Continuous Data Assimilation for Semilinear Parabolic Equations: A General Approach by Evolution Equations

CONTINUOUS D A T A ASSIMILA TION F OR SEMILINEAR P ARABOLIC EQUA TIONS: A GENERAL APPR O A CH BY EV OLUTION EQUA TIONS GIANMARCO DEL SAR TO, MA TTHIAS HIEBER, FILIPPO P ALMA, AND T AREK ZÖCHLING Abstract. This article develops a general framew ork for con tinuous deterministic data assimilation for semilinear parab olic equations by means of evolution equations. Introducing a nudged mo del driven by partial observ ations, the global w ell-posedness of the reference and the approximating systems is established under natural assumptions. In addition, it is shown that the appro ximating solution conv erges exp onen tially to the solution of the reference system, pro vided the observ ational resolution and the nudging parameter are suitably c hosen. The approach allows us to consider many systems, such as the Allen-Cahn, Cahn-Hilliard, Sellers-type energy balance, and bidomain systems, for the first time. 1. Intr oduction The approach to semilinear parab olic equations via the theory of evolution equations has a long and rich history and tradition. In many cases, one obtains via this method w ell-p osedness results for these equations, lo cally or globally , within the strong, weak, mild or v ariational setting, see e. g. [ 5 , 20 , 24 , 25 , 32 , 37 , 42 ]. In all of these approac hes, a precise knowledge of the initial data and its regularity is a fundamen tal ingredien t in the analysis of these problems. W e also refer to the setting of critical spaces for parab olic evolution equations, see e. g. [ 7 , 38 , 39 ]. In many practical situations, how ever, the initial state of a system is only partially known or en tirely una v ailable. In these cases, one aims to reconstruct or approximate the solution from partial observ ational data. This leads to the framew ork of data assimilation , whic h has b ecome an important research area in recen t y ears, b oth within more theoretical and more applied communities [ 26 , 27 , 40 ]. One often distinguishes b et ween discrete or contin uous-in-time data assimilation, with or without noise. W e refer here to the monographs [ 16 , 31 ], and to the articles [ 8 , 18 , 21 , 30 , 44 ]. A seminal contribution to contin uous data assimilation was made b y Azouani, Olson, and Titi [ 6 ], who prop osed an algorithm for deterministic data assimilation for the tw o-dimensional Navier-Stok es equations. They introduced a n udged system which, lik e the reference system, admits a unique, global, strong solution, and pro v ed that the n udged solution conv erges exp onen tially in time to the reference solution in b oth the L 2 - and the H 1 -norms. Man y other mo dels arising in mathematical physics ha v e also b een analysed within this framework, b oth in the context of strong and weak solutions, see e.g. [ 3 , 9 , 17 , 43 ]. Most existing analyses rely on the F aedo- Galerkin method to establish the existence of solutions to the nudged system, follo wed b y con vergence proofs based on a priori estimates and Gron wall-t yp e inequalities. This pap er aims to dev elop a general framework for contin uous deterministic data assimilation for semi- linear parab olic problems using the theory of ev olution equations. Our approach encompasses b oth an existence theory for the nudged system and a conv ergence result in a prescrib ed norm. Let us b egin by considering a semilinear evolution equation of the form (ASE) ( u ′ + Au = F ( u ) , u (0) = u 0 , where A is the generator of an analytic semigroup on a Banach space X and F is a giv en nonlinear mapping, ho wev er, the initial data is not known. In this scenario, uniquely determining the tra jectory of the system is imp ossible. The ob jective is now to construct a solution to an asso ciated problem p erturbed b y partial observ ations (the nudged system), with known initial data, and to show that the solution of this asso ciated equation 1 2 GIANMARCO DEL SAR TO, MA TTHIAS HIEBER, FILIPPO P ALMA, AND T AREK ZÖCHLING con verges exponentially , for large times, to the solution of the original problem. More precisely , let I δ ( u ( t )) b e av ailable measurements of the state. W e then study the corresp onding nudged system (1.1) ( v ′ + Av = F ( v ) − µ (I δ v − I δ ˜ u ) , v (0) = v 0 , where ˜ u denotes a suitably shifted solution of ( ASE ), as clarified in detail in the next section. T o establish long-term predictability of u , w e first verify that ( ASE ), p ossibly up to a p ositiv e time shift, is globally well-posed for an appropriate class of initial data. W e then form ulate general assumptions on the op erator A and the nonlinearity F ensuring global solv ability of ( ASE ) and pro ve that, for suitable choices of the parameters δ and µ , the n udged system is globally w ell-p osed and its solutions conv erge exponentially in time to those of ( ASE ), or of a p ositiv ely shifted v ariant. Note that w e do not ass ume the existence of a unique, global, strong solution to ( ASE ); rather, we sho w that our assumptions (A1), (A2) and (A3) b elo w are sufficien t to guarantee such a solution. W e will illustrate our abstract approach with several applications from fluid mec hanics and biomedical mo delling. In the con text of strong solutions, we consider the tw o-dimensional Na vier-Stokes equations, the three-dimensional primitive equations, a Sellers-t yp e energy balance model [ 34 , 41 ], and the tw o-dimensional bidomain mo del [ 12 , 28 ]. F or weak solutions, we treat the t wo-dimensional Na vier-Stokes equations, the one- dimensional Allen-Cahn equation [ 4 ], and the one- and tw o-dimensional Cahn-Hilliard equations [ 11 ]. W e note that our approach allows us to consider the contin uous data assimilation problem in the