ ξ ` δ Λ F ξξ ` H F ˙ ´ ε M ˚ p 2 F ξ Λ F ´ J Λ , F K F ξ q . (3.40) Applying p M ˚ M q ´ 1 (understo od mo de-b y-mo de on k ‰ 0) yields the ﬁrst inv ersion step F τ “ 1 ε p M ˚ M q ´ 1 M ˚ « p I ` ΥΛ q F ξ ` β 4 Λ F ξξ ξ ` δ Λ F ξξ ` H F ﬀ (3.41) ´ p M ˚ M q ´ 1 M ˚ « 2 F ξ Λ F ´ J Λ , F K F ξ ﬀ . On the F ourier side ( k ‰ 0), one c hecks that p M ˚ M q ´ 1 M ˚ “ a p Λ , B ξ q ` b p Λ , B ξ q H , where a p Λ , B ξ q and b p Λ , B ξ q are real F ourier m ultipliers with symbols a k : “ 2 p 1 ` Υ | k |q 4 p 1 ` Υ | k |q 2 ` δ 2 | k | 2 k 2 , b k : “ δ | k | 2 4 p 1 ` Υ | k |q 2 ` δ 2 | k | 2 k 2 , k ‰ 0 , or equiv alen tly , a p Λ , B ξ q “ 2 p I ` ΥΛ q 4 p I ` ΥΛ q 2 ´ δ 2 Λ 2 B 2 ξ , b p Λ , B ξ q “ δ Λ 2 4 p I ` ΥΛ q 2 ´ δ 2 Λ 2 B 2 ξ . (3.42) Substituting this iden tit y in to ( 3.41 ) yields F τ “ 1 ε ` a ` b H ˘ ˆ p I ` ΥΛ q F ξ ` β 4 Λ F ξξ ξ ` δ Λ F ξξ ` H F ˙ ´ ` a ` b H ˘ p 2 F ξ Λ F ´ J Λ , F K F ξ q , (3.43) with a and b giv en by ( 3.42 ). 20 D. ALONSO-OR ´ AN, R. GRANERO-BELINCH ´ ON, AND J. S. ZIEBELL 3.1.1. The unidir e ctional mo del for ( 3.30 ) . W e now deriv e the unidirectional reduction of ( 3.30 ) in the same manner as for the ﬁrst mo del. More precisely , we restrict to one horizontal space dimension and introduce the slow v ariables ξ “ x ´ t, τ “ εt, f p x, t q “ F p ξ , τ q . With these con v entions, up to O p ε 2 q we hav e f t “ ´ F ξ ` εF τ , f tt “ F ξξ ´ 2 εF ξτ ` O p ε 2 q , f xx “ F ξξ , f xxxx “ F ξξ ξ ξ , B t f xx “ p´B ξ ` ε B τ q F ξξ “ ´ F ξξ ξ ` εF τ ξ ξ . Substituting into equation ( 3.30 ), doing some straightforw ard computations and neglecting O p ε 2 q we ﬁnd the equation p I ` ΥΛ q ` F ξξ ´ 2 εF ξτ ˘ ` β 4 Λ F ξξ ξ ξ ` δ Λ F ξξ ξ ´ εδ Λ F τ ξ ξ ` Λ F (3.44) “ ε « 1 2 Λ ´ F 2 ξ ´ p Λ F q 2 ¯ ´ J Λ , F ξ K F ξ ` F ξξ Λ F ` J Λ , F K T F ´ F ξ HT F ` β 4 ´ J Λ , F K T F ξξ ξ ξ ´ F ξ Λ T F ξξ ξ ¯ ` δ ´ J Λ , F K T F ξξ ξ ´ F ξ Λ T F ξξ ¯ ﬀ . W e ﬁrst mov e the τ –dependent linear terms to the left-hand side and factor out the slow time deriv- ativ e. Namely , ( 3.44 ) can be rewritten as ε ´ 2 p I ` ΥΛ qB ξ ` δ Λ B 2 ξ ¯ F τ “ p I ` ΥΛ q F ξξ ` β 4 Λ F ξξ ξ ξ ` δ Λ F ξξ ξ ` Λ F (3.45) ´ ε « 1 2 Λ ´ F 2 ξ ´ p Λ F q 2 ¯ ´ J Λ , F ξ K F ξ ` F ξξ Λ F ` J Λ , F K T F ´ F ξ HT F ` β 4 ´ J Λ , F K T F ξξ ξ ξ ´ F ξ Λ T F ξξ ξ ¯ ` δ ´ J Λ , F K T F ξξ ξ ´ F ξ Λ T F ξξ ¯ ﬀ . F or con v enience, denote the linear operator acting on F τ b y M 1 : “ 2 p I ` ΥΛ qB ξ ` δ Λ B 2 ξ , M ˚ 1 : “ ´ 2 p I ` ΥΛ qB ξ ` δ Λ B 2 ξ , so that, M ˚ 1 M 1 “ ` δ Λ B 2 ξ ˘ 2 ´ 4 ` p I ` ΥΛ qB ξ ˘ 2 . WEAKL Y NONLINEAR MODELS F OR HYDR OELASTIC W A TER W A VES 21 Multiplying ( 3.45 ) by M ˚ 1 and applying p M ˚ 1 M 1 q ´ 1 (mo de-b y-mo de on k ‰ 0) yields the ﬁrst in v ersion step F τ “ 1 ε p M ˚ 1 M 1 q ´ 1 M ˚ 1 ” p I ` ΥΛ q F ξξ ` β 4 Λ F ξξ ξ ξ ` δ Λ F ξξ ξ ` Λ F ı (3.46) ´ p M ˚ 1 M 1 q ´ 1 M ˚ 1 « 1 2 Λ ´ F 2 ξ ´ p Λ F q 2 ¯ ´ J Λ , F ξ K F ξ ` F ξξ Λ F ` J Λ , F K T F ´ F ξ HT F ` β 4 ´ J Λ , F K T F ξξ ξ ξ ´ F ξ Λ T F ξξ ξ ¯ ` δ ´ J Λ , F K T F ξξ ξ ´ F ξ Λ T F ξξ ¯ ﬀ . On the F ourier side ( k ‰ 0), the op erator M 1 : “ 2 p I ` ΥΛ qB ξ ` δ Λ B 2 ξ has symbol m 1 p k q “ ´ δ | k | k 2 ` i 2 p 1 ` Υ | k |q k . Therefore, p M ˚ 1 M 1 q ´ 1 M ˚ 1 has asso ciated sym b ol m 1 p k q | m 1 p k q| 2 “ ´ δ | k | k 2 ´ i 2 p 1 ` Υ | k |q k δ 2 | k | 2 k 4 ` 4 p 1 ` Υ | k |q 2 k 2 Similarly as b efore, we obtain the decomp osition p M ˚ 1 M 1 q ´ 1 M ˚ 1 “ α p Λ , B ξ q ` γ p Λ , B ξ q H , (3.47) where α p Λ , B ξ q and γ p Λ , B ξ q are real F ourier m ultipliers with symbols α k : “ ´ δ | k | δ 2 | k | 4 ` 4 p 1 ` Υ | k |q 2 , γ k : “ 2 p 1 ` Υ | k |q | k | ` δ 2 | k | 4 ` 4 p 1 ` Υ | k |q 2 ˘ , k ‰ 0 . Equiv alently , at the op erator lev el, α p Λ , B ξ q “ ´ δ Λ δ 2 Λ 2 B 2 ξ ´ 4 p I ` ΥΛ q 2 , γ p Λ , B ξ q “ 2 p I ` ΥΛ q Λ ´ 1 δ 2 Λ 2 B 2 ξ ´ 4 p I ` ΥΛ q 2 . With this notation, we may write F τ “ 1 ε ` α ` γ H ˘ ” p I ` ΥΛ q F ξξ ` β 4 Λ F ξξ ξ ξ ` δ Λ F ξξ ξ ` Λ F ı (3.48) ´ ` α ` γ H ˘ « 1 2 Λ ´ F 2 ξ ´ p Λ F q 2 ¯ ´ J Λ , F ξ K F ξ ` F ξξ Λ F ` J Λ , F K T F ´ F ξ HT F ` β 4 ´ J Λ , F K T F ξξ ξ ξ ´ F ξ Λ T F ξξ ξ ¯ ` δ ´ J Λ , F K T F ξξ ξ ´ F ξ Λ T F ξξ ¯ ﬀ . 22 D. ALONSO-OR ´ AN, R. GRANERO-BELINCH ´ ON, AND J. S. ZIEBELL 4. Well-posedness resul t for the bidirectional model In this section we establish the w ell-p osedness theory for ( 3.24 ). In particular, we prov e the follo wing result: Theorem 4.1 (Lo cal well-posedness for small H 3 data) . Let Υ ą 0 and δ, β ą 0, and let ε P R b e ﬁxed. Assume that the initial data satisfy f p x, 0 q “ f 0 P H 3 0 p T 2 q , B t f p x, 0 q “ f 1 P H 1 0 p T 2 q , } f 0 } H 3 ď ε ˚ , for some ε ˚ “ ε ˚ p Υ , δ, β , | ε |q ą 0. Then there exists a time T “ T ` Υ , δ, β , | ε | , } f 1 } H 1 , } f 0 } H 3 ˘ ą 0 and a unique solution f of ( 3.24 ) on r 0 , T s suc h that f P L 8 p 0 , T ; H 3 0 p T 2 qq , f t P L 8 p 0 , T ; H 1 0 p T 2 qq . Pr o of of The or em 4.1 . W e begin b y rewriting ( 3.24 ) as a ﬁrst–order system in time. Set v : “ B t f , N p f , w q : “ J Λ , f K w ` ∇ x f ¨ R w Then ( 3.24 ) is equiv alen t to $ ’ & ’ % B t f “ v , ` I ` ΥΛ ˘ B t v ` δ Λ 3 v ` ´ Λ ` β 4 Λ 5 ¯ f ` N ` f , B t v ˘ “ ε Q p v q , (4.1) where the quadratic forcing is Q p v q : “ 1 2 Λ ´ v 2 ´ ˇ ˇ R v ˇ ˇ 2 ¯ ´ J Λ , v K v ´ ∇ x v ¨ R v . (4.2) Fix T ą 0. W e w ork in the class of functions X T : “ ! ¯ f P L 8 p 0 , T ; H 3 0 p T 2 qq : ¯ f p 0 q “ f 0 , } ¯ f } L 8 t H 3 x ď 2 } f 0 } H 3 x ) , Y T : “ ! ¯ v P L 8 p 0 , T ; H 1 0 p T 2 qq : ¯ v p 0 q “ f 1 u , and construct a solution b y a t w o-stage appro ximation procedure. Step 0: Str ate gy of the pr o of. The argumen t is somewhat delicate: we introduce tw o regularizations with parameters µ and λ , solv e an elliptic problem to identify the time deriv ativ e, construct the ev olution by a contraction mapping, and then pass to the limits using µ – and λ –uniform estimates. W e summarize these steps next and provide the detailed implemen tation afterw ards. Giv en p ¯ f , ¯ v q P X T ˆ Y T , w e solv e an auxiliary λ –regularized elliptic equation for an unkno wn U λ,µ r ¯ f , ¯ v s (Step 1). Since N p f , ¨q is linear in its second argument, this amounts to inv erting a p erturbation of p I ` ΥΛ q ; the smallness of } f 0 } H 3 (and hence of } ¯ f } L 8 t H 3 x ) ensures that the in v ersion is well-deﬁned and yields a unique U λ,µ in the c hosen Sob olev class. WEAKL Y NONLINEAR MODELS F OR HYDR OELASTIC W A TER W A VES 23 With this U λ,µ ﬁxed, w e solv e the corresponding λ –regularized ev olution for v λ,µ with initial datum v 0 and choose T so that the map ¯ v ÞÑ v λ,µ is a con traction in Y T (Step 2). In particular, the ﬁxed p oin t v λ,µ satisﬁes v λ,µ t “ U λ,µ b y construction. W e then pass to the limit λ Ñ 8 (with µ ﬁxed) using uniform-in- λ b ounds, obtaining a function v µ solving the µ –regularized equation (Step 3). Second, with v µ a v ailable, we recov er f µ b y solving f µ t “ v µ with initial datum f 0 , and close the coupling through a ﬁxed p oint argument in X T (Step 4), now only at the µ –regularized level. Finally , w e let µ Ñ 8 and use the corresponding uniform-in- µ a priori bounds to pass to the limit, yielding a solution f of the original system on r 0 , T s with f P L 8 p 0 , T ; H 3 0 q and f t P L 8 p 0 , T ; H 1 0 q (Step 5). Step 1: Solving the el liptic pr oblem for U . Fix T ą 0 and regularization parameters 0 ă λ, µ ď 1. Let p ¯ f , ¯ v q P X T ˆ Y T . In this step we will construct an auxiliary unknown U “ U λ,µ r ¯ f , ¯ v s P L 8 p 0 , T ; H 1 0 p T 2 qq , as the unique solution to the regularized elliptic problem p I ` ΥΛ q U “ F λ,µ r ¯ f , ¯ v s ´ J λ N p J µ ¯ f , J λ U q , in p 0 , T q ˆ T 2 , (4.3) where the forcing is F λ,µ r ¯ f , ¯ v s : “ ´ δ J λ Λ 3 J λ ¯ v ´ J µ ´ Λ ` β 4 Λ 5 ¯ J µ ¯ f ` ε J λ « 1 2 Λ ´ p J λ ¯ v q 2 ´ ˇ ˇ R J λ ¯ v ˇ ˇ 2 ¯ ´ J Λ , J λ ¯ v K J λ ¯ v ´ ∇ x p J λ ¯ v q ¨ R p J λ ¯ v q ﬀ , (4.4) and the bilinear coupling is N p f , g q : “ J Λ , f K g ` ∇ x f ¨ R g. (4.5) Since Λ and eac h Riesz transform R j annihilate constan ts, and J λ , J µ preserv e the zero F ourier mode, it follows that if ¯ f , ¯ v hav e zero spatial mean, then eac h term in ( 4.4 ) has zero mean. Similary , after a straigh tforw ard integration by parts the bilinear term q Λ , J µ ¯ f y p J λ U q and ∇ x p J µ ¯ f q ¨ R p J λ U q hav e zero mean. Therefore, w e seek U p t q P H 1 0 p T 2 q solving ( 4.3 ). Deﬁne the Banac h space U T : “ L 8 ` 0 , T ; H 1 0 p T 2 q ˘ , } U } U T : “ ess sup t Pp 0 ,T q } U p t q} H 1 . F or ﬁxed p ¯ f , ¯ v q we deﬁne the op erator Φ ¯ f , ¯ v : U T Ñ U T , b y p Φ ¯ f , ¯ v r ¯ U sqp t q : “ p I ` ΥΛ q ´ 1 ´ F λ,µ r ¯ f , ¯ v sp t q ´ J λ N p J µ ¯ f p t q , J λ ¯ U p t qq ¯ , t P p 0 , T q . (4.6) A