strong setting for b oth the energy balance model ( Subsection 3.3 ) and the t wo-dimensional bidomain mo del ( Subsec- tion 3.4 ) for the first time. In the w eak setting, we obtain by our metho d new results for the one-dimensional Allen-Cahn equation and for the one- and tw o-dimensional Cahn-Hilliard equations. The pap er is organised as follows. In Section 2 , w e introduce the general framework and develop an existence theory for the initial b oundary v alue problem asso ciated with ( ASE ). W e then establish general assumptions ensuring global-in-time solv ability of ( ASE ), up to a p ositiv e shift, and prov e our main results concerning the solv ability and conv ergence of the data assimilation system. In Section 3 , w e apply the theory in the strong setting to mo dels motiv ated b y climate science and biomedical applications. Finally , in Section 4 , we present some examples in the weak setting. 2. Preliminaries and Main resul ts Let ( V , H , V ∗ ) b e a Gelfand triple of real Hilb ert spaces, meaning the em b eddings V  → H  → V ∗ are dense and contin uous, and the pairing b et w een V and V ∗ satisfies ⟨ u, v ⟩ V , V ∗ = ( u, v ) H , ∀ u ∈ V , v ∈ H . W e assume that for the real interpolation space it holds ( V ∗ , V ) 1 2 , 2 = H , where ( · , · ) θ,p denotes the real in terp olation functor for θ ∈ (0 , 1) and p ∈ (1 , ∞ ) . W e further denote the complex interpolation space [ V ∗ , V ] β b y V β for β ∈ (0 , 1) . Moreo ver, for m ∈ N and q ∈ (1 , ∞ ) , we denote by L q (Ω) and H m,q (Ω) = W m,q (Ω) respectively the Lebesgue and Sob olev spaces and we refer to their norm as ∥ · ∥ q and ∥ · ∥ m,q . F or more information on function spaces we refer e.g. to [ 1 , 33 ]. Giv en a b ounded operator A : V → V ∗ and u 0 ∈ H , consider the follo wing semi-linear parab olic evolution equation (2.1) ( u ′ + Au = F ( u ) , t ∈ (0 , T ) , u (0) = u 0 . W e imp ose the following conditions: (A1): A is quasi-co erciv e, i. e., for all u ∈ V it holds that ⟨ Au, u ⟩ V ∗ , V ≥ α ∥ u ∥ 2 V − ω ∥ u ∥ 2 H for some α > 0 and ω ≥ 0 . CONTINUOUS DA T A ASSIMILA TION FOR SEMILINEAR EQUA TIONS: A GENERAL APPRO ACH BY EVOLUTION EQUA TIONS 3 (A2): F : V β → V ∗ satisfies for all u 1 , u 2 ∈ V β the estimate ∥ F ( u 1 ) − F ( u 2 ) ∥ V ∗ ≤ C k X j =1  1 + ∥ u 1 ∥ ρ j V β + ∥ u 2 ∥ ρ j V β  ∥ u 1 − u 2 ∥ V β j for a constant C > 0 and num bers k ∈ N , ρ j ≥ 0 , β ∈ ( 1 2 , 1) and β j ∈ ( 1 2 , β ] . (A3): F or all j = 1 , . . . , k assume ρ j  β j − 1 2  + β ≤ 1 . Remark 2.1. Man y fluid dynamics mo dels, suc h as the Navier-Stok es equations, are giv en by the abstract form ulation ( u ′ + Au = Φ( u, u ) , t ∈ (0 , T ) , u (0) = u 0 , where Φ : V β × V β → V ∗ is bilinear and b ounded, thus ∥ Φ( u, u ) ∥ V ∗ ≤ C ∥ u ∥ 2 V β . If β ≤ 3 / 4 , suc h bilinear function Φ naturally satisfies ( A2 ) and ( A3 ) with j = 1 , ρ = ρ 1 = 1 . Remark 2.2. Note that, from ( A1 ) , it follows that − A generates an analytic semigroup on V ∗ , which b y general theory admits maximal L 2 -regularit y . Based on the ab o ve assumptions, we recall the follo wing result on the lo cal well-posedness of the abstract ev olution equation ( 2.1 ), see [ 25 , Theorem 18.2.6]. Lemma 2.3. L et T > 0 and assume that ( A1 ) − ( A3 ) ar e satisfie d. Then for any u 0 ∈ H , ther e exists a = a ( u 0 ) ≤ T such that pr oblem ( 2.1 ) admits a unique solution (2.2) u ∈ L 2 (0 , a ; V ) ∩ H 1 (0 , a ; V ∗ ) ∩ BUC([0 , a ]; H ) . The solution exists on a maximal time interval [0 , a max ( u 0 )) and dep ends c ontinuously on the data. If the solution do es not exist glob al ly in time, i. e., if a max < T , the maximal existenc e time is char acterize d by a blow-up (2.3) lim t → a max ∥ u ∥ L 2 (0 ,t ; V ) ∩ H 1 (0 ,t ; V ∗ ) = ∞ . No w let T > 0 , u 0 ∈ H be giv en and suppose that ( A1 ) − ( A3 ) hold. W e denote by u the corresp onding maximal solution provided in Lemma 2.3 . T o guaran tee global-in-time existence, w e imp ose the follo wing crucial assumption. (A4): Let t + > 0 and u ∈ L 2 (0 , t ; V ) ∩ H 1 (0 , t ; V ∗ ) for all t < t + . Assume that lim t → t + ∥ F ( u ) ∥ L 2 (0 ,t ; V ∗ ) < ∞ and ∥ u ( t ) ∥ 2 H is in tegrable on (0 , ∞ ) . Denoting b y ω ≥ 0 the constan t sp ecified in ( A1 ) , the latter assumption guaran tees that the shifted problem (2.4) ( ˜ u ′ + ( A + ω ) ˜ u = F ( ˜ u ) , t ∈ (0 , T ) , ˜ u (0) = u 0 . admits a unique, global strong solution. Corollary 2.4 (Global well-posedness of ( 2.4 )) . Assume that ( A1 ) − ( A3 ) hold. L et u 0 ∈ H and assume that the solution ˜ u of ( 2.4 ) satisfies ( A4 ) . Then ther e is a unique, glob al solution ˜ u of ( 2.4 ) satisfying ˜ u ∈ L 2 (0 , ∞ ; V ) ∩ H 1 (0 , ∞ ; V ∗ ) . Pr o of. By Lemma 2.3 there is a unique maximal solution ˜ u of ( 2.4 ). In particular, Lemma 2.3 and ( A4 ) guaran tee lim t → a max ∥ ˜ u ∥ L 2 (0 ,t ; V ) ∩ H 1 (0 ,t ; V ∗ ) ≤ C lim t → a max  ∥ F ( ˜ u ) ∥ L 2 (0 ,t ; V ∗ ) + ∥ u 0 ∥ H  < ∞ and therefore ˜ u ∈ L 2 loc (0 , ∞ ; V ) ∩ H 1 loc (0 , ∞ ; V ∗ ) ∩ BUC((0 , ∞ ); H ) . W e note that the constant C > 0 is indep enden t of time since the operator A + ω I is inv ertible, thanks to ( A1 ) . T o extend ˜ u to T = ∞ we note that the spectrum of the shifted op erator A + ω lies en tirely in the righ t half of the complex plane, 4 GIANMARCO DEL SAR TO, MA TTHIAS HIEBER, FILIPPO P ALMA, AND T AREK ZÖCHLING that is, σ ( A + ω ) ⊂ C + . Specifically , since the semigroup generated b y − A − ω is analytic, it implies the exp onen tial deca y of the semigroup. This in turn implies that, given ∥ ˜ u ( t 0 ) ∥ H sufficien tly small for some t 0 > 0 , then the solution extends to T = ∞ . By ( A4 ) and the ab o ve considerations, ∥ ˜ u ∥ 2 H is integrable and con tinuous on (0 , ∞ ) and it follows that inf t ∈ (0 , ∞ ) ∥ ˜ u ( t ) ∥ 2 H = 0 . Hence, there is t 0 ≥ 0 such that ∥ ˜ u ( t 0 ) ∥ 2 H is sufficien tly small, which concludes the pro of. □ W e are now in a position to introduce an approximate system corresp onding to this solution ˜ u , the so-called Data Assimilation system . Let ˜ u denote the unique, global solu tion of ( 2.4 ). Consider the system ( D A ) ( v ′ + Av = F ( v ) − µ (I δ v − I δ ˜ u ) , t ∈ (0 , T ) , v (0) = v 0 , where µ > 0 denotes a constan t referred to as nudging p ar ameter and I δ denotes a linear op erator satisfying (2.5) ⟨ f − I δ f , g ⟩ V ∗ , V ≤ C δ ∥ f ∥ H ∥ g ∥ V for all f ∈ H and g ∈ V . The op erator I δ mo dels observ ational measuremen ts of the system ( 2.1 ) at a coarse spatial resolution with a scale δ . F rom the b ound ( 2.5 ) on the observ ation op erator I δ w e conclude the following result: Prop osition 2.5 (Global well-posedness of the Data Assimilation problem ( DA )) . Supp ose that the observation op er ator I δ satisfies ( 2.5 ) and v 0 , u 0 ∈ H . Mor e over, assume that ( A1 ) − ( A3 ) and ( A4 ) hold and denote by ˜ u ∈ L 2 (0 , ∞ ; V ) ∩ H 1 (0 , ∞ ; V ∗ ) the c orr esp onding glob al solution of e quation ( 2.4 ) . Then ther e exist µ 0 , δ 0 > 0 such that for al l µ > µ 0 and 0 < δ < δ 0 the pr oblem ( D A ) admits a unique, glob al solution v ∈ L 2 (0 , ∞ ; V ) ∩ H 1 (0 , ∞ ; V ∗ ) . Pr o of. F or an appropriate c hoice of µ , δ > 0 , the op erator A + µ I δ is readily seen to satisfy ( A1 ) . Sp ecifically , since for the op erator A assumption ( A1 ) holds, inv oking the bound ( 2.5 ) and Y oung’s inequalit y , for v ∈ V w e compute ⟨ ( A + µ I δ ) v , v ⟩ V ∗ , V = ⟨ Av , v ⟩ V ∗ , V + µ ⟨ I δ v − v , v ⟩ V ∗ , V + µ ⟨ v , v ⟩ V ∗ , V ≥ α ∥ v ∥ 2 V − C µδ ∥ v ∥ H ∥ v ∥ V + ( µ − ω ) ∥ v ∥ 2 H ≥  α − ε  ∥ v ∥ 2 V +  µ − µ 2 δ 2 4 ε − ω  ∥ v ∥ 2 H . Cho osing 0 < ε < α and δ > 0 appropriately small as w ell as µ > 0 sufficiently large guarantees that ⟨ ( A + µ I δ ) v , v ⟩ V ∗ , V ≥ α ′ ∥ v ∥ 2 V + β ∥ v ∥ 2 H for some α ′ > 0 and β ≥ 0 . Therefore, assumption ( A1 ) is satisfied by A + µI δ with ω = 0 . Since the non-linear term F satisfies ( A1 ) − ( A3 ) we obtain a unique lo cal maximal solution of the data assimilated problem ( D A ). Next, we verify ( A4 ) . T aking inner products of ( DA ) with v and adding µ ⟨ ˜ u − ˜ u, v ⟩ V ∗ , V on the right-hand side, yields in view of the ab o v e prop erty of A and ( 2.5 ) (2.6) ∂ t 1 2 ∥ v ∥ 2 H + α ′ ∥ v ∥ 2 V ≤ |⟨ F ( v ) , v ⟩ V ∗ , V | + µ |⟨ ˜ u − I δ ˜ u, v ⟩ V ∗ , V | + µ |⟨ ˜ u, v ⟩ V ∗ , V | ≤ C  ∥ F ( v ) ∥ 2 V ∗ + ∥ ˜ u ∥ 2 H + ∥ ˜ u ∥ 2 V ∗  + ε ∥ v ∥ 2 V . In tegrating in time, using Gronw all’s inequality implies in view of ( A4 ) and Corollary 2.4 lim t → a max ∥ v ∥ L 2 (0 ,t ; V ) ∩ L ∞ (0 ,t ; H ) < ∞ and therefore v ∈ L 2 loc (0 , ∞ ; V ) ∩ H 1 loc (0 , ∞ ; V ∗ ) ∩ BUC((0 , ∞ ); H ) . By similar arguments to those in the pro of of Corollary 2.4 the solution can b e extended to T = ∞ . □ Remark 2.6. It remains to verify that ( A4 ) holds for the data-assimilated solution v in all examples b elo w. Observe that the data-assimilation problem ( DA ) can b e rewritten as the forced problem ( 2.4 ) with the same nonlinearity and with A ′ : = A + µ I δ . As in the pro of of Prop osition 2.5 , one chec ks that A ′ satisfies ( A1 ), while the additional forcing term I δ ˜ u is con trolled by the interpolant b ounds in ( 2.5 ). Consequently , verifying ( A4 ) for the data-assimilation CONTINUOUS DA T A ASSIMILA TION FOR SEMILINEAR EQUA TIONS: A GENERAL APPRO ACH BY EVOLUTION EQUA TIONS 5 setting is straightforw ard in practice, and w e shall omit this chec k in the examples presented in Section 3 and Section 4 . Ha ving established global in time existence of a unique solution ˜ u to problem ( 2.1 ), as well as the global in time existence of a unique solution v to the approximate problem ( DA ), it is reasonable to consider the difference w := ˜ u − v , which corresp onds to the evolution equation (2.7) ( w ′ + Aw = F ( ˜ u ) − F ( v ) − µ I δ w − ω ˜ u, t ∈ (0 , ∞ ) , w (0) = w 0 . W e are now in a p osition to state our main theorems concerning the long time b eha viour of the difference w , measured in the H -and V -norm. Theorem 2.7 (Con vergence in the H -norm) . L et u 0 , v 0 ∈ H , supp ose that ( A1 ) − ( A4 ) ar e satisfie d and the observation op er ator I δ satisfies ( 2.5 ) . Then, ther e exist µ 0 , δ 0 > 0 such that for al l µ > µ 0 and 0 < δ < δ 0 the unique, glob al solution of ( 2.7 ) satisfies (2.8) ∥ w ( t ) ∥ H → 0 exp onential ly, as t → ∞ . In p articular, if ω = 0 in ( A1 ) , then the solution of the data assimilation system ( DA ) c onver ges exp onen- tial ly fast to the solution u of the original pr oblem ( 2.1 ) . W e w an t to stress that the con vergence established in Theorem 2.7 is strong. Sp ecifically , embedding H  → V ∗ naturally leads to the following weak er conv ergence result. Corollary 2.8 (Conv ergence in the V ∗ -norm) . Under the same assumptions of The or em 2.7 , the unique, glob al solution of ( 2.7 ) satisfies (2.9) ∥ w ( t ) ∥ V ∗ → 0 exp onential ly, as t → ∞ . Pr o of of The or em 2.7 . T aking inner pro ducts of ( 2.7 ) with w , yields ⟨ ∂ t w , w ⟩ V ∗ , V + ⟨ Aw , w ⟩ V ∗ , V = ⟨ F ( ˜ u ) − F ( v ) , w ⟩ V ∗ , V − µ ⟨ I δ w , w ⟩ V ∗ , V − ω ⟨ ˜ u, w ⟩ V ∗ , V . Using the co ercivit y of A in ( A1 ) , the assumptions on the pairing betw een V ∗ and V and adding the term µ ⟨ w − w , w ⟩ V ∗ , V on the right-hand side, yields 1 2 ∂ t ∥ w ∥ 2 H + α ∥ w ∥ 2 V ≤ ⟨ F ( u ) − F ( v ) , w ⟩ V ∗ , V + µ ⟨ w − I δ w , w ⟩ V ∗ , V − µ ∥ w ∥ 2 H − ω ⟨ ˜ u, w ⟩ V ∗ , V . First, w e estimate using Y oung’s inequality to obtain | − ω ⟨ ˜ u, w ⟩ V ∗ , V | ≤ ω ∥ ˜ u ∥ V ∗ · ∥ w ∥ V ≤ C ω ∥ ˜ u ∥ 2 V ∗ + ε ∥ w ∥ 2 V . Next, b y ( A3 ) , embedding V β  → V β j and Y oung’s inequality we estimate |⟨ F ( ˜ u ) − F ( v ) , w ⟩ V ∗ , V | ≤ ∥ F ( ˜ u ) − F ( v ) ∥ V ∗ · ∥ w ∥ V ≤ C k X j =1  1 + ∥ ˜ u ∥ ρ j V β + ∥ v ∥ ρ j V β  ∥ w ∥ V β j · ∥ w ∥ V ≤ C ( ε ) k X j =1  1 + ∥ ˜ u ∥ 2 ρ j V β + ∥ v ∥ 2 ρ j V β  ∥ w ∥ 2 V β + ε ∥ w ∥ 2 V . Using in terp olation and Y oung’s inequalities, we obtain further C ( ε ) k X j =1  1 + ∥ ˜ u ∥ 2 ρ j V β + ∥ v ∥ 2 ρ j V β  ∥ w ∥ 2 V β + ε ∥ w ∥ 2 V ≤ C ( ε ) k X j =1  1 + ∥ ˜ u ∥ 2 ρ j V β + ∥ v ∥ 2 ρ j V β  ∥ w ∥ 4 − 4 β H ∥ w ∥ 4 β − 2 V + ε ∥ w ∥ 2 V ≤ C ( ε ) k X j =1  1 + ∥ ˜ u ∥ ρ j 1 − β V β + ∥ v ∥ ρ j 1 − β V β  ∥ w ∥ 2 H + ε ∥ w ∥ 2 V . Using the b ound ( 2.5 ), the remaining term inv olving the in terp olation op erator can b e estimated b y | µ ⟨ w − I δ w , w ⟩ V ∗ , V | ≤ C µδ ∥ w ∥ H ∥ w ∥ V ≤ C ( ε ) µ 2 δ 2 ∥ w ∥ 2 H + ε ∥ w ∥ 2 V . 