ﬁxed p oin t U “ Φ ¯ f , ¯ v r U s is precisely a solution to ( 4.3 ) in U T . Let ¯ U P U T . Using the fact that }p I ` ΥΛ q ´ 1 G } H 1 ď } G } L 2 , G P H 1 0 p T 2 q , 24 D. ALONSO-OR ´ AN, R. GRANERO-BELINCH ´ ON, AND J. S. ZIEBELL the bilinear estimate } N p J µ ¯ f p t q , J λ ¯ U p t qq} H 1 ď C µ } ¯ f p t q} H 3 } ¯ U p t q} H 1 and the b oundedness of J λ on H 1 and J µ on H 3 , we obtain for a.e. t P p 0 , T q }p Φ ¯ f , ¯ v r ¯ U sqp t q} H 1 ≲ } F λ,µ r ¯ f , ¯ v sp t q} H 1 ` C µ } ¯ f p t q} H 3 } ¯ U p t q} H 1 . T aking the essential supremum o ver t gives } Φ ¯ f , ¯ v r ¯ U s} L 8 p 0 ,T ; H 1 q ď C p λ, µ, ¯ f , ¯ v q ` C µ } ¯ f } L 8 p 0 ,T ; H 3 q } ¯ U } L 8 p 0 ,T ; H 1 q , (4.7) where we set C p λ, µ, ¯ f , ¯ v q : “ C } F λ,µ r ¯ f , ¯ v s} L 8 p 0 ,T ; H 1 q . (4.8) This quan tity is ﬁnite for p ¯ f , ¯ v q P X T ˆ Y T b ecause J λ and J µ are smoothing and hence map L 8 t H s x in to L 8 t H s ` m x for any m ě 0. Next, let us show the con traction estimate. Let ¯ U p 1 q , ¯ U p 2 q P U T . Subtracting ( 4.6 ), Φ ¯ f , ¯ v r ¯ U p 1 q s ´ Φ ¯ f , ¯ v r ¯ U p 2 q s “ ´p I ` ΥΛ q ´ 1 J λ N ´ J µ ¯ f , J λ p ¯ U p 1 q ´ ¯ U p 2 q q ¯ . Hence, by a similar argument as before } Φ ¯ f , ¯ v r ¯ U p 1 q s ´ Φ ¯ f , ¯ v r ¯ U p 2 q s} U T ≲ } ¯ f } L 8 p 0 ,T ; H 3 q } ¯ U p 1 q ´ ¯ U p 2 q } U T . Assume now that } f 0 } H 3 p T 2 q ď ε ˚ , (4.9) where ε ˚ ą 0 will b e ﬁxed momentarily . Since ¯ f P X T w e ha ve } ¯ f } L 8 H 3 ď 2 } f 0 } H 3 , and th us } Φ ¯ f , ¯ v r ¯ U p 1 q s ´ Φ ¯ f , ¯ v r ¯ U p 2 q s} L 8 H 1 ď p 2 C } f 0 } H 3 q } ¯ U p 1 q ´ ¯ U p 2 q } L 8 H 1 . Cho ose ε ˚ ą 0 so small that 2 C ε ˚ ď 1 2 and C ε ˚ ď 1 2 . (Equiv alently , ε ˚ ď p 4 C q ´ 1 .) Then Φ ¯ f , ¯ v is a strict contraction on U T , hence b y Banach’s ﬁxed-p oint theorem there exists a unique U λ,µ r ¯ f , ¯ v s P L 8 p 0 , T ; H 1 0 p T 2 qq suc h that U λ,µ r ¯ f , ¯ v s “ Φ ¯ f , ¯ v r U λ,µ r ¯ f , ¯ v ss , i.e. U λ,µ r ¯ f , ¯ v s solv es ( 4.3 ). Moreov er, taking ¯ U “ U in ( 4.7 ) and using } ¯ f } L 8 t H 3 x ď 2 } f 0 } H 3 ď ε ˚ , we obtain } U } L 8 p 0 ,T ; H 1 q ď C p λ, µ, ¯ f , ¯ v q ` C ε ˚ } U } L 8 p 0 ,T ; H 1 q . Hence C ε ˚ ď 1 2 implies } U } L 8 p 0 ,T ; H 1 q ď 2 C p λ, µ, ¯ f , ¯ v q . (4.10) So, we conclude that for eac h ﬁxed T ą 0 and 0 ă λ, µ ď 1, and for each p ¯ f , ¯ v q P X T ˆ Y T with mean zero and satisfying ( 4.9 ), the elliptic ﬁxed-p oint problem ( 4.3 ) admits a unique solution U λ,µ r ¯ f , ¯ v s P L 8 p 0 , T ; H 1 0 p T 2 qq , and this solution satisﬁes the b ound ( 4.10 ). WEAKL Y NONLINEAR MODELS F OR HYDR OELASTIC W A TER W A VES 25 Step 2: Solving the e quation for v . In Step 1 w e constructed, under the smallness assumption } f 0 } H 3 ď ε ˚ , a unique auxiliary ﬁeld U λ,µ r ¯ f , ¯ v s P U T “ L 8 p 0 , T ; H 1 0 p T 2 qq , solving the elliptic problem ( 4.3 ). In this step w e solv e for a v elocity v “ v λ,µ suc h that v t “ U λ,µ r ¯ f , v s . Giv en ¯ v P Y T , deﬁne U ¯ f p ¯ v q : “ U λ,µ r ¯ f , ¯ v s P U T , the elliptic solution from Step 1 associated with p ¯ f , ¯ v q . Then w e deﬁne Ψ ¯ f : Y T Ñ Y T b y p Ψ ¯ f r ¯ v sqp t q : “ v 0 ` ż t 0 U ¯ f p ¯ v qp s q ds, t P r 0 , T s . (4.11) Let ¯ v P Y T . Since U ¯ f p ¯ v q P U T , the in tegral in ( 4.11 ) is w ell-deﬁned in H 1 , and } Ψ ¯ f r ¯ v sp t q} H 1 ď } v 0 } H 1 ` ż t 0 } U ¯ f p ¯ v qp s q} H 1 ds ď } v 0 } H 1 ` T } U ¯ f p ¯ v q} L 8 p 0 ,T ; H 1 q . Hence Ψ ¯ f r ¯ v s P Y T pro vided we con trol } U ¯ f p ¯ v q} L 8 H 1 . Such a bound follows from Step 1, since for eac h t the ﬁxed p oint U p t q solv es p I ` ΥΛ q U “ F λ,µ r ¯ f , ¯ v s ´ J λ N p J µ ¯ f , J λ U q , and Step 1 provides an estimate of the form } U ¯ f p ¯ v q} L 8 p 0 ,T ; H 1 q ď C p λ, µ, ¯ f , ¯ v q , (4.12) where C p λ, µ, ¯ f , ¯ v q is ﬁnite for p ¯ f , ¯ v q P X T ˆ Y T (and dep ends on λ, µ through smo othing). Let ¯ v p 1 q , ¯ v p 2 q P Y T , and set U p i q : “ U ¯ f p ¯ v p i q q “ U λ,µ r ¯ f , ¯ v p i q s P U T . Subtracting the elliptic equations ( 4.3 ) satisﬁed b y U p 1 q and U p 2 q yields, for a.e. t P p 0 , T q , p I ` ΥΛ q ` U p 1 q ´ U p 2 q ˘ “ ´ F λ,µ r ¯ f , ¯ v p 1 q s ´ F λ,µ r ¯ f , ¯ v p 2 q s ¯ ´ J λ ´ N p J µ ¯ f , J λ U p 1 q q ´ N p J µ ¯ f , J λ U p 2 q q ¯ . (4.13) The diﬀerence in the coupling term is linear in U p 1 q ´ U p 2 q , and by the bilinear estimate from the preliminaries (applied in H 1 using that J λ is b ounded on H 1 ) we hav e › › J λ ` N p J µ ¯ f , J λ U p 1 q q ´ N p J µ ¯ f , J λ U p 2 q q ˘ › › H 1 ≲ } ¯ f } H 3 } U p 1 q ´ U p 2 q } H 1 . (4.14) It remains to estimate the forcing diﬀerence in H 1 . Insp ecting ( 4.4 ), the ¯ v –dep endence is contained only in the terms ´ δ J λ Λ 3 J λ ¯ v and ε J λ Q p J λ ¯ v q , where Q p w q : “ 1 2 Λ ´ w 2 ´ | R w | 2 ¯ ´ J Λ , w K w ´ ∇ x w ¨ R w . 26 D. ALONSO-OR ´ AN, R. GRANERO-BELINCH ´ ON, AND J. S. ZIEBELL Since J λ is smo othing, the F ourier multiplier J λ Λ 3 J λ is b ounded from H 1 to H 1 , with op erator norm dep ending on λ ; hence › › J λ Λ 3 J λ p ¯ v p 1 q ´ ¯ v p 2 q q › › H 1 ď C λ } ¯ v p 1 q ´ ¯ v p 2 q } H 1 . (4.15) F or the quadratic term, w e use that J λ : H 1 Ñ H m for every m , and in particular } J λ h } H 3 ď C λ } h } H 1 . Since H 3 p T 2 q ã Ñ W 1 , 8 p T 2 q , standard pro duct/commutator bounds imply that Q is locally Lipsc hitz from H 3 to H 1 : for every R ą 0, } Q p w p 1 q q ´ Q p w p 2 q q} H 1 ď C ` } w p 1 q } H 3 ` } w p 2 q } H 3 ˘ } w p 1 q ´ w p 2 q } H 3 (4.16) and therefore, for ¯ v p 1 q , ¯ v p 2 q with } ¯ v p i q } H 1 ď R , } Q p J λ ¯ v p 1 q q ´ Q p J λ ¯ v p 2 q q} H 1 ď C λ,R } ¯ v p 1 q ´ ¯ v p 2 q } H 1 , (4.17) for some constan t C λ,R dep ending on λ and R . Since J λ is b ounded on H 1 , we then also hav e › › J λ p Q p J λ ¯ v p 1 q q ´ Q p J λ ¯ v p 2 q qq › › H 1 ď C λ,R } ¯ v p 1 q ´ ¯ v p 2 q } H 1 . (4.18) Com bining ( 4.15 ) and ( 4.18 ) gives › › F λ,µ r ¯ f , ¯ v p 1 q s ´ F λ,µ r ¯ f , ¯ v p 2 q s › › H 1 ď C λ,R } ¯ v p 1 q ´ ¯ v p 2 q } H 1 . (4.19) No w apply p I ` ΥΛ q ´ 1 to ( 4.13 ). Using that p I ` ΥΛ q ´ 1 is b ounded on H 1 0 p T 2 q and combining ( 4.14 ) and ( 4.19 ), w e obtain } U p 1 q ´ U p 2 q } H 1 ď C λ,R } ¯ v p 1 q ´ ¯ v p 2 q } H 1 ` C } ¯ f } H 3 } U p 1 q ´ U p 2 q } H 1 . (4.20) Under the same smallness assumption as in Step 1, } ¯ f } L 8 p 0 ,T ; H 3 q ď 2 } f 0 } H 3 ď ε ˚ with ε ˚ c hosen so that C ε ˚ ď 1 2 , we can absorb the last term in ( 4.20 ) and obtain the Lipschitz b ound } U ¯ f p ¯ v p 1 q q ´ U ¯ f p ¯ v p 2 q q} L 8 p 0 ,T ; H 1 q ď C λ,R } ¯ v p 1 q ´ ¯ v p 2 q } L 8 p 0 ,T ; H 1 q , (4.21) for ¯ v p 1 q , ¯ v p 2 q in the closed ball t} v } Y T ď R u . Let ¯ v p 1 q , ¯ v p 2 q P Y T with } ¯ v p i q } Y T ď R , and set v p i q “ Ψ ¯ f r ¯ v p i q s . Using ( 4.11 ) and ( 4.21 ), } v p 1 q p t q ´ v p 2 q p t q} H 1 ď ż t 0 } U ¯ f p ¯ v p 1 q qp s q ´ U ¯ f p ¯ v p 2 q qp s q} H 1 ds ď t C λ,R } ¯ v p 1 q ´ ¯ v p 2 q } Y T . T aking the essential supremum o ver t P p 0 , T q giv es } Ψ ¯ f r ¯ v p 1 q s ´ Ψ ¯ f r ¯ v p 2 q s} Y T ď T C λ,R } ¯ v p 1 q ´ ¯ v p 2 q } Y T . (4.22) Cho osing T ą 0 so that T C λ,R ď 1 2 , we conclude that Ψ ¯ f is a strict contraction on the closed ball t} v } Y T ď R u , provided R is chosen so that Ψ ¯ f maps the ball into itself. This follo ws from ( 4.12 ) and } v } Y T ď } v 0 } H 1 ` T } U ¯ f p v q} L 8 p 0 ,T ; H 1 q . Hence Banach’s ﬁxed point theorem yields a unique v λ,µ P Y T suc h that v λ,µ “ Ψ ¯ f r v λ,µ s . Moreo ver, w e hav e B t v λ,µ “ U λ,µ r ¯ f , v λ,µ s in L 8 p 0 , T ; H 1 p T 2 qq , (4.23) WEAKL Y NONLINEAR MODELS F OR HYDR OELASTIC W A TER W A VES 27 thereb y iden tifying the elliptic unknown with the time deriv ativ e of v . In particular, substituting ( 4.23 ) in to the elliptic iden tit y ( 4.3 ) (with ¯ v “ v λ,µ ) shows that v λ,µ satisﬁes the λ –regularized evolution equation p I ` ΥΛ q B t v λ,µ ` δ J λ Λ 3 J λ v λ,µ ` J µ ´ Λ ` β 4 Λ 5 ¯ J µ ¯ f ` J λ N ` J µ ¯ f , J λ B t v λ,µ ˘ “ ε J λ « 1 2 Λ ´ p J λ v λ,µ q 2 ´ ˇ ˇ R J λ v λ,µ ˇ ˇ 2 ¯ ´ q Λ , J λ v λ,µ y J λ v λ,µ ´ ∇ x p J λ v λ,µ q ¨ R p J λ v λ,µ q ﬀ , (4.24) in L 8 p 0 , T ; H 1 p T 2 qq . Step 3: Passing to the limit as λ Ñ 0 . Fix 0 ă µ ď 1 and let v λ b e the solutions constructed in Step 2 on r 0 , T s , where v λ P L 8 p 0 , T ; H 1 0 p T 2 qq . 1 W e claim that, for T ą 0 chosen as in Step 2, the family t v λ u satisﬁes the uniform estimates sup 0 ď t ď T } v λ p t q} H 1 ď C, ż T 0 } J λ v λ p t q} 2 H 2 dt ď C , ż T 0 }B t v λ p t q} 2 H ´ 1 { 2 dt ď C , (4.25) with a constan t C indep endent of λ . T esting the regularized equation ( 4.24 ) against Λ v λ yields an inequalit y of the form 1 2 d dt ´ } v λ } 2 9 H 1 { 2 ` Υ } v λ } 2 9 H 1 ¯ ` δ } J λ v λ } 2 9 H 2 ď R λ 1 p t q , (4.26) where R λ 1 collects the quadratic terms and the coupling term N p J µ ¯ f , J λ U λ q . More precisely , R λ 1 p t q “ ´ ż T 2 ´ J µ ´ Λ ` β 4 Λ 5 ¯ J µ ¯ f p t q ¯ Λ v λ p t q dx ´ ż T 2 ´ J λ N ` J µ ¯ f p t q , J λ v λ t p t q ˘ ¯ Λ v λ p t q dx ` ε ż T 2 J λ « 1 2 Λ ´ p J λ v λ p t qq 2 ´ ˇ ˇ R J λ v λ p t q ˇ ˇ 2 ¯ ´ q Λ , J λ v λ p t q y J λ v λ p t q ´ ∇ x p J λ v λ p t qq ¨ R p J λ v λ p t qq ﬀ Λ v λ p t q dx “ : M 1 p t q ` M 2 p t q ` M 3 p t q . Using } ¯ f } L 8 t H 3 x ď 2 } f 0 } H 3 and Cauc h y–Sch warz together with the b oundedness/smo othing of J µ on Sob olev spaces, w e hav e | M 1 p t q| ≲ } ¯ f p t q} H 3 } v λ p t q} H 1 ď C µ,β } f 0 } H 3 } v λ p t q} H 1 . T o estimate M 3 p t q w e use that J λ is self-adjoin t and bounded on all H s , H 2 p T 2 q ã Ñ W 1 , 8 p T 2 q , and the b oundedness of Riesz transforms on Sobolev spaces. W e obtain | M 3 p t q| ≲ } J λ v λ p t q} H 2 } v λ p t q} L 2 } v λ p t q} H 1 ` } J λ v λ p t q} H 2 } v λ p t q} 2 H 1 . (4.27) T o estimate M 2 p t q we use that J λ is self-adjoint on L 2 and write M 2 p t q : “ ´ ż T 2 ´ J λ N p J µ ¯ f p t q , J λ v λ t p t qq ¯ Λ v λ p t q dx “ ´ ż T 2 N p J µ ¯ f p t q , J λ v λ t p t qq Λ J λ v λ p t q dx. 1 Here and below we suppress the dep endence on µ and on ¯ f to lighten notation. 28 D. ALONSO-OR ´ AN, R. GRANERO-BELINCH ´ ON, AND J. S. ZIEBELL Expanding N p f , g q “ r Λ , f s g ` ∇ f ¨ R g , in tegrating by parts and Sob olev em bedding w e ﬁnd that | M 2 p t q| ď C   J λ v λ   H 2   J λ v λ t   L 2   ¯ f   L 8 ´ 2 ż T 2 ¯ f Λ J λ v λ Λ J λ v λ t dx, ď C   J λ v λ   H 2   v λ t   L 2   ¯ f   L 8 ` C   ¯ f Λ J λ v λ   H 1   Λ J λ v λ t   H ´ 1 , ď C   J λ v λ   H 2   v λ t   L 2   ¯ f   H 2 where in the second inequality we make use of the H ´ 1 ´ H 1 dualit y . In particular, since } ¯ f } L 8 t H 2 x ď } ¯ f } L 8 t H 3 x ď 2 } f 0 } H 3 , | M 2 p t q| ≲ µ } f 0 } H 3 } v λ t p t q} L 2 } J λ v λ p t q} H 2 . Therefore, using Y oung’s inequalit y we ﬁnd that | R λ 1 p t q| ď δ 4 } J λ v λ p t q} 2 H 2 ` C δ,µ,β ´ 1 ` } f 0 } 2 H 3 ¯´ } v λ p t q} 4 H 1 ` } v λ t p t q} 2 L 2 ` 1 ¯ . (4.28) Th us, 1 2 d dt ´ } v λ } 2 9 H 1 { 2 ` Υ } v λ } 2 9 H 1 ¯ ` δ 2 } J λ v λ } 2 9 H 2 ď C δ,µ,β ´ 1 ` } f 0 } 2 H 3 ¯´ } v λ p t q} 4 H 1 ` } v λ t p t q} 2 L 2 ` 1 ¯ . (4.29) Next, we test ( 4.24 ) against Λ ´ 1 v λ t . Using that I ` ΥΛ is self-adjoin t and commutes with Λ ´ 1 , and that J λ is self-adjoint and comm utes with Λ, we obtain 1 2 } v λ t p t q} 2 9 H ´ 1 { 2 ` Υ 2 } v λ t p t q} 2 L 2 ` δ 2 d dt } J λ v λ p t q} 2 9 H 1 ď R λ 2 p t q , (4.30) where R λ 2 collects the quadratic terms and the coupling term N p J µ ¯ f , J λ v λ t q tested against Λ ´ 1 v λ t . Arguing exactly as in the estimate of R λ 1 , we obtain the b ound R λ 2 p t q ď C } v λ p t q} 4 H 1 ` Υ 4 } v λ t p t q} 2 L 2 , (4.31) with C indep endent of λ . Absorbing the last term into the left-hand side of ( 4.30 ) yields } v λ t p t q} 2 9 H ´ 1 { 2 ` Υ 2 } v λ t p t q} 2 L 2 ` δ d dt } J λ v λ p t q} 2 9 H 1 ď C } v λ p t q} 4 H 1 . (4.32) Set E λ 1 p t q : “ } v λ p t q} 2 9 H 1 { 2 ` Υ } v λ p t q} 2 9 H 1 , Y p t q : “ sup 0 ď s ď t } v λ p s q} 2 H 1 . Since v λ has zero mean, Poincar ´ e implies E λ 1 p t q „ } v λ p t q} 2 H 1 . F rom ( 4.29 ) w e ha ve, for all t P r 0 , T s , d dt E λ 1 p t q ` δ 2 } J λ v λ p t q} 2 9 H 2 ď C 0 ´ } v λ p t q} 4 H 1 ` } v λ t p t q} 2 L 2 ` 1 ¯ , (4.33) where C 0 “ C 0 p Υ , δ, µ, β q ` 1 ` } f 0 } 2 H 3 ˘ and is indep enden t of λ . In tegrating ( 4.33 ) on r 0 , t s gives sup 0 ď s ď t E λ 1 p s q ` δ 2 ż t 0 } J λ v λ p s q} 2 9 H 2 ds ď E λ 1 p 0 q ` C 0 ż t 0 ´ } v λ p s q} 4 H 1 ` } v λ t p s q} 2 L 2 ` 1 ¯ ds. (4.34) F rom ( 4.32 ) we hav e, for all t P r 0 , T s , } v λ t p t q} 2 9 H ´ 1 { 2 ` Υ 2 } v λ t p t q} 2 L 2 ` δ d dt } J λ v λ p t q} 2 9 H 1 ď C } v λ p t q} 4 H 1 , (4.35) WEAKL Y NONLINEAR MODELS F OR HYDR OELASTIC W A TER W A VES 29 with C indep endent of λ . Integrating ( 4.35 ) on r 0 , t s yields ż t 0 } v λ t p s q} 2 9 H ´ 1 { 2 ds ` Υ 2 ż t 0 } v λ t p s q} 2 L 2 ds ` δ } J λ v λ p t q} 2 9 H 1 ď δ } J λ v 0 } 2 9 H 1 ` C ż t 0 } v λ p s q} 4 H 1 ds. (4.36) In particular, using } J λ v 0 } 9 H 1 ď } v 0 } 9 H 1 and } v λ p s q} 4 H 1 ď Y p t q 2 for s ď t , ż t 0 } v λ t p s q} 2 L 2 ds ≲ } v 0 } 2 H 1 ` t Y p t q 2 , ż t 0 } v λ t p s q} 2 H ´ 1 { 2 ds ≲ } v 0 } 2 H 1 ` t Y p t q 2 , (4.37) uniformly in λ . Insert ( 4.37 ) and ş t 0 } v λ p s q} 4 H 1 ds ď t Y p t q 2 in to ( 4.34 ). Using E λ 1 p 0 q „ } v 0 } 2 H 1 and E λ 1 p s q „ } v λ p s q} 2 H 1 , we obtain Y p t q ` ż t 0 } J λ v λ p s q} 2 H 2 ds ď C 1 ` 1 ` } v 0 } 2 H 1 ˘ ` C 2 t Y p t q 2 , t P r 0 , T s , (4.38) where C 1 , C 2 are indep enden t of λ (and depend on Υ , δ, µ, β and } f 0 } H 3 ). In particular we ha ve the p olynomial inequality Y p t q ď C 1 ` 1 ` } v 0 } 2 H 1 ˘ ` C 2 t Y p t q 2 , t P r 0 , T s . (4.39) Cho ose T ą 0 suc h that C 2 T ´ 2 C 1 p 1 ` } v 0 } 2 H 1 q ¯ ď 1 2 . A standard con tin uity/bo otstrapping argumen t applied to ( 4.39 ) yields sup 0 ď t ď T } v λ p t q} 2 H 1 “ sup 0 ď t ď T Y p t q ď 2 C 1 ` 1 ` } v 0 } 2 H 1 ˘ , (4.40) with a b ound indep endent of λ . Plugging ( 4.40 ) bac k in to ( 4.38 ) and ( 4.36 ) gives the remaining uniform b ounds: sup 0 ď t ď T } v λ p t q} H 1 ď C , ż T 0 } J λ v λ p t q} 2 H 2 dt ď C , ż T 0 } v λ t p t q} 2 H ´ 1 { 2 dt ď C , (4.41) for a constan t C indep endent of λ . By ( 4.41 ) we hav e the λ –uniform bounds v λ b ounded in L 8 p 0 , T ; H 1 p T 2 qq , B t v λ b ounded in L 2 p 0 , T ; H ´ 1 { 2 p T 2 qq . Fix θ P p 0 , 1 2 q . Since the embedding H 1 p T 2 q ã Ñ H 1 ´ θ p T 2 q is compact and H 1 ´ θ p T 2 q ã Ñ H ´ 1 { 2 p T 2 q is contin uous, the Aubin–Lions lemma yields, after extracting a subsequence (not relab eled), the existence of v such that v λ Ñ v strongly in L 2 p 0 , T ; H 1 ´ θ p T 2 qq , v λ á ˚ v in L 8 p 0 , T ; H 1 p T 2 qq . (4.42) Moreo ver, up to a further subsequence, there exists U such that B t v λ á U w eakly in L 2 p 0 , T ; H ´ 1 { 2 p T 2 qq . (4.43) In particular, v P L 8 p 0 , T ; H 1 p T 2 qq X W 1 , 2 p 0 , T ; H ´ 1 { 2 p T 2 qq , and hence v P C ` r 0 , T s ; H s p T 2 q ˘ for every s P r´ 1 2 , 1 q . (4.44) 30 D. ALONSO-OR ´ AN, R. GRANERO-BELINCH ´ ON, AND J. S. ZIEBELL Since v λ P L 8 p 0 , T ; H 1 q and B t v λ is b ounded in L 2 p 0 , T ; H ´ 1 { 2 q , after extraction we ha ve B t v λ á U w eakly in L 2 p 0 , T ; H ´ 1 { 2 q for some U . F or an y φ P C 8 c pp 0 , T q ˆ T 2 q , ż T 0 x U λ , φ y dt “ ´ ż T 0 x v λ , B t φ y dt. W e now pass to the limit λ Ñ 0 in the regularized equation ( 4.24 ). The linear terms conv erge by ( 4.42 ) and the b oundedness of F ourier m ultipliers on Sob olev spaces. F or the nonlinear terms we use the regularization and the uniform bounds ( 4.25 ). Consequen tly , the limit v µ r ¯ f s P L 8 p 0 , T ; H 1 0 p T 2 qq satisﬁes p I ` ΥΛ qB t v ` δ Λ 3 v ` J µ ´ Λ ` β 4 Λ 5 ¯ J µ ¯ f ` N p J µ ¯ f , B t v q “ ε Q p v q , (4.45) Step 4: Solving the e quation for f . Fix 0 ă µ ď 1 and let T ą 0 b e the time obtained in Step 3. In particular, for ev ery ¯ f P X T w e dispose of a function v µ r ¯ f s P L 8 ` 0 , T ; H 1 0 p T 2 q ˘ X W 1 , 2 ` 0 , T ; H ´ 1 { 2 p T 2 q ˘ , v µ “ v 0 , whic h satisﬁes p I ` ΥΛ q B t v µ r ¯ f s ` δ Λ 3 v µ r ¯ f s ` J µ ´ Λ ` β 4 Λ 5 ¯ J µ ¯ f ` N ` J µ ¯ f , B t v µ r ¯ f s ˘ “ ε Q ` v µ r ¯ f s ˘ , (4.46) where Q p w q : “ 1 2 Λ ´ w 2 ´ | R w | 2 ¯ ´ r Λ , w s w ´ ∇ x w ¨ R w . W e recall that X T : “ ! ¯ f P L 8 ` 0 , T ; H 3 0 p T 2 q ˘ : ¯ f p 0 q “ f 0 , } ¯ f } L 8 p 0 ,T ; H 3 q ď 2 } f 0 } H 3 ) , whic h records the desired H 3 con trol. F or ¯ f P X T , deﬁne p T µ r ¯ f sqp t q : “ f 0 ` ż t 0 J µ J µ v µ r ¯ f sp s q ds, t P r 0 , T s . (4.47) Since v µ r ¯ f s P L 8 p 0 , T ; H 1 q , the map T µ r ¯ f s P X T and B t p T µ r ¯ f sq “ J µ J µ v µ r ¯ f s in L 8 ` 0 , T ; H 1 p T 2 q ˘ . (4.48) Th us, if f “ T µ r f s , then f t “ J µ J µ v µ r f s and inserting ¯ f “ f in ( 4.46 ) yields the coupled µ –regularized system p I ` ΥΛ q v t ` δ Λ 3 v ` J µ ´ Λ ` β 4 Λ 5 ¯ J µ f ` N p J µ f , v t q “ ε Q p v q , (4.49) f t “ J µ J µ v . (4.50) Let ¯ f p 1 q , ¯ f p 2 q P X T , and set v p i q : “ v µ r ¯ f p i q s and r v : “ v p 1 q ´ v p 2 q . Subtracting ( 4.46 ) for ¯ f p 1 q and ¯ f p 2 q giv es p I ` ΥΛ qB t r v ` δ Λ 3 r v “ ´ J µ ´ Λ ` β 4 Λ 5 ¯ J µ p ¯ f p 1 q ´ ¯ f p 2 q q ´ ´ N p J µ ¯ f p 1 q , B t v p 1 q q ´ N p J µ ¯ f p 2 q , B t v p 2 q q ¯ ` ε ´ Q p v p 1 q q ´ Q p v p 2 q q ¯ . (4.51) WEAKL Y NONLINEAR MODELS F OR HYDR OELASTIC W A TER W A VES 31 Using bilinearity of N in its second argumen t we split N p J µ ¯ f p 1 q , B t v p 1 q q ´ N p