6 GIANMARCO DEL SAR TO, MA TTHIAS HIEBER, FILIPPO P ALMA, AND T AREK ZÖCHLING Absorbing the highest order norms, we conclude the inequality 1 2 ∂ t ∥ w ∥ 2 H + α ∥ w ∥ 2 V ≤ C  k X j =1  1 + ∥ ˜ u ∥ ρ j 1 − β V β + ∥ v ∥ ρ j 1 − β V β  + µ 2 δ 2 − µ  ∥ w ∥ 2 H + C ω ∥ ˜ u ∥ 2 V ∗ for a suitable constan t C > 0 . Note that ˜ u and v are the given global solutions of ( 2.4 ) and ( DA ) resp ectiv ely . In particular, by ( A3 ) we ha ve L 2 (0 , ∞ ; V ) ∩ H 1 (0 , ∞ ; V ∗ )  → H 1 − β (0 , ∞ ; V β )  → L ρ j 1 − β (0 , ∞ ; V β ) and therefore there exist time-indep enden t constants C 1 , C 2 > 0 such that Z t 0 k X j =1  1 + ∥ ˜ u ∥ ρ j 1 − β V β + ∥ v ∥ ρ j 1 − β V β  d s ≤ C 1 t + C 2 . Hence, in tegrating in time, and using Gronw all’s inequality yields 1 2 ∥ w ( t ) ∥ 2 H + α Z t 0 ∥ w ( s ) ∥ 2 V d s ≤ e C 2  1 2 ∥ w 0 ∥ 2 H + C ω ∥ ˜ u ∥ 2 L 2 t V ∗  · e C 1 (1+ µ 2 δ 2 − µ ) t for all t > 0 . The desired exp onen tial conv ergence follows from c ho osing µ , δ > 0 such that 1 + µ 2 δ 2 − µ < 0 and using ∥ ˜ u ∥ 2 L 2 t V ∗ ≤ C for a constant C > 0 indep enden t of time. □ 3. Illustra tions of the General Framew ork - strong solutions As previously noted, the general framework we develop incorp orates a range of mo dels from mathematical ph ysics, whic h will b e discussed in detail b elo w. W e begin b y presenting the general structure gov erning the con vergence of strong solutions of the appro ximate system ( D A ) to those of the original system ( 2.1 ), or the shifted system ( 2.4 ), dep ending on the v alue of ω from ( A1 ) . Let Ω ⊂ R n b e an op en set. Set V ∗ = L 2 (Ω) , V = D( A ) and H = (L 2 (Ω) , D( A )) 1 2 , 2 . Define the pairing b et w een L 2 (Ω) and D( A ) by (3.1) ⟨ u, v ⟩ L 2 (Ω) , D( A ) : = ( u, Av ) 2 + ( u, v ) 2 for all u ∈ L 2 (Ω) and v ∈ D( A ) . Here ( · , · ) 2 denotes the standard L 2 -inner pro duct. Concerning the in terp olation op erator I δ , we make the standard assumption (3.2) ∥ f − I δ f ∥ 2 ≤ C δ ∥ f ∥ H for all f ∈ H , see also [ 6 ]. This assumption will b e used throughout all examples in the strong setting. A straightforw ard calculation sho ws that this implies the b ound ( 2.5 ) on the measurement op erator I δ . Note that in contrast to [ 6 , 35 ] we do not require any additional assumptions on the observ ations. W e are now in a position to discuss several examples drawn from the existing literature, as well as introduce new ones that, to the b est of our knowledge, ha v e not b een previously treated, thereby presenting the first data assimilation results for these cases. 3.1. 2D-Na vier-Stok es equations. [ 6 ] Let Ω ⊂ R 2 b e a b ounded domain with smooth b oundary . The 2D-Navier-Stok es equations with no-slip b oundary conditions are given by the following set of equations ( 2D-NSE )          ∂ t u + ( u · ∇ ) u − ∆ u + ∇ p = 0 , in (0 , T ) × Ω , div u = 0 , in (0 , T ) × Ω , u = 0 , in (0 , T ) × ∂ Ω , u (0) = u 0 , where u : Ω → R 2 denotes the velocity field and p : Ω → R the pressure. T o rewrite ( 2D-NSE ) in the form of an abstract evolution equation ( 2.1 ), we introduce the Stokes op erator A realized in the sp ace of w eakly div ergence-free vector fields L 2 σ (Ω) b y Au : = − P ∆ u, D( A ) : = H 2 (Ω; R 2 ) ∩ H 1 0 (Ω; R 2 ) ∩ L 2 σ (Ω; R 2 ) . CONTINUOUS DA T A ASSIMILA TION FOR SEMILINEAR EQUA TIONS: A GENERAL APPRO ACH BY EVOLUTION EQUA TIONS 7 Here, P denotes the tw o-dimensional Helmholtz pro jector from L 2 (Ω; R 2 ) onto the solenoidal v ector fields L 2 σ (Ω; R 2 ) . Then ( 2D-NSE ) can b e rewritten as an abstract evolution equation ( 2.1 ), where A is the Stokes op erator and F ( u ) = − ( u · ∇ ) u . In the following, w e verify that ( A1 ) − ( A3 ) and ( A4 ) are satisfied. First, w e obtain in view of the orthogonality of P and Poincaré’s inequality ⟨ Au, u ⟩ L 2 (Ω) , D( A ) = ∥ Au ∥ 2 2 + ∥∇ u ∥ 2 2 ≥ α ∥ u ∥ 2 D( A ) for some α > 0 and all u ∈ D( A ) . In particular, ( A1 ) is satisfied with ω = 0 . Next, observ e that the non-linear term F is bilinear and therefore it suffices to v erify the condition in Remark 2.1 . Indeed, by Hölder’s inequalit y and Sob olev embedding, we obtain ∥ ( u · ∇ ) u ∥ 2 ≤ C ∥ u ∥ 4 · ∥∇ u ∥ 4 ≤ C ∥ u ∥ 2 H 3 / 2 (Ω) . W e conclude that ( A2 ) , ( A3 ) are satisfied with β = 3 4 and ρ = 1 . T o verify ( A4 ) , w e note that the b oundedness of the non-linear term is well kno wn by the fundamental w ork of Ladyzhensk ay a [ 29 ], for all u 0 ∈ H 1 0 (Ω; R 2 ) ∩ L 2 σ (Ω; R 2 ) . F or the square-in tegrability of the H 1 -norm of u , w e refer also to Pro di [ 36 , Lemma 3], where he prov ed the v alidity of the energy equalit y . W e denote by v the appro ximate solution of ( DA ), guaran teed b y Prop osition 2.5 . An application of Theorem 2.7 then yields the following result. Corollary 3.1 (Data Assimilation for ( 2D-NSE )) . L et u 0 , v 0 ∈ H 1 0 (Ω; R 2 ) ∩ L 2 σ (Ω; R 2 ) . Then, ther e exist µ 0 , δ 0 > 0 such that for al l µ > µ 0 and 0 < δ < δ 0 ∥ ( u − v )( t ) ∥ H 1 (Ω) → 0 exp onential ly, as t → ∞ . Remark 3.2. In this subsection, w e consider ( 2D-NSE ) with no-slip b oundary conditions. The analysis can also be extended to other types of b oundary conditions, such as pur e slip, outflow , or fr e e conditions b y suitably adapting the domain of the Stokes op erator; see, for example, [ 37 , Section 7]. It is imp ortan t to note, how ev er, that in these alternative settings, ( A1 ) may only b e v alid under the additional condition that ω > 0 . 