J µ ¯ f p 2 q , B t v p 2 q q “ N p J µ p ¯ f p 1 q ´ ¯ f p 2 q q , B t v p 1 q q ` N p J µ ¯ f p 2 q , B t r v q . T esting ( 4.51 ) against Λ r v and arguing as in Step 3 (using that J µ ¯ f p 2 q is smo oth and b ounded in W 1 , 8 with constants dep ending on µ and } ¯ f p 2 q } H 3 ) yields, for a.e. t P p 0 , T q , d dt ´ } r v } 2 9 H 1 { 2 ` Υ } r v } 2 9 H 1 ¯ ` δ } r v } 2 9 H 2 ď C µ,R } J µ p ¯ f p 1 q ´ ¯ f p 2 q q} 2 H 3 ` C µ,R } r v } 2 H 1 , (4.52) where R : “ sup ¯ f P X T } ¯ f } L 8 H 3 ď 2 } f 0 } H 3 and C µ,R dep ends on µ , the mo del parameters and R , but not on T . Since r v p 0 q “ 0, Gr¨ on w all implies } v µ r ¯ f p 1 q s ´ v µ r ¯ f p 2 q s} L 8 p 0 ,T ; H 1 q ď C µ,R T } ¯ f p 1 q ´ ¯ f p 2 q } L 8 p 0 ,T ; H 1 q . (4.53) T o sho w the con traction of T µ in X T , let ¯ f p 1 q , ¯ f p 2 q P X T . Using ( 4.47 ) and ( 4.53 ), } T µ r ¯ f p 1 q s ´ T µ r ¯ f p 2 q s} X T ď C µ T } v µ r ¯ f p 1 q s ´ v µ r ¯ f p 2 q s} L 8 H 1 ď T 2 C µ,R } ¯ f p 1 q ´ ¯ f p 2 q } L 8 H 1 . Since } h } H 1 ď } h } H 3 and we can tak e 0 ă T µ ă 1, w e also ha v e } T µ r ¯ f p 1 q s ´ T µ r ¯ f p 2 q s} X T ď T C µ,R } ¯ f p 1 q ´ ¯ f p 2 q } X T . Cho ose T µ P p 0 , T s such that T µ C µ,R ď 1 2 . Then T µ is a strict contraction on X T µ , and Banach’s ﬁxed p oin t theorem yields a unique f µ P X T µ suc h that f µ “ T µ r f µ s . Setting v µ : “ v µ r f µ s , we obtain a pair p f µ , v µ q satisfying ( 4.49 )–( 4.50 ) on p 0 , T µ q , with f µ p 0 q “ f 0 and v µ p 0 q “ v 0 . A t this stage w e kno w f µ P L 8 p 0 , T µ ; H 3 q and f µ t “ v µ P L 8 p 0 , T µ ; H 1 q . In Step 5 w e pro ve µ –uniform a priori estimates at the H 3 lev el for the coupled system ( 4.49 )–( 4.50 ). In particular, there exists a time T ˚ ą 0, dep ending only on } f 0 } H 3 ` } v 0 } H 1 and the parameters p Υ , δ, β q but indep endent of µ , suc h that, for T ˚ ď T µ , sup 0 ď t ď T ˚ } f µ p t q} H 3 ď 2 } f 0 } H 3 . Th us f µ P X T ˚ and f µ t P L 8 p 0 , T ˚ ; H 1 q , which is the regularity required to pass to the limit µ Ñ 0 in Step 5. Step 5: Passing to the limit as µ Ñ 0 . Let p f µ , v µ q b e the µ –regularized solution from Step 4, with v µ “ f µ t . Rep eating the energy multipliers from Step 3 (now with ¯ f “ f µ and v “ v µ ) yields, for a.e. t P p 0 , T q , 1 2 d dt ´ } f µ t } 2 H 1 { 2 ` Υ } f µ t } 2 H 1 ` } f µ } 2 H 1 ` β 4 } f µ } 2 H 3 ¯ ď } f µ t } 4 H 1 ` C } J µ f µ } 2 H 2 } f µ tt } 2 L 2 , (4.54) 1 2 } f µ tt } 2 H ´ 1 { 2 ` Υ 4 } f µ tt } 2 L 2 ` δ 2 d dt } f µ t } 2 H 1 ď } f µ t } 4 H 1 . (4.55) 32 D. ALONSO-OR ´ AN, R. GRANERO-BELINCH ´ ON, AND J. S. ZIEBELL Deﬁne Y µ p t q : “ } f µ t p t q} 2 H 1 { 2 ` Υ } f µ t p t q} 2 H 1 ` } f µ p t q} 2 H 1 ` β 4 } f µ p t q} 2 H 3 . Since J µ is b ounded on H s and } g } H 2 ≲ } g } H 3 , we hav e } J µ f µ } 2 H 2 ≲ } f µ } 2 H 3 ≲ Y µ , } f µ t } 2 H 1 ≲ Y µ . In tegrating ( 4.55 ) on r 0 , t s and using Υ ą 0 giv es ż t 0 } f µ tt p s q} 2 L 2 ds ≲ Υ } v 0 } 2 H 1 ` ż t 0 } f µ t p s q} 4 H 1 ds ≲ 1 ` ż t 0 ` Y µ p s q ˘ 2 ds. (4.56) No w in tegrate ( 4.54 ) on r 0 , t s and estimate ż t 0 } J µ f µ } 2 H 2 } f µ tt } 2 L 2 ď ´ sup 0 ď s ď t Y µ p s q ¯ ż t 0 } f µ tt p s q} 2 L 2 ds, then use ( 4.56 ). W riting Z µ p t q : “ sup 0 ď s ď t Y µ p s q we obtain Z µ p t q ď C 0 ` C 1 t p Z µ p t qq 2 ` C 2 t p Z µ p t qq 3 , t P r 0 , T s , (4.57) where C 0 „ 1 ` Y µ p 0 q and C 1 , C 2 dep end only on the mo del parameters and on } f 0 } H 3 ` } v 0 } H 1 , and are indep endent of µ . A standard contin uity argument applied to ( 4.57 ) yields a time T ˚ ą 0 (dep ending only on p f 0 , v 0 q , but not on µ ) such that sup 0 ď t ď T ˚ Y µ p t q ď 2 Y µ p 0 q for all 0 ă µ ď 1 . (4.58) In particular, uniformly in 0 ă µ ď 1, t f µ u is b ounded in L 8 p 0 , T ˚ ; H 3 p T 2 qq , t f µ t u is b ounded in L 8 p 0 , T ˚ ; H 1 p T 2 qq . (4.59) t f µ tt u is b ounded in L 2 p 0 , T ˚ ; H ´ 1 { 2 p T 2 qq . (4.60) By ( 4.59 ) the family t f µ u is equi-Lipsc hitz in time with v alues in H 1 and b ounded in L 8 t H 3 x . As in Step 3, Simon’s compactness theorem (using H 3 ⋐ H 2 ` θ ã Ñ H 1 ) yields, after extracting a subsequence, f µ Ñ f strongly in C pr 0 , T ˚ s ; H 2 ` θ p T 2 qq for any θ P p 0 , 1 q , and f µ á ˚ f in L 8 p 0 , T ˚ ; H 3 q . Moreov er, f µ t á ˚ f t in L 8 p 0 , T ˚ ; H 1 q and f µ tt á f tt w eakly in L 2 p 0 , T ˚ ; H ´ 1 { 2 q . Using these conv ergences, the fact that J µ Ñ I strongly on H s as µ Ñ 0, and the contin uity of N p¨ , ¨q and Q p¨q at this regularity , w e may pass to the limit µ Ñ 0 in the µ –regularized second-order equation and conclude that f solves ( 3.24 ) on p 0 , T ˚ q ˆ T 2 , with f p¨ , 0 q “ f 0 and f t p¨ , 0 q “ v 0 . In particular, f P L 8 p 0 , T ˚ ; H 3 0 p T 2 qq , f t P L 8 p 0 , T ˚ ; H 1 0 p T 2 qq . WEAKL Y NONLINEAR MODELS F OR HYDR OELASTIC W A TER W A VES 33 Step 6: Uniqueness. W e now show that the solution constructed ab ov e is unique (in the natural class giv en b y Theorem 4.1 ). Recall that we ha ve obtained a pair p f , v q satisfying p I ` ΥΛ q v t ` δ Λ 3 v ` ´ Λ ` β 4 Λ 5 ¯ f ` N ` f , v t ˘ “ ε Q p v q , (4.61) f t “ v . (4.62) Let p f p 1 q , v p 1 q q and p f p 2 q , v p 2 q q b e tw o solutions of ( 4.61 )–( 4.62 ) on r 0 , T s with the same initial data. Setting r f : “ f p 1 q ´ f p 2 q , r v : “ v p 1 q ´ v p 2 q , w e obtain a coupled system for p r f , r v q with zero initial data. T esting the r v –equation against Λ r v (and, when needed, using the relation r f t “ r v ), and arguing as in the a priori estimates of Steps 3–5, we deriv e an energy inequalit y for a suitable quantit y Z p t q controlling } r f } H 3 and } r v } H 1 . In particular, for t P r 0 , T s one obtains an estimate of the form Z p t q ď C 1 t p Z p t qq 2 ` C 2 t p Z p t qq 3 , Z p 0 q “ 0 , (4.63) where the constants C 1 , C 2 dep end only on the parameters of the model and on the a priori b ounds for the tw o solutions on r 0 , T s . F or T ą 0 chosen as in the existence argument (so that the right-hand side can b e absorb ed), the p olynomial inequality ( 4.63 ) forces Z p t q ” 0 on r 0 , T s . Consequently r f ” 0 and r v ” 0, and hence f p 1 q ” f p 2 q and v p 1 q ” v p 2 q on r 0 , T s . Therefore the solution is unique in the stated class, which completes the pro of. □ Remark 4.2. Let us emphasize that w e do not currently obtain an analogue of Theorem 4.1 for the alternativ e bidirectional mo del ( 3.30 ). While one can construct regularized/approximate solutions for that equation b y standard pro cedures, we are unable to close the corresp onding a priori estimates uniformly at the lev el required to pass to the limit. Establishing a well-posedness theory for this second bidirectional closure therefore remains an open problem. 5. Well-posedness resul ts for the unidirectional models In this section w e study the Cauc h y problem for the unidirectional models deriv ed in Subsection 3.1 as an O p ε q approximation of the full h ydroelastic system. The analysis relies on energy estimates adapted to the nonlo cal op erator structure of the mo del, together with standard multilinear and commutator b ounds for the F ourier m ultipliers appearing in the equation. W e ﬁrst establish a lo cal w ell-p osedness result for the unidirectional model ( 3.43 ), stated as follows. Theorem 5.1 (Lo cal well-posedness) . Let Υ ą 0 and δ, β ą 0. Assume that F 0 P H 2 p T q has zero spatial mean and consider the unidirectional mo del ( 3.43 ). Then there exist T ą 0, dep ending only on } F 0 } H 2 p T q and the parameters of the mo del, and a unique solution F P C pr 0 , T s ; H 2 p T qq , F p¨ , 0 q “ F 0 . 34 D. ALONSO-OR ´ AN, R. GRANERO-BELINCH ´ ON, AND J. S. ZIEBELL Pr o of of The or em 5.1 . The argumen t combines classical a priori energy estimates with a standard regularization sc heme based on molliﬁers, cf. [ 25 ]. W e ﬁrst deriv e the required a priori b ounds, then construct solutions at the regularized level and pass to the limit, and ﬁnally establish uniqueness. F or clarit y , w e decomp ose the pro of in to several steps. Step 1: c onservation of the me an. Let x F p τ qy : “ 1 2 π ş T F p ξ , τ q dξ . W e claim that any suﬃcien tly smo oth real-v alued solution of ( 3.43 ) satisﬁes x F p τ qy “ x F 0 y for all τ P r 0 , T s . Indeed, taking the spatial mean of ( 3.43 ), the linear part has zero av erage (it is a com bination of ξ –deriv ativ es and F ourier m ultipliers annihilating constan ts). With the con v en tion a p 0 q “ 1 2 and b p 0 q “ 0, w e obtain B τ x F y “ ´ 1 2 A 2 F ξ Λ F ´ J Λ , F K F ξ E . Expanding the comm utator, 2 F ξ Λ F ´ J Λ , F K F ξ “ 2 F ξ Λ F ´ Λ p F F ξ q ` F Λ F ξ , and using that x Λ p¨qy “ 0, it remains to show that x 2 F ξ Λ F ` F Λ F ξ y “ 0. In one dimension Λ “ H B ξ , hence Λ F “ H F ξ and Λ F ξ “ H F ξξ . Therefore, x F ξ Λ F y “ 1 2 π ż T F ξ H F ξ dξ “ 0 , x F Λ F ξ y “ 1 2 π ż T F H F ξξ dξ “ ´ 1 2 π ż T F ξ H F ξ dξ “ 0 , Hence B τ x F y “ 0 and the mean is conserv ed. In particular, if x F 0 y “ 0, then F p¨ , τ q remains mean-zero for all times of existence. Step 2: a priori estimates. In order to derive the a priori estimates it is conv enien t to work with a symmetrized formulation of the evolution, namely ε p M ˚ M q F τ “ M ˚ ˆ p I ` ΥΛ q F ξ ` β 4 Λ F ξξ ξ ` δ Λ F ξξ ` H F ˙ ´ ε M ˚ p 2 F ξ Λ F ´ J Λ , F K F ξ q . (5.1) where M ˚ M “ 4 p I ` ΥΛ q 2 ´ δ 2 Λ 2 B 2 ξ , M ˚ “ 2 p I ` ΥΛ q ´ δ Λ B ξ . WEAKL Y NONLINEAR MODELS F OR HYDR OELASTIC W A TER W A VES 35 T aking the L 2 p T q inner pro duct of ( 5.1 ) with F and in tegrating by parts in ξ , we obtain the ev olution iden tity d dτ ´ } F } 2 L 2 ` 2Υ } F } 2 9 H 1 { 2 ` Υ 2 } F } 2 9 H 1 ` δ 2 4 } F } 2 9 H 2 ¯ “ ż T p I ` ΥΛ q ´ p I ` ΥΛ q F ξ ¯ F dξ ` β 4 ż T p I ` ΥΛ q Λ F ξξ ξ F dξ ` δ ż T p I ` ΥΛ q Λ F ξξ F dξ ` ż T p I ` ΥΛ q H F F dξ ` δ 2 ż T p I ` ΥΛ q F ξ Λ F ξ dξ ` β δ 8 ż T Λ F ξξ ξ Λ F ξ dξ ` δ 2 2 ż T Λ F ξξ Λ F ξ dξ ` δ 2 ż T H F Λ F ξ dξ ´ 2 ż T p I ` ΥΛ q ` F ξ Λ F ˘ F dξ ` ż T p I ` ΥΛ q ` J Λ , F K F ξ ˘ F dξ ´ δ ż T F ξ p Λ F qp Λ F ξ q dξ ` δ 2 ż T ` J Λ , F K F ξ ˘ p Λ F ξ q dξ “ 12 ÿ j “ 1 I j . Using that A : “ I ` ΥΛ and Λ are self-adjoint on L 2 p T q , that B ξ comm utes with A and Λ, and that H ˚ “ ´ H , we obtain the cancellations I 1 “ I 2 “ I 4 “ I 7 “ 0. Moreov er, we also ﬁnd that I 6 “ β δ 8 ż T p Λ F ξ q ξξ p Λ F ξ q dξ “ ´ β δ 8 } F } 2 9 H 3 p T q , I 3 “ ´ δ ´ } F } 2 9 H 3 { 2 ` Υ } F } 2 9 H 2 ¯ , I 8 “ ´ δ 2 } F } 2 9 H 1 p T q . W e now b ound the nonlinear contributions I 9 , I 10 , I 11 , I 12 . Let A : “ I ` ΥΛ. By self-adjointness of A , I 9 “ ´ 2 ż T A p F ξ Λ F q F dξ “ ´ 2 x F ξ Λ F , AF y , I 10 “ ż T A pr Λ , F s F ξ q F dξ “ xr Λ , F s F ξ , AF y . Hence, integrating by parts and using Cauc h y–Sc h w arz, | I 9 | ď ∥ F ∥ L 8 ∥ F ∥ 2 9 H 2 p T q (5.2) Similarly , to b ound I 10 w e can expand the comm utator and use in tegration b y parts to ﬁnd that | I 10 | ď ∥ F ∥ L 8 ∥ F ∥ 2 9 H 2 p T q (5.3) F or I 11 and I 12 w e similarly ha ve I 11 “ δ 2 ż T F ξξ p Λ F q 2 dξ , I 12 “ δ 2 ż T p Λ p F F ξ q ´ F Λ F ξ q Λ F ξ dξ . By means of Gagliardo-Nirenberg in terp olation inequality ( 1.13 ) | I 11 | ď δ } F } 9 H 2 p T q } Λ F } 2 L 4 ď C δ ∥ F ∥ L 8 } F } 2 9 H 2 p T q (5.4) On the other hand, using Cauch y-Sc h w arz inequalit y and Moser estimate ( 1.12 ) w e ﬁnd that I 12 ď δ 4   B ξ Λ ` F 2 ˘   L 2 ∥ Λ F ξ ∥ L 2 ` δ 2 ∥ F ∥ L 8 ∥ Λ F ξ ∥ 2 L 2 ď C ∥ F ∥ L 8 ∥ F ∥ 2 9 H 2 p T q (5.5) Therefore, combining all the previous estimates, we hav e sho wn that d dτ ´ } F } 2 L 2 ` 2Υ } F } 2 9 H 1 { 2 ` Υ 2 } F } 2 9 H 1 ` δ 2 4 } F } 2 9 H 2 ¯ ` β δ 8 } F } 2 9 H 3 p T q ď C ∥ F ∥ L 8 ∥ F ∥ 2 9 H 2 p T q (5.6) 36 D. ALONSO-OR ´ AN, R. GRANERO-BELINCH ´ ON, AND J. S. ZIEBELL Deﬁne E p τ q : “ } F } 2 L 2 ` 2Υ } F } 2 9 H 1 { 2 ` Υ 2 } F } 2 9 H 1 ` δ 2 4 } F } 2 9 H 2 . Moreo ver b y means of the Sobolev em bedding ( 1.11 ), we hav e that } F } L 8 ≲ } F } H 2 ≲ E p τ q 1 { 2 . Therefore the energy inequality implies E 1 p τ q ` β δ 8 } F } 2 9 H 3 ď C } F } L 8 } F } 2 9 H 2 ≲ C E p τ q 3 { 2 , and in particular E 1 p τ q ď C E p τ q 3 { 2 . Solving this diﬀeren tial inequality gives E p τ q ´ 1 { 2 ě E p 0 q ´ 1 { 2 ´ C 2 τ , so for 0 ď τ ď T ˚ : “ 1 C E p 0 q ´ 1 { 2 w e ha ve the uniform b ound E p τ q ď 4 E p 0 q . Step 3: existenc e via mol liﬁc ation. W e deﬁne the molliﬁed unknown F ν as the unique solution to the regularized Cauch y problem $ ’ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ ’ % B τ F ν “ J ν # 1 ε ` a ` b H ˘ ´ p I ` ΥΛ qB ξ p J ν F ν q ` β 4 Λ B 3 ξ p J ν F ν q ` δ Λ B 2 ξ p J ν F ν q ` H p J ν F ν q ¯ ´ ` a ` b H ˘ ´ 2 B ξ p J ν F ν q Λ p J ν F ν q ´ J Λ , J ν F ν K B ξ p J ν F ν q ¯ + , F ν p¨ , 0 q “ J ν F 0 . (5.7) Since J ν is smo othing, the righ t-hand side of ( 5.7 ) deﬁnes a lo cally Lipschitz vector ﬁeld on H 2 p T q . Therefore, b y the Picard–Lindel¨ of theorem in Banach spaces, there exists T ν ą 0 and a unique solution F ν P C 1 pr 0 , T ν s ; H 2 p T qq . Because J ν is self-adjoin t on L 2 p T q and comm utes with B ξ , Λ, H , p I ` ΥΛ q and the multipliers a, b , the a priori energy computation in Step 2 can b e repeated for the molliﬁed system ( 5.7 ). The only mo diﬁcation is that J ν ma y b e shifted b etw een factors using x J ν G, F ν y “ x G, J ν F ν y . Consequen tly , the same cancellations and sign identities hold, and w e obtain the same diﬀeren tial inequality for the molliﬁed energy E ν p τ q with constan ts indep endent of ν . Denoting E ν p τ q : “ } F ν } 2 L 2 ` 2Υ } F ν } 2 9 H 1 { 2 ` Υ 2 } F ν } 2 9 H 1 ` δ 2 4 } F ν } 2 9 H 2 , w e obtain, for τ P r 0 , T ν q , d dτ E ν p τ q ď C p E ν p τ qq 3 { 2 , (5.8) with a constan t C indep endent of ν . Solving ( 5.8 ) yields a time T ˚ “ T ˚ p E p 0 qq ą 0 suc h that sup 0 ď τ ď T ˚ E ν p τ q ď 4 E p 0 q for all ν ě 1 . (5.9) In particular, T ν ě T ˚ and t F ν u is b ounded in L 8 pr 0 , T ˚ s ; H 2 p T qq uniformly in ν . By the uniform bound ( 5.9 ) and the molliﬁed equation ( 5.7 ), the sequence t F ν u is b ounded in L 8 pr 0 , T ˚ s ; H 2 p T qq and tB τ F ν u is b ounded in L 8 pr 0 , T ˚ s ; H 1 p T qq , uniformly in ν . Hence, after ex- tracting a subsequence, F ν Ñ F strongly in C pr 0 , T ˚ s ; H 1 ` θ p T qq for every θ P p 0 , 1 q and weakly- ˚ WEAKL Y NONLINEAR MODELS F OR HYDR OELASTIC W A TER W A VES 37 in L 8 pr 0 , T ˚ s ; H 2 p T qq . Using the contin uit y of F ourier multipliers on Sob olev spaces and standard pro duct/comm utator stabilit y at the H 2 lev el, we may pass to the limit in the molliﬁed equation and conclude that F solves ( 3.43 ) on r 0 , T ˚ s with F p¨ , 0 q “ F 0 , and moreo v er F P C pr 0 , T ˚ s ; H 2 p T qq . Step 4: uniqueness. Let F and G b e tw o solutions of ( 5.1 ) on r 0 , T s suc h that F , G P L 8 pr 0 , T s ; H 2 p T qq , F p¨ , 0 q “ G p¨ , 0 q “ F 0 . Set W : “ F ´ G . Subtracting ( 5.1 ) (written for F and G ) yields p M ˚ M q W τ “ M ˚ ˆ p I ` ΥΛ q W ξ ` β 4 Λ W ξξ ξ ` δ Λ W ξξ ` H W ˙ ´ M ˚ ` N p F q ´ N p G q ˘ , (5.10) where N p U q : “ 2 U ξ Λ U ´ J Λ , U K U ξ . T esting ( 5.10 ) against W in L 2 p T q and rep eating the energy computation of Step 2 gives d dτ E W p τ q ` β δ 8 } W } 2 9 H 3 p T q ≲ ˇ ˇ x M ˚ p N p F q ´ N p G qq , W y ˇ ˇ , (5.11) with E W p τ q : “ } W } 2 L 2 ` 2Υ } W } 2 9 H 1 { 2 ` Υ 2 } W } 2 9 H 1 ` δ 2 4 } W } 2 9 H 2 . Moreo ver, expanding with F “ G ` W w e hav e N p F q ´ N p G q “ 2 W ξ Λ F ` 2 G ξ Λ W ´ J Λ , W K F ξ ´ J Λ , G K W ξ . (5.12) Using H 2 p T q ã Ñ W 1 , 8 p T q and the standard commutator b ound } J Λ , f K g } L 2 ≲ } f ξ } L 8 } g } L 2 , the nonlinear diﬀerence is Lipschitz in the energy space, namely ˇ ˇ x M ˚ p N p F q ´ N p G qq , W y ˇ ˇ ≲ ` } F } H 2 ` } G } H 2 ˘ E W p τ q . Therefore, d dτ E W p τ q ď C ` 1 ` } F } H 2 ` } G } H 2 ˘ E W p τ q . By Step 2, F and G are b ounded in L 8 pr 0 , T s ; H 2 p T qq and E W p 0 q “ 0. Hence Gr¨ onw all’s inequality implies E W p τ q ” 0 on r 0 , T s , so W ” 0 and F “ G . □ In the sequel, we establish global-in-time existence for the unidirectional model ( 3.43 ) under a small- ness assumption on the initial datum. More precisely , we hav e the follo wing result. Theorem 5.2 (Global existence and decay for small data) . Let Υ ą 0 and δ, β ą 0. Assume that F 0 P H 2 p T q has zero spatial mean. There exists ε ˚ ą 0, dep ending only on Υ , δ, β , such that if } F 0 } H 2 p T q ď ε ˚ , then the solution F giv en by Theorem 5.1 exists globally in time, F P C ` r 0 , 8q ; H 2 p T q ˘ , Moreo ver, there exist constan ts c, C ą 0, depending only on Υ , δ, β , suc h that for all τ ě 0, E p τ q ď C E p 0 q e ´ cτ , E p τ q : “ } F } 2 L 2 ` 2Υ } F } 2 9 H 1 { 2 ` Υ 2 } F } 2 9 H 1 ` δ 2 4 } F } 2 9 H 2 . 