3.2. 3D-Primitiv e Equations. [ 19 , 35 ] Let Ω = T 2 × (0 , 1) ⊂ R 3 . Here, T 2 denotes the unit square in R 2 with p eriodic b oundary conditions. The 3D-Primitiv e equations are given by the following set of equations ( 3D-PE )          ∂ t v + ( v · ∇ H ) v + w · ∂ z v − ∆ v + ∇ H p = 0 , in (0 , T ) × Ω , ∂ z p = 0 , in (0 , T ) × Ω , div u = 0 , in (0 , T ) × Ω , v (0) = v 0 . supplemen ted by the b oundary conditions (3.3) v | T 2 ×{ 1 } = 0 , ( ∂ z v ) | T 2 ×{ 0 } = 0 and w | T 2 ×{ 0 }∪ T 2 ×{ 1 } = 0 . Here u = ( v , w ) : Ω → R 3 denotes the v elo cit y field, p : Ω → R the pressure and w e use ∇ H , div H for the horizon tal gradient and the horizon tal divergence, that is ∇ H := ( ∂ x , ∂ y ) T and div H := ∇ H · . Note that the v ertical velocity w is fully determined by the divergence free condition and the b oundary conditions, that is, w = − Z z 0 div H v ( · , ξ ) d ξ . T o rewrite ( 3D-PE ) in the form of an abstract ev olution equation, we introduce the hydr ostatic Stokes op er ator A H realized in the space of hydr ostatic al ly solenoidal ve ctor fields L 2 σ (Ω) = { v ∈ C ∞ (Ω; R 2 ) : div H v = 0 } ∥·∥ L 2 (Ω) with v = Z 1 0 v ( · , ξ ) d ξ as follo ws A H v : = − P H ∆ v , D( A H ) : = { v ∈ H 2 (Ω; R 2 ) ∩ L 2 σ (Ω; R 2 ) : v | T 2 ×{ 1 } = ( ∂ z v ) | T 2 ×{ 0 } = 0 } . Here, P H denotes the hydr ostatic Helmholtz pr oje ction . F or an extensiv e discussion of the h ydrostatic Stok es op erator and related results we refer to [ 22 ]. Hence we write ( 3D-PE ) in the form of an abstract ev olution 8 GIANMARCO DEL SAR TO, MA TTHIAS HIEBER, FILIPPO P ALMA, AND T AREK ZÖCHLING equation ( 2.1 ) b y choosing A to b e the h ydrostatic Stokes op erator and setting F ( v ) = − ( v · ∇ H ) v − w · ∂ z v . By orthogonalit y of P H and P oincaré’s inequality , we verify ⟨ A H v , v ⟩ L 2 (Ω) , D( A H ) = ∥ A H v ∥ 2 2 + ∥∇ v ∥ 2 2 ≥ α ∥ v ∥ 2 D( A H ) for some α > 0 and all v ∈ D( A H ) , whic h implies that ( A1 ) is satisfied with ω = 0 . Moreov er, arguing as in [ 22 , Section 5], we see that ( A2 ) and ( A3 ) are satisfied with β = 3 4 and ρ = 1 . Finally , ( A4 ) is satisfied b y [ 22 , Section 6] for all v 0 ∈ H 1 (Ω; R 2 ) ∩ L 2 σ (Ω; R 2 ) sub ject to suitable compatibility conditions, that are sp ecified in Corollary 3.3 . Denoting by ˆ v the solution of the approximate system ( D A ), guaranteed b y Prop osition 2.5 , an application of Theorem 2.7 yields the following result. Corollary 3.3 (Data assimilation of ( 3D-PE )) . L et v 0 , ˆ v 0 ∈ H 1 (Ω; R 2 ) ∩ L 2 σ (Ω; R 2 ) satisfying the c omp atibility c onditions v | T 2 ×{ 1 } = ˆ v | T 2 ×{ 1 } = 0 . Then, ther e exist µ 0 , δ 0 > 0 such that for al l µ > µ 0 and 0 < δ < δ 0 ∥ ( v − ˆ v )( t ) ∥ H 1 (Ω) → 0 exp onential ly, as t → ∞ . 3.3. Energy Balance Mo del coupled to an activ e fluid. [ 13 , 14 ] Energy Balance Mo dels (EBMs) are conceptual climate mo dels that describe the evolution of the Earth’s temp erature based on the fundamental principle of radiativ e balance. They are instrumental for in v estigating core climate dynamics, such as bistability and ice-alb edo feedback, by capturing essen tial physics without the computational complexity of general circulation mo dels ( [ 10 , 34 , 41 ]). Let Ω = T 2 × (0 , 1) ⊂ R 3 b e as in Subsection 3.2 . W e consider the follo wing system of equations, in tro duced in [ 13 , 14 ] (EBM-PE)                                ∂ t v + ( v · ∇ H ) v + w · ∂ z v − ∆ v + ∇ H p = Z z 0 ∇ H τ ( · , ξ ) d ξ , in (0 , T ) × Ω , ∂ z p = 0 , in (0 , T ) × Ω , div u = 0 , in (0 , T ) × Ω , ∂ t τ + ( v · ∇ H ) τ + w · ∂ z τ − ∆ τ = 0 , in (0 , T ) × Ω , τ | T 2 ×{ 1 } = ρ, in (0 , T ) × T 2 , ∂ t ρ + ( v · ∇ H ) ρ − ∆ H ρ + ( ∂ z τ ) | T 2 ×{ 1 } = Q ( t, x ) β ( ρ ) − | ρ | 3 ρ, in (0 , T ) × T 2 , v (0) = v 0 , τ (0) = τ 0 , where v 0 and τ 0 are the initial conditions, supplemented by the b oundary conditions (3.4) v | T 2 ×{ 1 } = 0 , ( ∂ z v ) | T 2 ×{ 0 } = 0 , w | T 2 ×{ 0 }∪ T 2 ×{ 1 } = 0 and τ | T 2 ×{ 0 } = 0 . Here, v : Ω → R 2 denotes the v elocity field, p : Ω → R denotes the pressure, τ : Ω → R denotes the temp erature and ρ = τ | T 2 ×{ 1 } : T 2 → R denotes the temp erature ev aluated at the surface. Moreov er, Q ∈ C 1 b ( R + × T 2 ) represents the p ositiv e solar radiation and β is the Lipschitz con tin uous co-alb edo, hence resulting in a Sellers-type EBM, which is parametrised by β ( ρ ) = β 1 + ( β 2 − β 1 ) 1 + tanh( ρ − ρ ref ) 2 , with 0 < β 1 < β 2 corresp onding to the co-albedo v alues for ice-cov ered and ice-free conditions, resp ectiv ely , and ρ ref b eing the temperature at whic h ice b ecomes white. F or more details we refer to [ 13 ]. T o rewrite ( EBM-PE ) as the abstract evolution equation ( 2.1 ), we introduce the op erator matrix A realized in the space V ∗ = L 2 σ (Ω; R 2 ) × L 2 (Ω) × L 2 ( T 2 ) b y A : =   A H 0 0 0 − ∆ 0 0 γ ∂ z − ∆ H   , D( A ) = D( A H ) × { ( τ , ρ ) ∈ H 2 (Ω) × H 2 ( T 2 ) : τ | T 2 ×{ 1 } = ρ, τ | T 2 ×{ 0 } = 0 } , CONTINUOUS DA T A ASSIMILA TION FOR SEMILINEAR EQUA TIONS: A GENERAL APPRO ACH BY EVOLUTION EQUA TIONS 9 where A H denotes the hydrostatic Stokes op erator as introduced in Subsection 3.2 and γ denotes the trace op erator. Note that with this choice of b oundary conditions, the op erator A is inv ertible due to its low er- triangular structure. Next, set F  ( v , τ , ρ ) ⊤  =   − ( v · ∇ H v ) − w · ∂ z v + R z 0 ∇ H τ ( · , ξ ) d ξ − ( v · ∇ H τ ) − w · ∂ z τ − ( v · ∇ H ) ρ + Q ( t, x ) β ( ρ ) − | ρ | 3 ρ.   In view of integration by parts and the inv ertibility of A , we calculate for all x = ( v , τ , ρ ) ∈ D( A ) ⟨ A x , x ⟩ V ∗ , D( A ) = ∥ A x ∥ 2 V ∗ + ∥∇ v ∥ 2 2 + ∥∇ τ ∥ 2 2 + ∥∇ H ρ ∥ 2 2 ≥ α ∥ x ∥ 2 D( A ) for some α > 0 . Sp ecifically , ( A1 ) is v alid with ω = 0 . Moreov er, [ 13 , Lemma 5.5] implies that ( A2 ) and ( A3 ) are satisfied. Finally , ( A4 ) is v alid b y [ 13 , Theorem 4.1] for all initial data x 0 = ( v 0 , τ 0 , ρ 0 ) satisfying (3.5) v 0 as in Corollary 3.3 and τ 0 ∈ H 1 (Ω) such that ρ 0 = τ 0 | T 2 ×{ 1 } ∈ H 1 ( T 2 ) . T o v erify that the H 1 -norm of x ( t ) is square-in tegrable on (0 , ∞ ) , we first take the inner pro duct of ( EBM-PE ) 4 with τ and use Hölder’s and Y oung’s inequality to obtain 1 2 ∂ t  ∥ τ ∥ 2 2 + ∥ ρ ∥ 2 2  + ∥∇ τ ∥ 2 2 + ∥∇ H ρ ∥ 2 2 + ∥ ρ ∥ 5 5 = Z T 2 Q ( t, x ) β ( ρ ) · ρ ≤ C ∥ Q ( t, x ) ∥ 5 / 4 5 / 4 + ε ∥ ρ ∥ 5 5 . In tegrating in time, using Gronw all’s inequality and assuming Q ( t, x ) ∈ L 5 / 4 ( R + × T 2 ) yields the result ∥ τ ∥ 2 H 1 and ∥ ρ ∥ 2 H 1 are in tegrable on (0 , ∞ ) . Moreov er, taking the inner pro duct of ( EBM-PE ) 1 with v and arguing similarly implies that ∥ v ∥ 2 H 1 is integrable on (0 , ∞ ) . Denoting b y ˆ x = ( ˆ v , ˆ τ , ˆ ρ ) the solution of the appro ximate system ( DA ), Theorem 2.7 implies the following conv ergence result. Corollary 3.4 (Data assimilation for ( EBM-PE )) . L et x 0 and ˆ x 0 satisfy ( 3.5 ) . Then, ther e exist µ 0 , δ 0 > 0 such that for al l µ > µ 0 and 0 < δ < δ 0 ∥ ( x − ˆ x )( t ) ∥ H 1 (Ω) × H 1 (Ω) × H 1 ( T 2 ) → 0 exp onential ly, as t → ∞ . 3.4. 2D-Bidomain problem with FitzHugh-Nagumo transp ort. The bidomain mo del offers a contin uum framework for the electrical activity in cardiac tissue, representing it as ov erlapping intra and extracellular domains. Coupled with FitzHugh-Nagumo kinetics, whic h provide a simplified yet p ow erful caricature of action potential dynamics, the system is a cornerstone of computational cardiology for simulating cardiac w av e propagation and in vestigating the mec hanisms of arrhythmia. F or an exhaustiv e ov erview of the mo del, w e refer to [ 12 , 28 ]. W e assume that Ω ⊂ R 2 is a b ounded domain with smo oth b oundary . The system w e will fo cus on reads as (2D-BIDOMAIN)                ∂ t u − div ( a 1 ∇ u 1 ) = f ( u, w ) , in (0 , T ) × Ω , ∂ t u + div ( a 2 ∇ u 2 ) = f ( u, w ) , in (0 , T ) × Ω , ∂ t w = g ( u, w ) , in (0 , T ) × Ω , u 1 − u 2 = u, in (0 , T ) × Ω , u (0) = u 0 , w (0) = w 0 . endo wed with b oundary conditions: (3.6) a i ∇ u i · ν = 0 on ∂ Ω × (0 , T ) , i = 1 , 2 , where ν is the outer unit normal v ector of the surface ∂ Ω . Here, ( u, w ) : Ω → R 2 . Moreo ver, w e assume that a i = a i ( x ) , i = 1 , 2 , b elong to W 1 , ∞ (Ω; R 2 × 2 ) , are uniformly p ositiv e definite on Ω and there exists γ ∈ H 1 ( ∂ Ω) such that ν ( x ) · a 2 ( x ) = γ ( x ) ν ( x ) · a 1 ( x ) . In particular, we deduce that γ ( x ) ≥ γ 0 > 0 , for all x ∈ ∂ Ω . Concerning the non-linear right-hand sides f and g , we assume they are transport terms of FitzHugh-Nagumo t yp e given by f ( u, w ) := − u 3 + ( a + 1) u 2 − ( a + δ ) u − w and g ( u, w ) := − bw + cu for a ∈ (0 , 1) and b, c, δ > 0 . 10 GIANMARCO DEL SAR TO, MA TTHIAS HIEBER, FILIPPO P ALMA, AND T AREK ZÖCHLING T o rewrite system ( 2D-BIDOMAIN ) as an abstract evolution equation ( 2.1 ), we first in tro duce the bidomain op erator A , realized in L 2 0 (Ω) , the space of mean v alue free functions, by A :=  div ( a 1 ∇ ) − 1 + div ( a 2 ∇ ) − 1  − 1 , D( A ) := { u ∈ H 2 (Ω) ∩ L 2 0 (Ω) : ν · a 1 ∇ u = ν · a 2 ∇ u = 0 on ∂ Ω } , and extend its definition to L 2 (Ω) . W e then define the op erator matrix A , realized in the space V ∗ = L 2 (Ω) × L 2 (Ω) , b y A :=  ε + A 0 0 b  , D( A ) = D( A ) × L 2 (Ω) and set F ( u, w ) = ( − u 3 + ( a + 1) u 2 − w − ( a − ε + δ ) u, cu ) . F or more details, we refer to [ 23 ]. T o verify ( A1 ) , we calculate for x = ( u, v ) ∈ D( A ) ⟨ A x , x ⟩ V ∗ , D( A ) = ∥ A x ∥ 2 V ∗ + ∥ ( A + ε ) 1 / 2 u ∥ 2 2 + b ∥ w ∥ 2 2 ≥ α ∥ x ∥ 2 D( A ) for some α > 0 , where we used the relation ∥ ( A + ε ) 1 / 2 u ∥ 2 2 ≥ C ∥ u ∥ 2 H 1 in the last step. Next, as a p olynomial of order 3 , F naturally satisfies ( A2 ) and ( A3 ) with β = 1 / 3 and ρ = 2 , see also [ 23 ]. In particular, [ 23 , Theorem 4.1] guaran tees the non blow-up condition in ( A4 ) . T o v erify that the H 1 -norm of the solution is square-integrable on (0 , ∞ ) , we take inner pro ducts with u and c − 1 w resp ectiv ely and adding the resulting equations yields 1 2 ∂ t  ∥ u ∥ 2 2 + 1 c ∥ w ∥ 2 2  + ∥  A + ε  1 / 2 u ∥ 2 2 + ( a − ε + δ ) ∥ u ∥ 2 2 + b c ∥ w ∥ 2 2 + ∥ u ∥ 4 4 = ( a + 1) ∥ u ∥ 3 3 . Using the relation ∥  A + ε  1 / 2 u ∥ 2 2 ≥ C ∥ u ∥ 2 H 1 and the estimate, ( a + 1) ∥ u ∥ 3 3 ≤ ( a + 1) ∥ u ∥ 2 · ∥ u ∥ 2 4 ≤ ( a + 1) 2 4 ∥ u ∥ 2 2 + ∥ u ∥ 4 4 whic h follows from Hölder and Y oung inequalities. W e conclude that 1 2 ∂ t  ∥ u ∥ 2 2 + 1 c ∥ w ∥ 2 2  + C ∥ u ∥ 2 H 1 + b c ∥ w ∥ 2 2 +  a − ε + δ − ( a + 1) 2 4  ∥ u ∥ 2 2 ≤ 0 . W e make the following assumption: a − ε + δ − ( a + 1) 2 4 ≥ 0 . Denoting b y ˆ x = ( ˆ u, ˆ w ) the solution to the approximated system ( DA ), guaranteed by Prop osition 2.5 , an application of Theorem 2.7 implies the following conv ergence result for the solution x of the ε shifted system, for each ε > 0 . Corollary 3.5 (Data assimilation for ( 2D-BIDOMAIN )) . L et x 0 , ˆ x 0 ∈ H 1 (Ω) × L 2 (Ω) . Then, ther e exist µ 0 , δ 0 > 0 such that for al l µ > µ 0 and 0 < δ < δ 0 ∥ ( x − ˆ x )( t ) ∥ H 1 (Ω) × L 2 (Ω) → 0 exp onential ly, as t → ∞ . 