38 D. ALONSO-OR ´ AN, R. GRANERO-BELINCH ´ ON, AND J. S. ZIEBELL Pr o of of The or em 5.2 . Let F b e the lo cal H 2 p T q solution giv en by Theorem 5.1 . By Step 1 in the pro of of Theorem 5.1 , the spatial mean is conserv ed, so x F p τ qy “ 0 for all times of existence. Recall the energy from Step 2, E p τ q : “ } F } 2 L 2 ` 2Υ } F } 2 9 H 1 { 2 ` Υ 2 } F } 2 9 H 1 ` δ 2 4 } F } 2 9 H 2 , and the diﬀeren tial inequality obtained there: E 1 p τ q ` β δ 8 } F p τ q} 2 9 H 3 p T q ď C } F p τ q} L 8 p T q } F p τ q} 2 9 H 2 p T q , (5.13) where C dep ends only on Υ , δ, β . Since x F y “ 0, all F ourier modes satisfy | k | ě 1, hence } F } 9 H 2 ď } F } 9 H 3 . Moreo v er, in one dimension H 2 p T q ã Ñ L 8 p T q and E p τ q „ } F p τ q} 2 H 2 , so } F p τ q} L 8 ≲ } F p τ q} H 2 ≲ E p τ q 1 { 2 . (5.14) Substituting these b ounds into ( 5.13 ) yields E 1 p τ q ` β δ 8 } F } 2 9 H 3 ď C 1 E p τ q 1 { 2 } F } 2 9 H 3 , (5.15) for some C 1 “ C 1 p Υ , δ, β q . Cho ose ε ˚ ą 0 so that } F 0 } H 2 ď ε ˚ implies C 1 E p 0 q 1 { 2 ď β δ 16 . Then, by contin uity of E p τ q , the b ound C 1 E p τ q 1 { 2 ď β δ 16 p ersists for as long as the solution exists, and ( 5.15 ) impro v es to E 1 p τ q ` β δ 16 } F p τ q} 2 9 H 3 p T q ď 0 . (5.16) In particular, E p τ q is nonincreasing and hence E p τ q ď E p 0 q for all times of existence. Therefore } F p τ q} H 2 sta ys bounded uniformly , and the con tin uation criterion from the lo cal w ell-p osedness the- orem preven ts ﬁnite-time breakdo wn; consequently the solution extends for all τ ě 0. Finally , since x F y “ 0, we hav e the co ercivity E p τ q ď C 0 } F p τ q} 2 9 H 3 for some C 0 “ C 0 p Υ , δ q , and com bining with ( 5.16 ) gives E 1 p τ q ` c E p τ q ď 0 , c : “ β δ 16 C 0 ą 0 , whic h implies E p τ q ď E p 0 q e ´ cτ . □ W e conclude this section b y establishing an analogous well-posedness result for the second unidirec- tional mo del ( 3.48 ). More precisely , w e prov e that Theorem 5.3 (Global well-posedness for small H 3 data) . Let Υ ą 0 and δ, β ą 0. Assume that F 0 P H 3 p T q has zero spatial mean and consider the second unidirectional mo del ( 3.48 ). There exists ε ˚ “ ε ˚ p Υ , δ, β q ą 0 such that, if } F 0 } H 3 p T q ď ε ˚ , then the corresp onding unique solution exists globally in time and F P C pr 0 , 8q ; H 3 p T qq , F p¨ , 0 q “ F 0 . WEAKL Y NONLINEAR MODELS F OR HYDR OELASTIC W A TER W A VES 39 Remark 5.4. The smallness h yp othesis in Theorem 5.3 is not only used to extend solutions glob- ally . In fact, for the second unidirectional mo del ( 3.48 ) we do not know whether large-data lo cal w ell-p osedness holds in H 3 p T q . The reason is that the nonlinear terms are highly singular (in par- ticular those m ultiplied b y β and δ ), and in our energy metho d they can b e controlled only by a b o otstrap/absorption argument that relies on } F } H 3 b eing small compared to the linear dissipation. Pr o of of The or em 5.3 . The argumen t follows the same general strategy as the pro of of Theorem 5.1 . In particular, one ma y construct solutions by a standard F riedrichs molliﬁcation sc heme, obtain a time of existence indep endent of the regularization parameter via uniform a priori b ounds, pass to the limit by compactness, and conclude uniqueness by an energy estimate on the diﬀerence of tw o solutions. T o a v oid repetition, we do not repro duce these classical appro ximation, compactness, and uniqueness steps here. Instead, we focus on the only new input for the presen t model, namely the a priori energy estimates. Once this estimate is established, the existence and uniqueness statemen ts follo w v erbatim from the same scheme used in Theorem 5.1 . Similarly as in Theorem 5.1 w e can sho w that the mean zero assumption is preserved b y the ev olution. Indeed, assume x F 0 y “ 0 and let F b e a smo oth solution of ( 3.48 ). T aking spatial means, it suﬃces to note that x H g p 0 q “ 0 and that α p Λ , B ξ q carries a factor Λ (hence also annihilates the zero mo de), so p α ` γ H q g is alw a ys mean-zero. Therefore B τ x F y “ 0 and x F p τ qy ” 0. F or the a priori bounds it is con venien t to w ork with the symmetrized form ulation, rather than with ( 3.48 ) written through α ` γ H . Throughout the proof w e ﬁx ε “ 1. Set A : “ I ` ΥΛ , M ˚ 1 “ ´ 2 A B ξ ` δ Λ B 2 ξ , M ˚ 1 M 1 “ δ 2 Λ 2 B 4 ξ ´ 4 A 2 B 2 ξ . (5.17) W e ma y rewrite the second unidirectional mo del ( 3.48 ) in the symmetrized form p M ˚ 1 M 1 q F τ “ M ˚ 1 ´ AF ξξ ` β 4 Λ F ξξ ξ ξ ` δ Λ F ξξ ξ ` Λ F ¯ ´ M ˚ 1 N 1 p F q , (5.18) where the nonlinear term N 1 p F q is given by N 1 p F q : “ 1 2 Λ ´ F 2 ξ ´ p Λ F q 2 ¯ ´ J Λ , F ξ K F ξ ` F ξξ Λ F ` J Λ , F K T F ´ F ξ H p T F q ` β 4 ´ J Λ , F K T F ξξ ξ ξ ´ F ξ Λ p T F ξξ ξ q ¯ ` δ ´ J Λ , F K T F ξξ ξ ´ F ξ Λ p T F ξξ q ¯ . (5.19) T ake the L 2 p T q inner product of ( 5.18 ) with F . Using that A and Λ are self-adjoin t on L 2 p T q and comm ute with B ξ , and in tegrating by parts w e ﬁnd that d dτ ` 2 } AF ξ } 2 L 2 ` δ 2 2 } F } 2 9 H 3 p T q ˘ “ @ M ˚ 1 L 1 p F q , F D ´ @ M ˚ 1 N 1 p F q , F D (5.20) where L 1 p F q : “ AF ξξ ` β 4 Λ F ξξ ξ ξ ` δ Λ F ξξ ξ ` Λ F . 40 D. ALONSO-OR ´ AN, R. GRANERO-BELINCH ´ ON, AND J. S. ZIEBELL Computing the linear terms of the righ t hand side of ( 5.20 ) w e readily c hec k that ´ @ M ˚ 1 L 1 p F q , F D “ 8 ÿ i “ 1 J i , (5.21) with J 1 : “ 2 ż T A p AF ξξ q F ξ dξ , J 2 : “ δ ż T Λ p AF ξξ q F ξξ dξ , J 3 : “ β 2 ż T A ´ Λ F ξξ ξ ξ ¯ F ξ dξ , J 4 : “ δ β 4 ż T Λ ´ Λ F ξξ ξ ξ ¯ F ξξ dξ , J 5 : “ 2 δ ż T A p Λ F ξξ ξ q F ξ dξ , J 6 : “ δ 2 ż T Λ p Λ F ξξ ξ q F ξξ dξ , J 7 : “ 2 ż T A p Λ F q F ξ dξ , J 8 : “ δ ż T Λ p Λ F q F ξξ dξ . Therefore, using once again that A and Λ are self-adjoint on L 2 p T q obtain that J 1 “ J 3 “ J 6 “ J 7 “ 0 . Moreo ver, w e also ﬁnd that J 2 ` J 5 “ ´ δ    A 1 { 2 F    2 9 H 5 { 2 , J 4 “ ´ δ β 4 ∥ F ∥ 2 9 H 4 , J 8 “ ´ δ ∥ F ∥ 2 9 H 2 . Th us, noticing that } AF ξ } 2 L 2 “ } F } 2 9 H 1 ` 2Υ } F } 2 9 H 3 { 2 ` Υ 2 } F } 2 9 H 2 (5.22) and denoting b y E p τ q “ } F } 2 9 H 1 ` 2Υ } F } 2 9 H 3 { 2 ` Υ 2 } F } 2 9 H 2 ` δ 2 4 } F } 2 9 H 3 p T q w e ﬁnd the energy estimate d dτ E p τ q ` δ 2    A 1 { 2 F    2 9 H 5 { 2 ` δ β 8 ∥ F ∥ 2 9 H 4 ` δ 2 ∥ F ∥ 2 9 H 2 “ ´ 1 2 @ M ˚ 1 N 1 p F q , F D (5.23) Next, we need to estimate the non-linear terms. More precisely , w e ha v e to control K j “ 2 ż T A B ξ ` N 1 ,j p F q ˘ F dξ ´ δ ż T Λ B 2 ξ ` N 1 ,j p F q ˘ F dξ . WEAKL Y NONLINEAR MODELS F OR HYDR OELASTIC W A TER W A VES 41 where K 1 “ 2 ż T A B ξ „ 1 2 Λ ´ F 2 ξ ´ p Λ F q 2 ¯ ȷ F dξ ´ δ ż T Λ B 2 ξ „ 1 2 Λ ´ F 2 ξ ´ p Λ F q 2 ¯ ȷ F dξ , K 2 “ 2 ż T A B ξ r ´ J Λ , F ξ K F ξ s F dξ ´ δ ż T Λ B 2 ξ r ´ J Λ , F ξ K F ξ s F dξ , K 3 “ 2 ż T A B ξ r F ξξ Λ F s F dξ ´ δ ż T Λ B 2 ξ r F ξξ Λ F s F dξ , K 4 “ 2 ż T A B ξ r J Λ , F K T F s F dξ ´ δ ż T Λ B 2 ξ r J Λ , F K T F s F dξ , K 5 “ 2 ż T A B ξ r´ F ξ H p T F qs F dξ ´ δ ż T Λ B 2 ξ r´ F ξ H p T F qs F dξ , K 6 “ 2 ż T A B ξ „ β 4 ´ J Λ , F K T F ξξ ξ ξ ´ F ξ Λ p T F ξξ ξ q ¯ ȷ F dξ ´ δ ż T Λ B 2 ξ „ β 4 ´ J Λ , F K T F ξξ ξ ξ ´ F ξ Λ p T F ξξ ξ q ¯ ȷ F dξ , K 7 “ 2 ż T A B ξ ” δ ´ J Λ , F K T F ξξ ξ ´ F ξ Λ p T F ξξ q ¯ı F dξ ´ δ ż T Λ B 2 ξ ” δ ´ J Λ , F K T F ξξ ξ ´ F ξ Λ p T F ξξ q ¯ı F dξ . Using the algebra prop erty ( 1.12 ), estimate ( 1.15 ) and Sob olev em b edding ( 1.11 ) we ﬁnd that | K 1 | ≲ ` } F ξ } 2 9 H 1 ` } Λ F } 2 9 H 1 ˘ ´ } AF ξ } L 2 ` } F } 9 H 3 ¯ ≲ E 3 2 p τ q , | K 2 | ≲ ` } F ξ } 2 9 H 1 ` } F ξ } L 8 } Λ F ξ } L 2 ˘ ´ } AF ξ } L 2 ` } F } 9 H 3 ¯ ≲ E 3 2 p τ q , | K 3 | ≲ ` } F } 9 H 2 } Λ F } L 8 ˘ ´ } AF ξ } L 2 ` } F } 9 H 3 ¯ ≲ E 3 2 p τ q , | K 4 | ≲ ` } F } 9 H 1 } T F } 9 H 1 ` } F } L 8 } Λ T F } L 2 ˘ ´ } AF ξ } L 2 ` } F } 9 H 3 ¯ ≲ E 3 2 p τ q , | K 5 | ≲ ` } F ξ } L 8 } HT F } L 2 ˘ ´ } AF ξ } L 2 ` } F } 9 H 3 ¯ ≲ E 3 2 p τ q . (5.24) T o conclude the a priori estimates, we are left with K 6 , K 7 whic h are the more singular terms. Ex- panding the comm utator and in tegrating b y parts, we readily c hec k that K 6 “ 2 ż T « β 4 ´ Λ ` F T F ξξ ξ ξ ˘ ´ F Λ ` T F ξξ ξ ξ ˘ ´ F ξ Λ ` T F ξξ ξ ˘ ¯ ﬀ A B ξ F dξ ´ δ ż T « β 4 ´ Λ ` F T F ξξ ξ ξ ˘ ´ F Λ ` T F ξξ ξ ξ ˘ ´ F ξ Λ ` T F ξξ ξ ˘ ¯ ﬀ Λ B 2 ξ F dξ . Rather than estimating these in tegrals directly , we ﬁrst integrate b y parts once more in the ﬁrst t wo contributions. This reveals the appropriate deriv ative distribution needed to close the estimate. 