4. Illustra tions of the General Framew ork - weak solutions In this final section, we present the general framework for analysing the conv ergence of w eak solutions of ( D A ) to those of ( 2.1 ). T o this end, w e revisit the system ( 2D-NSE ) and introduce new examples. Let Ω ∈ R n b e an op en set. Set H = L 2 (Ω) and assume the pairing b et ween V ∗ and V satisfies (4.1) ⟨ u, v ⟩ V ∗ , V = ⟨ u, v ⟩ H = ( u, v ) 2 for all u ∈ L 2 (Ω) and v ∈ V . Concerning the interpolation op erator I δ , w e imp ose (4.2) ∥ f − I δ f ∥ V ∗ ≤ C δ ∥ f ∥ 2 for all f ∈ L 2 (Ω) , Note that the condition for the measuremen ts I δ is relaxed compared to the strong setting. Ho wev er, in most practical examples, such as those in [ 9 , 44 ] the stronger condition ( 3.2 ) is assumed. CONTINUOUS DA T A ASSIMILA TION FOR SEMILINEAR EQUA TIONS: A GENERAL APPRO ACH BY EVOLUTION EQUA TIONS 11 4.1. 2D-Na vier-Stok es equations - revisited. Consider the v ariational formulation of the ( 2D-NSE ) on a b ounded domain Ω , whose strong form ulation w as discussed in Subsection 3.1 . F or this purpose, let ϕ ∈ H 1 0 (Ω; R 2 ) with div ϕ = 0 and define the w eak Stok es op erator by A w : D( A w ) := H 1 0 (Ω; R 2 ) ∩ L 2 σ (Ω; R 2 ) → H − 1 σ (Ω; R 2 ) , ⟨ A w u, ϕ ⟩ H − 1 σ , D( A w ) := ( ∇ u, ∇ ϕ ) 2 for all u ∈ D( A w ) . Setting ⟨ F w ( u ) , ϕ ⟩ H − 1 σ , D( A w ) := ( u ⊗ u, ∇ ϕ ) 2 results in the v ariational formulation (w eak-2D-Stokes) ( u ′ + A w u = F w ( u ) , t ∈ (0 , T ) , u (0) = u 0 . T o verify ( A1 ) , we calculate for all u ∈ D( A w ) in view of Poincaré’s inequality ⟨ Au, u ⟩ H − 1 σ , D( A w ) = ∥∇ u ∥ 2 2 ≥ α ∥ u ∥ 2 D( A w ) for some α > 0 . Moreo ver, by Hölder’s inequality and Sob olev embedding we obtain ( u ⊗ u, ∇ ϕ ) 2 ≤ C ∥ u ⊗ u ∥ 2 · ∥ ϕ ∥ H 1 ≤ C ∥ u ∥ 2 4 · ∥ ϕ ∥ H 1 ≤ C ∥ u ∥ 2 H 1 / 2 · ∥ ϕ ∥ H 1 . Hence, ( A2 ) and ( A3 ) are satisfied with β = 3 4 and ρ = 1 . Since u ∈ D( A w ) is an eligible test function, we can test the v ariational formulation of ( 2D-NSE ) with u , resulting in the energy equalit y 1 2 ∥ u ( t ) ∥ 2 2 + Z t 0 ∥∇ u ∥ 2 2 d s = 1 2 ∥ u 0 ∥ 2 2 for all t ∈ (0 , ∞ ) . This readily implies that ( A4 ) is v alid. Denoting b y v the appro ximate solution of ( DA ), guaran teed b y Prop osition 2.5 , an application of Theorem 2.7 then yields the following result. Corollary 4.1 (Data Assimilation for ( weak-2D-Stok es )) . L et u 0 , v 0 ∈ L 2 σ (Ω; R 2 ) . Then, ther e exist µ 0 , δ 0 > 0 such that for al l µ > µ 0 and 0 < δ < δ 0 ∥ ( u − v )( t ) ∥ L 2 (Ω) → 0 exp onential ly, as t → ∞ . 4.2. 1D-Allen-Cahn equation. [ 4 ] The Allen-Cahn equation is a seminal reaction-diffusion mo del that arises from a Ginzburg-Landau free energy functional to describ e phase separation pro cesses. It is a prototypical example of a system featuring fron t propagation, where the interface motion is driv en b y mean curv ature, making it a fundamen tal testbed for studying pattern formation and metastability in materials science ( [ 4 ]). Let Ω = (0 , 1) ⊂ R . Consider the one-dimensional Allen-Cahn mo del given by the equations (1D-Allen-Cahn)      ∂ t u = ∂ xx u + u − u 3 , in (0 , T ) × Ω , u = 0 , in (0 , T ) × ∂ Ω , u (0) = u 0 , T o reform ulate ( 1D-Allen-Cahn ) as an abstract ev olution equation ( 2.1 ) w e define the w eak Diric hlet Lapla- cian b y ∂ w xx : D( ∂ w xx ) = H 1 0 (Ω) → H − 1 (Ω) , ⟨ ∂ w xx u, ϕ ⟩ H − 1 , D( ∂ w xx ) = ( ∂ x u, ∂ x ϕ ) 2 for all u, ϕ ∈ D( ∂ w xx ) and set ⟨ F w ( u ) , ϕ ⟩ H − 1 , D( ∂ w xx ) := ( u, ϕ ) 2 − ( u 3 , ϕ ) 2 for all u, ϕ ∈ D( ∂ w xx ) . T o verify ( A1 ) , note that by Poincaré’s inequality we hav e ⟨ ∂ w xx u, u ⟩ H − 1 , D( ∂ w xx ) = ∥ ∂ x u ∥ 2 2 ≥ α ∥ u ∥ 2 H 1 for all u ∈ D( ∂ w xx ) , for a constant α > 0 . Regarding ( A2 ) , ( A3 ) , w e estimate using Hölder’s inequalit y and the em b edding L r (Ω)  → H − 1 (Ω) , whic h holds for any r ∈ (1 , ∞ ) ∥ F ( u ) − F ( v ) ∥ H − 1 ≤ C  ∥ u − v ∥ 2 + ∥ ( u 2 + v 2 ) | u − v |∥ r  ≤ C  ∥ u − v ∥ 2 + ( ∥ u ∥ 2 3 r + ∥ v ∥ 2 3 r ) ∥ u − v ∥ 3 r  . 12 GIANMARCO DEL SAR TO, MA TTHIAS HIEBER, FILIPPO P ALMA, AND T AREK ZÖCHLING In particular, ( A1 ) , ( A2 ) are v alid for β = 2 / 3 , ρ = 2 and r ∈ (1 , 2) arbitrary . T o v erify ( A4 ) , we test ( 1D-Allen-Cahn ) with u , which is an eligible test function, to obtain 1 2 ∂ t ∥ u ∥ 2 2 + ∥ ∂ x u ∥ 2 2 + ∥ u ∥ 4 4 = ∥ u ∥ 2 2 ≤ κ ∥ ∂ x u ∥ 2 2 with κ ∈ (0 , 1) where w e used P oincaré’s inequalit y in the last step; note that κ ∈ (0 , 1) since κ = λ − 1 1 , where λ 1 = π 2 is the smallest eigenv alue of the Diric hlet Laplacian on (0 , 1) , see [ 15 ]. Integrating in time and Gronw all’s inequalit y then imply that ( A4 ) is v alid. Denoting by v the approximate solution of ( DA ), guaranteed by Prop osition 2.5 , an application of Theorem 2.7 then yields the following result. Corollary 4.2 (Data Assimilation for ( 1D-Allen-Cahn )) . L et u 0 , v 0 ∈ L 2 (Ω) . Then, ther e exist µ 0 , δ 0 > 0 such that for al l µ > µ 0 and 0 < δ < δ 0 ∥ ( u − v )( t ) ∥ L 2 (Ω) → 0 exp onential ly, as t → ∞ . 4.3. 1D and 2D Cahn-Hilliard equation. [ 11 ] The Cahn-Hilliard equation is a foundational fourth-order partial differen tial equation that models phase separation in binary mixtures while, crucially , conserving the total mass of each comp onen t. It describ es complex coarsening dynamics, suc h as spino dal decomposition, where a system lo w ers its free energy by forming distinct phase domains, making it indisp ensable in materials science and the study of fluid mixtures ( [ 11 ]). Let Ω ⊂ R d , with d = 1 , 2 , b e an op en and b ounded domain with a smo oth b oundary . W e consider the d -dimensional Cahn-Hilliard equation (4.3)            ∂ t u + ∆ 2 u = ∆ f ( u ) , in (0 , T ) × Ω , ∇ u · ν = 0 , in (0 , T ) × ∂ Ω , ∇ (∆ u ) · ν = 0 , in (0 , T ) × ∂ Ω , u (0) = u 0 , where ν denotes the out ward unit v ector. The nonlinear term is given b y f ( u ) = u 3 − u , whic h corresp onds to the deriv ativ e of the standard double-w ell p oten tial, but more general f can b e c hosen, see for instance [ 2 , Section 5.1]. T o cast ( 4.3 ) in the abstract ev olution framework ( 2.1 ), we consider the spaces V = H 2 N (Ω) =  u ∈ H 2 (Ω) | ∇ u · ν = 0 , ∇ (∆ u ) · ν = 0 in ∂ Ω  , H = L 2 (Ω) , V ∗ =  H 2 N (Ω)  ∗ . The linear op erator A : V → V ∗ and the nonlinear op erator F : V → V ∗ are giv en by ⟨ Au, v ⟩ V ∗ , V = Z Ω ∆ u ∆ v dx, ⟨ F ( u ) , ϕ ⟩ V ∗ , V = Z Ω (∆ f ( u )) ϕ dx. W e now verify the necessary assumptions on A and F . The operator A is quasi-co erciv e. Indeed, by elliptic regularit y theory for the Laplacian op erator with Neumann b oundary conditions, we find that there exists α > 0 such that ⟨ Au, u ⟩ V ∗ , V = ∥ ∆ u ∥ 2 2 ≥ α ∥ u ∥ 2 V − ∥ u ∥ 2 H . Th us ( A1 ) holds with ω = 1 . T o verify the lo cal Lipsc hitz condition for F , we estimate ∥ F ( u 1 ) − F ( u 2 ) ∥ V ∗ . Using integration b y parts and the b oundary conditions, we hav e ∥ F ( u 1 ) − F ( u 2 ) ∥ V ∗ = sup ∥ φ ∥ V ≤ 1 |⟨ F ( u 1 ) − F ( u 2 ) , ϕ ⟩| = sup ∥ φ ∥ V ≤ 1     Z Ω ∇ ( f ( u 1 ) − f ( u 2 )) · ∇ ϕdx     = sup ∥ φ ∥ V ≤ 1     Z Ω ( f ( u 1 ) − f ( u 2 )) · ∆ ϕ dx     . No w, observe that, for the sp ecific choice f ( u ) = u 3 − u we can find a b ound for the difference | f ( u 1 ) − f ( u 2 ) | ≤ C (1 + u 2 1 + u 2 2 ) | u 1 − u 2 | , CONTINUOUS DA T A ASSIMILA TION FOR SEMILINEAR EQUA TIONS: A GENERAL APPRO ACH BY EVOLUTION EQUA TIONS 13 where C is a positive constant that will change line by line. Applying this b ound, the triangle inequality and Hölder’s inequality yields ∥ F ( u 1 ) − F ( u 2 ) ∥ V ∗ ≤ C sup ∥ φ ∥ V ≤ 1 Z Ω (1 + | u 1 | 2 + | u 2 | 2 ) | u 1 − u 2 || ∆ ϕ | dx ≤ ∥ (1 + | u 1 | 2 + | u 2 | 2 )( u 1 − u 2 ) ∥ 2 ≤ ∥ 1 + | u 1 | 2 + | u 2 | 2 ∥ p 1 ∥ u 1 − u 2 ∥ p 2 ≤ C (1 + ∥ u 1 ∥ 2 2 p 1 + ∥ u 2 ∥ 2 2 p 1 ) ∥ u 1 − u 2 ∥ p 2 , where 1 2 = 1 p 1 + 1 p 2 . Cho osing p 1 = 3 and p 2 = 6 , w e obtain ∥ F ( u 1 ) − F ( u 2 ) ∥ V ∗ ≤ C (1 + ∥ u 1 ∥ 2 6 + ∥ u 2 ∥ 2 6 ) ∥ u 1 − u 2 ∥ 6 . By Sobolev em b edding theorems, for the in terp olation space V β = [ V ∗ , V ] β  → H 4 β − 2 (Ω) , w e hav e V β  → L 6 (Ω) for β ≥ 7 / 12 if d = 1 , and for β ≥ 2 / 3 if d = 2 . Thus, c ho osing β j = β = 2 / 3 and ρ = 2 , b oth (A2) - (A3) are satisfied. Consequently , by Lemma 2.3 , there exists a unique lo cal solution to ( 4.3 ) in the class L 2 t (H 2 x ) ∩ H 1 t (H − 2 x ) ∩ BUC t (L 2 x ) on a maximal time interv al [0 , t + ( u 0 )) . T o establish global existence, we consider the shifted problem (4.4) ( ˜ u ′ + ( A + I ) ˜ u = F ( ˜ u ) , ˜ u (0) = u 0 , and v erify the conditions of ( A4 ) . The functional E [ ˜ u ] = Z Ω 1 2 |∇ ˜ u | 2 + 1 4 | ˜ u | 4 dx is a Ly apuno v functional for ( 4.4 ). In other words, if u ∈ L 2 t (H 2 x ) ∩ H 1 t (H − 2 x ) ∩ BUC t (L 2 x ) denotes a maximal solution of ( 4.4 ), then it can b e chec k ed that ∂ t E [ ˜ u ( t )] ≤ −∥∇ ϕ ( t ) ∥ 2 2 − ∥ ˜ u ( t ) ∥ 4 4 , where ϕ := f ( u ) − ∆ u. F ollo wing the previous computations, we hav e ∥ F ( ˜ u ( t )) ∥ V ∗ ≤ ∥ f ( ˜ u ( t )) ∥ 2 = ∥ ˜ u 3 ( t ) − ˜ u ( t ) ∥ 2 ≤ ∥ ˜ u ( t ) ∥ 3 6 + ∥ ˜ u ( t ) ∥ 2 . By the em b eddings H 2 (Ω)  → L 6 (Ω)  → L 2 (Ω) , whic h hold for d = 1 , 2 in the bounded domain Ω , w e deduce, thanks to the Lyapuno v functional, that lim t → t + ∥ F ( ˜ u ( t )) ∥ V ∗ < ∞ . Next, w e verify that ∥ ˜ u ( t ) ∥ 2 2 ∈ L 1 (0 , ∞ ) . T aking the inner pro duct of ( 4.4 ) 1 with ˜ u gives 1 2 ∂ t ∥ ˜ u ( t ) ∥ 2 2 + ∥ ∆ ˜ u ( t ) ∥ 2 2 + ∥ ˜ u ( t ) ∥ 2 2 = (∆ f ( ˜ u ) , ˜ u ) 2 . In tegrating the right-hand side by parts, we get (∆ f ( ˜ u ) , ˜ u ) 2 = − ( f ′ ( ˜ u ) ∇ ˜ u, ∇ ˜ u ) 2 = ((1 − 3 ˜ u 2 ) ∇ ˜ u, ∇ ˜ u ) 2 = ∥∇ ˜ u ∥ 2 2 − 3 Z Ω ˜ u 2 |∇ ˜ u | 2 dx. This leads to the inequality 1 2 ∂ t ∥ ˜ u ( t ) ∥ 2 2 + ∥ ∆ ˜ u ( t ) ∥ 2 2 + ∥ ˜ u ( t ) ∥ 2 2 + 3 Z Ω ˜ u 2 |∇ ˜ u | 2 dx ≤ ∥∇ ˜ u ∥ 2 2 . Using in tegration by parts, Cauch y-Sch warz inequalit y , and Y oung inequalit y on the righ t-hand side, w e get ∥∇ ˜ u ∥ 2 2 ≤ − Z Ω ˜ u ∆ ˜ u dx ≤ ∥ ˜ u ∥ 2 ∥ ∆ ˜ u ∥ 2 ≤ 1 2 ∥ ˜ u ∥ 2 2 + 1 2 ∥ ∆ ˜ u ∥ 2 2 . Substituting this back and absorbing terms gives 1 2 ∂ t ∥ ˜ u ( t ) ∥ 2 2 + 1 2 ∥ ∆ ˜ u ( t ) ∥ 2 2 + 1 2 ∥ ˜ u ( t ) ∥ 2 2 + 3 Z Ω ˜ u 2 |∇ ˜ u | 2 dx ≤ 0 . 14 GIANMARCO DEL SAR TO, MA TTHIAS HIEBER, FILIPPO P ALMA, AND T AREK ZÖCHLING In tegrating this differential inequality in time from 0 to ∞ , w e conclude that ∥ ˜ u ( t ) ∥ 2 2 ∈ L 1 (0 , ∞ ) . Denoting b y v the appro ximate solution of ( DA ), guaranteed b y Prop osition 2.5 , w e apply Theorem 2.7 and we obtain the follo wing result. Corollary 4.3 (Data Assimilation for ( 4.3 )) . L et u 0 , v 0 ∈ L 2 (Ω) . Then, ther e exist µ 0 , δ 0 > 0 such that for al l µ > µ 0 and 0 < δ < δ 0 ∥ ( ˜ u − v )( t ) ∥ L 2 (Ω) → 0 exp onential ly, as t → ∞ . A c kno wledgemen ts. Gianmarco Del Sarto, Matthias Hieber and T arek Zö c hling ac knowledge the supp ort from the DFG pro ject FOR 5528. The researc h of Filipp o Palma is carried on under the auspices of GNFM-INdAM. References [1] R. Adams and J. F ournier. 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Appl. Math. Optim. , 85(2):5, 19, 2022. Technische Universit ä t Darmst adt, F achbereich Ma thema tik, Schlossgar tenstr. 7, 64289 Darmst adt, Ger- many Email addr ess : delsarto@mathematik.tu-darmstadt.de Technische Universit ä t Darmst adt, F achbereich Ma thema tik, Schlossgar tenstr. 7, 64289 Darmst adt, Ger- many Email addr ess : hieber@mathematik.tu-darmstadt.de Universit à degli Studi della Camp ania L. V anvitelli, Dip a r timento di Ma tema tica e Fisica, Via Viv aldi 43, 81100 Caser t a, It al y Email addr ess : filippo.palma@unicampania.it Technische Universit ä t Darmst adt, F achbereich Ma thema tik, Schlossgar tenstr. 7, 64289 Darmst adt, Ger- many Email addr ess : zoechling@mathematik.tu-darmstadt.de