42 D. ALONSO-OR ´ AN, R. GRANERO-BELINCH ´ ON, AND J. S. ZIEBELL Indeed, we can write K 6 “ K 6 ,A ` K 6 ,δ where K 6 ,A “ “ β 2 ż T F T F ξξ ξ ξ Λ A B ξ F dξ ´ β 2 ż T T F ξξ ξ ξ Λ ` F A B ξ F ˘ dξ ´ β 2 ż T F ξ Λ p T F ξξ ξ q A B ξ F dξ and K 6 ,δ “ β δ 4 ż T F T F ξξ ξ ξ Λ 4 F dξ ` β δ 4 ż T T F ξξ ξ ξ Λ ` F Λ B 2 ξ F ˘ dξ ` β δ 4 ż T F ξ Λ p T F ξξ ξ q Λ B 2 ξ F dξ . Therefore, using Cauc h y-Sch warz, the algebra prop erty ( 1.12 ) and estimate ( 1.15 ) we conclude that | K 6 ,A | ≲ ∥ F ∥ L 8 ∥T F ∥ 9 H 4 ∥ AF ξ ∥ L 2 ` ∥ T F ∥ 9 H 4 ∥ F ∥ 9 H 1 ∥ AF ξ ∥ 9 H 1 ` ∥ F ξ ∥ L 8 ∥T F ∥ 9 H 4 ∥ AF ξ ∥ L 2 and | K 6 ,δ | ≲ ∥ F ∥ L 8 ∥T F ∥ 2 9 H 4 ` ∥ T F ∥ 2 9 H 4 ∥ F ∥ 9 H 1 ` ∥ F ξ ∥ L 8 ∥T F ∥ 9 H 4 ∥ F ∥ 9 H 3 . Com bining both estimates, recalling ( 5.22 ) and using the Sobolev em b edding ( 1.11 ) yields | K 6 | ď | K 6 ,A | ` | K 6 ,δ | ≲ a E p τ q ∥ F ∥ 2 9 H 4 (5.25) The same estimate holds for the term K 7 . More precisely , by the same argument we hav e that | K 7 | ≲ a E p τ q ∥ F ∥ 2 9 H 4 (5.26) Therefore, collecting estimates ( 5.23 ), ( 5.24 ), ( 5.25 ) and ( 5.26 ) w e hav e shown that d dτ E p τ q ` δ 2    A 1 { 2 F    2 9 H 5 { 2 ` δ β 8 ∥ F ∥ 2 9 H 4 ` δ 2 ∥ F ∥ 2 9 H 2 ď C 1 E p τ q 3 2 ` C 2 a E p τ q ∥ F ∥ 2 9 H 4 (5.27) Deﬁne the b ootstrap time T ˚ : “ sup ! T ą 0 : sup 0 ď τ ď T E p τ q ď 4 E p 0 q ) . On r 0 , T ˚ s we hav e a E p τ q ď 2 a E p 0 q , hence C 2 a E p τ q } F } 2 9 H 4 ď 2 C 2 a E p 0 q } F } 2 9 H 4 . Assume the smallness condition a E p 0 q ď ε ˚ : “ δ β 32 C 2 . (5.28) Then on r 0 , T ˚ s the last term in ( 5.27 ) is absorb ed b y δ β 8 } F } 2 9 H 4 , and w e obtain E 1 p τ q ` δ 2 } A 1 { 2 F } 2 9 H 5 { 2 ` δ β 16 } F } 2 9 H 4 ` δ 2 } F } 2 9 H 2 ď C 1 E p τ q 3 { 2 , τ P r 0 , T ˚ s . (5.29) Dropping the nonnegativ e dissipation terms yields E 1 p τ q ď C 1 E p τ q 3 { 2 , so that E p τ q ´ 1 { 2 ě E p 0 q ´ 1 { 2 ´ C 1 2 τ , τ P r 0 , T ˚ s . In particular, E p τ q ď 4 E p 0 q on r 0 , T ˚ s , which impro ves the bo otstrap assumption and forces T ˚ “ 8 . Hence sup τ ě 0 E p τ q ď 4 E p 0 q and the corresponding H 3 solution extends globally in time. Uniqueness follo ws by rep eating the same energy estimate on the diﬀerence of tw o solutions as in the pro of of Theorem 5.1 □ WEAKL Y NONLINEAR MODELS F OR HYDR OELASTIC W A TER W A VES 43 Appendix A. Geometric formulas for graph surf aces F or con v enience we collect here the basic geometric iden tities asso ciated with the free surface Γ p t q “ tp x 1 , x 2 , η p x, t qq : x “ p x 1 , x 2 q P L S 1 ˆ L S 1 u , expressing all quan tities directly in terms of the height function η p x, t q , where x “ p x 1 , x 2 q denotes the horizontal v ariables. The surface is parametrised b y X p x, t q “ p x 1 , x 2 , η p x, t qq , so that diﬀeren tiation with resp ect to x yields the tangent vectors B x i X “ e i ` pB x i η q e 3 , i “ 1 , 2 . The ﬁrst fundamen tal form is then g ij “ B x i X ¨ B x j X “ δ ij ` B x i η B x j η , with determinant | g | “ 1 ` | ∇ x η | 2 , α : “ 1 ` | ∇ x η | 2 , and inv erse metric g ij “ α ij “ p g ij q ´ 1 . The corresp onding upw ard-pointing unit normal takes the familiar form n p x, t q “ 1 ? α ˆ ´ η x 1 ´ η x 2 1 ˙ , and the induced surface area element is ? α dx . Diﬀeren tiating the parametrisation twice, B x i x j X “ p 0 , 0 , B x i x j η q , and pro jecting onto the normal yields the second fundamental form, b ij “ B x i x j X ¨ n “ η x i x j ? α . In terms of g ij and b ij , the mean curv ature of the graph is H p η q “ 1 2 g ij b ij “ 1 2 ? α ´ α 11 η x 1 x 1 ` 2 α 12 η x 1 x 2 ` α 22 η x 2 x 2 ¯ , while the Gauss curv ature is giv en by the classical identit y K p η q “ det p b ij q det p g ij q “ η x 1 x 1 η x 2 x 2 ´ p η x 1 x 2 q 2 p 1 ` | ∇ x η | 2 q 2 . Finally , for any scalar function f “ f p x, t q deﬁned on the surface, the Laplace–Beltrami operator reduces to the divergence-form expression ∆ Γ f “ 1 ? α B x i ` ? α α ij B x j f ˘ , where all deriv atives are horizon tal. These formulas p ermit the elastic op erator E p η q to b e expressed en tirely in terms of η and its deriv atives on the reference torus. 44 D. ALONSO-OR ´ AN, R. GRANERO-BELINCH ´ ON, AND J. S. ZIEBELL Appendix B. Nondimensionalisa tion of the Fluid–Structure System In this app endix we provide a complete nondimensionalisation of the system go v erning the coupled ev olution of the ﬂuid p oten tial ϕ and the plate displacement η . B.1. Choice of c haracteristic scales. Let L denote the typical horizon tal wa v elength and H the t ypical v ertical amplitude of the elastic sheet. W e write x “ p x 1 , x 2 q for the horizon tal v ariables and x 3 for the v ertical co ordinate. W e introduce the dimensionless spatial and temporal v ariables x “ L ˜ x, x 3 “ L ˜ x 3 , t “ d L g ˜ t, and the nondimensional surface displacemen t, p otential, and surface p otential η p x, t q “ H ˜ η p ˜ x, ˜ t q , ϕ p x, x 3 , t q “ H a g L ˜ ϕ p ˜ x, ˜ x 3 , ˜ t q , ψ p x, t q “ H a g L ˜ ψ p ˜ x, ˜ t q . W e denote the dimensionless steepness parameter by ε : “ H L . In what follo ws all deriv atives with respect to ˜ x are written with tildes. T ransformations of deriv atives follo w from B t “ c g L B ˜ t , ∇ x “ 1 L ˜ ∇ x , ∆ x “ 1 L 2 ˜ ∆ x , and therefore B t η “ H c g L B ˜ t ˜ η , B 2 t η “ H g L B 2 ˜ t ˜ η . B.2. Scaling of the b ending op erator and dynamic b oundary conditions. The elastic oper- ator in the mo del is E p η q “ B 2 ∆ Γ H p η q ` B H p η q 3 ´ B H p η q K p η q , where B ą 0 denotes the bending stiﬀness. 2 W e no w express eac h geometric quantit y more explicitly in terms of the dimensionless v ariables. Recall that η p x, t q “ H ˜ η p ˜ x, ˜ t q , x i “ L ˜ x i , i “ 1 , 2 , so that B x i η “ ε B ˜ x i ˜ η , B 2 x i x j η “ H L 2 B 2 ˜ x i ˜ x j ˜ η , ε “ H L . The induced metric and its determinant are g ij “ δ ij ` B x i η B x j η “ δ ij ` ε 2 B ˜ x i ˜ η B ˜ x j ˜ η , | g | “ 1 ` | ∇ x η | 2 “ 1 ` ε 2 | ˜ ∇ x ˜ η | 2 “ : ˜ α, and the in v erse metric entries are α 11 “ 1 ` ε 2 pB ˜ x 2 ˜ η q 2 ˜ α , α 22 “ 1 ` ε 2 pB ˜ x 1 ˜ η q 2 ˜ α , α 12 “ ´ ε 2 B ˜ x 1 ˜ η B ˜ x 2 ˜ η ˜ α . 2 Dimensionally , B has units of energy , r B s “ ML 2 T ´ 2 , so that the integrand in the b ending energy E b r η s “ ş B 2 H p η q 2 a 1 ` | ∇ x η | 2 dx has units of energy per unit area. WEAKL Y NONLINEAR MODELS F OR HYDR OELASTIC W A TER W A VES 45 Mean curv ature in terms of ˜ η . Using the form ula (see Appendix A ) H p η q “ 1 2 a 1 ` | ∇ x η | 2 ` α 11 B 2 x 1 x 1 η ` 2 α 12 B 2 x 1 x 2 η ` α 22 B 2 x 2 x 2 η ˘ , and substituting B 2 x i x j η “ H L 2 B 2 ˜ x i ˜ x j ˜ η and 1 ` | ∇ x η | 2 “ ˜ α , w e obtain H p η q “ H L 2 1 2 ? ˜ α ` α 11 B 2 ˜ x 1 ˜ x 1 ˜ η ` 2 α 12 B 2 ˜ x 1 ˜ x 2 ˜ η ` α 22 B 2 ˜ x 2 ˜ x 2 ˜ η ˘ . It is con v enient to introduce the dimensionless functional H p ˜ η ; ε q : “ 1 2 ? ˜ α ` α 11 B 2 ˜ x 1 ˜ x 1 ˜ η ` 2 α 12 B 2 ˜ x 1 ˜ x 2 ˜ η ` α 22 B 2 ˜ x 2 ˜ x 2 ˜ η ˘ , so that H p η q “ H L 2 H p ˜ η ; ε q . Gaussian curv ature in terms of ˜ η . F rom K p η q “ B 2 x 1 x 1 η B 2 x 2 x 2 η ´ pB 2 x 1 x 2 η q 2 p 1 ` | ∇ x η | 2 q 2 , w e obtain B 2 x 1 x 1 η B 2 x 2 x 2 η ´ pB 2 x 1 x 2 η q 2 “ H 2 L 4 ` B 2 ˜ x 1 ˜ x 1 ˜ η B 2 ˜ x 2 ˜ x 2 ˜ η ´ pB 2 ˜ x 1 ˜ x 2 ˜ η q 2 ˘ , p 1 ` | ∇ x η | 2 q 2 “ ˜ α 2 . Th us K p η q “ H 2 L 4 B 2 ˜ x 1 ˜ x 1 ˜ η B 2 ˜ x 2 ˜ x 2 ˜ η ´ pB 2 ˜ x 1 ˜ x 2 ˜ η q 2 ˜ α 2 “ : H 2 L 4 K p ˜ η ; ε q , where K p ˜ η ; ε q “ B 2 ˜ x 1 ˜ x 1 ˜ η B 2 ˜ x 2 ˜ x 2 ˜ η ´ pB 2 ˜ x 1 ˜ x 2 ˜ η q 2 ˜ α 2 . Laplace–Beltrami of H p η q . Recall ∆ Γ f “ 1 a 1 ` | ∇ x η | 2 B x i ` a 1 ` | ∇ x η | 2 α ij B x j f ˘ , where B x i acts on the horizontal v ariables x “ p x 1 , x 2 q . F or f “ H p η q , with H p η q “ H L 2 H p ˜ η ; ε q , we ha ve B x j H “ H L 2 B x j H “ H L 3 B ˜ x j H p ˜ η ; ε q . Hence a 1 ` | ∇ x η | 2 α ij B x j H “ ? ˜ α α ij H L 3 B ˜ x j H , and B x i ` a 1 ` | ∇ x η | 2 α ij B x j H ˘ “ 1 L B ˜ x i ˆ ? ˜ α α ij H L 3 B ˜ x j H ˙ “ H L 4 B ˜ x i ´ ? ˜ α α ij B ˜ x j H ¯ . Dividing by a 1 ` | ∇ x η | 2 “ ? ˜ α we arrive at ∆ Γ H p η q “ H L 4 L Γ p ˜ η ; ε q , 46 D. ALONSO-OR ´ AN, R. GRANERO-BELINCH ´ ON, AND J. S. ZIEBELL where L Γ p ˜ η ; ε q : “ 1 ? ˜ α B ˜ x i ´ ? ˜ α α ij B ˜ x j H p ˜ η ; ε q ¯ . Putting everything together. With the abov e notation, H p η q “ H L 2 H p ˜ η ; ε q , K p η q “ H 2 L 4 K p ˜ η ; ε q , ∆ Γ H p η q “ H L 4 L Γ p ˜ η ; ε q . Therefore, B 2 ∆ Γ H p η q “ B H 2 L 4 L Γ p ˜ η ; ε q , B H p η q 3 “ B ˆ H L 2 ˙ 3 H p ˜ η ; ε q 3 “ B H L 4 ε 2 H p ˜ η ; ε q 3 , ´ B H p η q K p η q “ ´ B ˆ H L 2 H ˙ ˆ H 2 L 4 K ˙ “ ´ B H L 4 ε 2 H p ˜ η ; ε q K p ˜ η ; ε q . F actoring out the common prefactor B H L 4 w e obtain E p η q “ B H L 4 E p ˜ η ; ε q where the dimensionless op erator E is given explicitly b y E p ˜ η ; ε q “ 1 2 L Γ p ˜ η ; ε q ` ε 2 ´ H p ˜ η ; ε q 3 ´ H p ˜ η ; ε q K p ˜ η ; ε q ¯ , with H p ˜ η ; ε q “ 1 2 ? ˜ α ` α 11 B 2 ˜ x 1 ˜ x 1 ˜ η ` 2 α 12 B 2 ˜ x 1 ˜ x 2 ˜ η ` α 22 B 2 ˜ x 2 ˜ x 2 ˜ η ˘ , K p ˜ η ; ε q “ B 2 ˜ x 1 ˜ x 1 ˜ η B 2 ˜ x 2 ˜ x 2 ˜ η ´ pB 2 ˜ x 1 ˜ x 2 ˜ η q 2 ˜ α 2 , L Γ p ˜ η ; ε q “ 1 ? ˜ α B ˜ x i ´ ? ˜ α α ij B ˜ x j H p ˜ η ; ε q ¯ , and ˜ α “ 1 ` ε 2 | ˜ ∇ x ˜ η | 2 , α ij “ α ij p ε, ˜ ∇ x ˜ η q . In particular, E dep ends only on ˜ η and its ﬁrst and second deriv ativ es; no additional geometric unkno wns are in tro duced. 3 Remark B.1. In many asymptotic regimes one assumes small steepness, ε ! 1. Expanding ˜ α “ 1 ` ε 2 | ˜ ∇ x ˜ η | 2 “ 1 ` O p ε 2 q , α ij “ δ ij ` O p ε 2 q , w e obtain, to leading order, H p ˜ η ; ε q “ 1 2 ∆ ˜ x ˜ η ` O ` ε 2 | ˜ ∇ x ˜ η | | ˜ ∇ 2 x ˜ η | ˘ , and therefore L Γ p ˜ η ; ε q “ ∆ ˜ x H p ˜ η ; ε q ` O p ε 2 q “ 1 2 ∆ 2 ˜ x ˜ η ` O ` ε 2 P p ˜ η q ˘ , 3 In particular, the nondimensional elastic op erator E p η ; ε q depends only on η and on its ﬁrst and second spatial deriv atives; no additional geometric variables or unknowns are introduced. WEAKL Y NONLINEAR MODELS F OR HYDR OELASTIC W A TER W A VES 47 where P p ˜ η q denotes a polynomial expression in ˜ ∇ x ˜ η , ˜ ∇ 2 x ˜ η , ˜ ∇ 3 x ˜ η , and ˜ ∇ 4 x ˜ η . Consequently E p ˜ η ; ε q “ 1 4 ∆ 2 ˜ x ˜ η ` O ` ε 2 P p ˜ η q ˘ , so that the principal part of the geometric b ending op erator is indeed biharmonic. This justiﬁes the app earance of operators of the type ∆ 2 x η as leading-order models in small-slop e asymptotic regimes, with the fully nonlinear curv ature eﬀects enco ded in higher-order corrections. W e no w turn to the nondimensionalisation of the dynamic boundary condition for the surface p otential ψ . In dimensional v ariables this equation reads, on the free surface, B t ψ “ ρ s h ρ f B 2 t η ` 1 ρ f E p η q ` γ ρ f ∆ x B t η ` B x 3 ϕ ` ∇ ϕ ¨ a 1 ` | ∇ x η | 2 n ˘ ´ 1 2 | ∇ ϕ | 2 ´ g η , on Γ p t q . Using the scalings introduced ab o v e, each term may b e rewritten in nondimensional form. F or the inertial term w e obtain ρ s h ρ f B 2 t η “ ρ s h ρ f H g L B 2 ˜ t ˜ η “ H g ˆ ρ s h ρ f L ˙ B 2 ˜ t ˜ η . F or the elastic con tribution, we use 1 ρ f E p η q “ 1 ρ f B H L 4 E p ˜ η ; ε q “ H g ˆ B ρ f g L 3 ˙ ε E p ˜ η ; ε q . The damping term scales as γ ρ f ∆ x B t η “ γ ρ f 1 L 2 ˜ ∆ x ` H b g L B ˜ t ˜ η ˘ “ H g ˜ γ ρ f a g L 3 ¸ ˜ ∆ x B ˜ t ˜ η . F or the Bernoulli terms we use ∇ ϕ “ a g LH 1 L ˜ ∇ ˜ ϕ “ H ? g ˜ ∇ ˜ ϕ, B x 3 ϕ “ H ? g B ˜ x 3 ˜ ϕ, and | ∇ ϕ | 2 “ H g | ˜ ∇ ˜ ϕ | 2 , g η “ H g ˜ η . Moreo ver, on the free surface the kinematic condition implies ∇ ϕ ¨ a 1 ` | ∇ x η | 2 n η “ H ? g ˜ ∇ ˜ ϕ ¨ b 1 ` ε 2 | ˜ ∇ x ˜ η | 2 ˜ n, where ˜ n is the rescaled unit normal. Altogether, the nonlinear Bernoulli term b ecomes B x 3 ϕ ` ∇ ϕ ¨ a 1 ` | ∇ x η | 2 n η ˘ “ H g ε B ˜ x 3 ˜ ϕ ` ˜ ∇ ˜ ϕ ¨ b 1 ` ε 2 | ˜ ∇ x ˜ η | 2 ˜ n ˘ , and the quadratic velocity contribution is 1 2 | ∇ ϕ | 2 “ H g 2 | ˜ ∇ ˜ ϕ | 2 . Dividing the entire dynamic b oundary condition by the factor H g , we arriv e at the nondimensional ev olution equation B ˜ t ˜ ψ “ ´ Υ B 2 ˜ t ˜ η ´ β ε E p ˜ η ; ε q ` δ ˜ ∆ x B ˜ t ˜ η ` ε B ˜ x 3 ˜ ϕ ` ˜ ∇ ˜ ϕ ¨ b 1 ` ε 2 | ˜ ∇ x ˜ η | 2 ˜ n ˘ ´ ε 2 | ˜ ∇ ˜ ϕ | 2 ´ ˜ η, on Γ p t q 48 D. ALONSO-OR ´ AN, R. GRANERO-BELINCH ´ ON, AND J. S. ZIEBELL where we hav e introduced the dimensionless parameters Υ : “ ρ s h ρ f L , β : “ B ρ f g L 3 , δ : “ γ ρ f a g L 3 , ε : “ H L . Remark B.2 (Physical meaning of the nondimensional parameters) . The quantit y ε is the usual steepness parameter, measuring the ratio b etw een vertical amplitude and horizon tal wa v elength. The parameter Υ represen ts an eﬀectiv e mass ratio b etw een the plate and a column of ﬂuid of depth L ; it con trols the relative strength of plate inertia in the coupling. The parameter β is a bending Bond n umber, comparing the c haracteristic b ending forces to gravitational restoring forces at lengthscale L , with an additional factor ε in front of E reﬂecting the amplitude of the deformation. Finally , δ is a dimensionless damping parameter, enco ding the relativ e imp ortance of Kelvin–V oigt dissipation compared to the natural gra vit y time scale a L { g . A cknowledgements D.A.-O. has b een partially s upported b y grant R YC2023-045563-I (MICIU/AEI/10.13039/501100011033 and ESF+). D.A.-O. and R.G.-B. are supp orted by the pro ject “An´ alisis Matem´ atico Aplicado y Ecua- ciones Diferenciales” Grant PID2022-141187NB-I00 funded b y MCIN/AEI/10.13039/501100011033/FEDER, UE. J. S. Zieb ell ackno wledges that part of this work w as carried out during a research visit to the Univ ersity of Cantabria. References [1] T. Alazard, I. Kuka vica, and A. T uﬀaha, Global-in-time we ak solutions for an inviscid fr ee surfac e ﬂuid–structur e pr oblem without damping , Ann. PDE 11 (2025), article no. 15. [2] T. Alazard, C. Shao, and H. Y ang, Glob al wel l-p ose dness of a 2D ﬂuid-structur e inter action pr oblem with fr e e surfac e , arXiv:2504.00213 (2025). [3] S. Alb en and M. J. Shelley , Flapping states of a ﬂag in an inviscid ﬂuid: bistability and the tr ansition to chaos , Phys. Rev. Lett. 100 (2008), no. 7, 074301. [4] D. Alonso-Or´ an, Asymptotic shal low mo dels arising in magnetohydr o dynamics , W ater W av es 3 (2021), 371–398. [5] D. Alonso-Or´ an, A. Dur´ an, and R. Granero-Belinc h´ on, Derivation and wel l-p osedness for asymptotic mo dels of c old plasmas , Nonlinear Anal. 244 (2024). [6] D. Alonso-Or´ an and R. Granero-Belinch´ on, A ﬂuid-solid inter action prob lem in p orous media. , (2026). [7] D. P . Am brose and S. Liu, Wel l-p ose dness of two-dimensional hydr o elastic vortex shee t evolution , J. Diﬀerential Equations 260 (2016), 1910–1954. [8] M. S. Aydın, I. Kuk a vica, and A. T uﬀaha, A fr e e b oundary inviscid mo del of ﬂow–structur e inter action , arXiv:2501.12515 (2025). [9] P . Baldi and J. F. T oland, Bifur c ation and se c ondary bifur c ation of he avy p erio dic hydr o elastic tr avel ling waves , Interfaces F ree Bound. 12 (2010), no. 1, 1–22. [10] A. Balakrishna, A. Kuk avica, A. Muha, and M. T uﬀaha, Inviscid ﬂuid inter acting with a nonlinear two-dimensional plate , Interfaces F ree Bound., to app ear (2024). [11] M. Buk al and B.Muha Rigor ous derivation of a line ar sixth-or der thin-ﬁlm e quation as a re duc e d mo del for thin ﬂuid – thin structur e inter action pr oblems , Applied Mathematics & Optimization V ol. 84 , 2245-2288 (2021). WEAKL Y NONLINEAR MODELS F OR HYDR OELASTIC W A TER W A VES 49 [12] M. Bukal and B.Muha Justiﬁcation of a nonlinear sixth-or der thin-ﬁlm e quation as the r e duc e d mo del for a ﬂuid – structure inter action pr oblem , Nonlinearity V ol. 35 8 , 4695–4726, (2022). [13] P . G. Ciarlet, An Intr o duction to Diﬀer ential Geometry with Applic ations to Elasticity , Springer, 2005. [14] A. Cheng, R. Granero-Belinch´ on, S. Shk oller, and J. Wilk ening, Rigor ous asymptotic mo dels of water waves , W ater W aves 1 (2019), 71–130. [15] D. Coutand and S. Shkoller, Motion of an elastic solid inside an inc ompressible visc ous ﬂuid , Arch. Ration. Mech. Anal. 176 (2005), 25–102. [16] R. Granero-Belinch´ on and S. Scrobogna, Asymptotic mo dels for fre e b oundary ﬂow in p or ous media , Physica D, 392 (2019), 1–16. [17] R. Granero-Belinch´ on and S. Scrobogna, Mo dels for damp e d water waves , SIAM J. Appl. Math. 79 (2019), no. 6, 2530–2550. [18] R. Granero-Belinch´ on and S. Shkoller, A mo del for R ayleigh–T aylor mixing and interfac e turn-over , Multiscale Model. Simul. 15 (2017), no. 1, 274–308. [19] A. Kuka vica and M. T uﬀaha, Wel l-p ose dness for a class of nonlinear ﬂuid–structur e inter action problems , J. Diﬀerential Equations 247 (2009), 169–200. [20] I. Kuk avica and A. T uﬀaha, A fr ee b oundary inviscid mo del of ﬂow–structure inter action , J. Diﬀerential Equations 413 (2024), 851–912. [21] I. Kuk a vica and A. T uﬀaha, An inviscid fre e b oundary ﬂuid-wave mo del , J. Evol. Equ. 23 (2023), 41. [22] I. Kuk avica, A. T uﬀaha, and M. Ziane, Str ong solutions to a nonline ar ﬂuid–structur e interaction system , J. Diﬀerential Equations 247 (2009), no. 5, 1452–1478. [23] I. Kuk avica, A. T uﬀaha, and M. Ziane, Str ong solutions for a ﬂuid–structur e inter action system , Adv. Diﬀerential Equations 15 (2010), no. 3–4, 231–254. [24] V. M´ ac ha, B. Muha, ˇ S. Neˇ casov´ a, A. Ro y , and S. T rifunovi ´ c, Existenc e of a we ak solution to a nonline ar ﬂuid– structur e inter action pr oblem with he at exchange , Commun. Partial Diﬀerential Equations 47 (2022), no. 8, 1591– 1635. [25] A. J. Ma jda and A. L. Bertozzi, V orticity and Inc ompr essible Flow , Cam bridge T exts in Applied Mathematics, Cambridge University Press, Cambridge, 2001. [26] B. Muha and S. Cani´ c, Existenc e of a weak solution to a nonlinear ﬂuid–structur e inter action pr oblem mo deling the ﬂow of an inc ompr essible, visc ous ﬂuid in a cylinder with deformable wal ls , Arch. Ration. Mec h. Anal. 207 (2013), 919–968. [27] P . I. Plotniko v and J. F. T oland, Mo del ling nonline ar hydr o elastic waves , Phil. T rans. R. So c. A 369 (2011), 2942–2956. [28] V. A. Squire, J. P . Dugan, P . W adhams, P . J. Rottier, and A. K. Liu, Of oc e an waves and se a ic e , Ann u. Rev. Fluid Mech. 27 (1995), no. 1, 115–168. [29] J. F. T oland, Ste ady p erio dic hydr o elastic waves , Arch. Ration. Mech. Anal. 189 (2008), no. 2, 325–362. [30] J. F. T oland, He avy hydro elastic tr avel ling waves , Proc. R. So c. Lond. Ser. A Math. Phys. Eng. Sci. 463 (2007), no. 2085, 2371–2397. [31] L. W an and J. Y ang, L ow r e gularity wel l-p ose dness for two-dimensional hydr o elastic waves , (2025).

Weakly nonlinear models for hydroelastic water waves

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