Closed-form finite-time blow-up and stability for a $(1+2)$D system (E1) derived from the 2D inviscid Boussinesq equations

CLOSED-F ORM FINITE-TIME BLO W-UP AND ST ABILITY F OR A (1 + 2) D SYSTEM (E1) DERIVED FR OM THE 2D INVISCID BOUSSINESQ EQUA TIONS Y A OMING SHI Abstract. In p olar v ariables ( x, θ ) on a planar sector, we study a (1 + 2)D system (E1) deriv ed from the t wo-dimensional in viscid Boussinesq equations. Rigorous reduction from Boussinesq. Under a parit y/symmetry ansatz on the whole plane (o dd/ev en reflection across the axes), we sho w that the v elo city-pressure form of the 2D inviscid Boussinesq system admits an exact reform ulation in terms of Hou–Li t ype new v ariables ( u, v , g ). In the reform ulated system (E1), the v ortex stretc h- ing terms are greatly simplified ( uv , v 2 − u 2 , − g 2 ). This prompts us to treat ( u, v , g ) as the v orticit y building blo cks . Our first main result is the disco very of explicit smo oth solutions that blow up in finite time 0 < T < ∞ while a natural weigh ted energy remains uniformly b ounde d for all t ∈ [0 , T ]. The construction pro ceeds in three steps. (1) W e identify spec ial ridge r ays θ 0 = ± π / 4 suc h that, under the divergence-free constrain t, system (E1) reduces on each ridge to a (1 + 1)D Constantin–Lax–Ma jda type c onve ction-fr e e reaction system in ( t, x ); see Theorem 2.5. (2) W e then embed these (1 + 1)D closed-form ridge solutions into the full sector x ∈ [0 , ∞ ), θ ∈ [ − π / 4 , π / 4] by in troducing carefully tuned θ -dep enden t seed data, pro ducing an explicit background profile that blows up only at ( x, θ ) = (0 , ± π / 4). (3) Finally , we derive the perturbation equations around this background and pro v e line ar and nonline ar stability in high-regularity weigh ted Sob olev norms. Consequen tly , the constructed background profiles are stable finite-time blow-up solutions of (E1). W e can similarly solv e the problem in wedge 2: x ∈ [0 , ∞ ), θ ∈ [ π 4 , 3 π 4 ] and wedge 3: x ∈ [0 , ∞ ), θ ∈ [ − 3 π 4 , − π 4 ]. Th us, by symmetry , w e co ver the right half-plane in Cartesian v ariables (equiv alen tly , x ∈ [0 , ∞ ) and θ ∈ [ − π 2 , π 2 ] in p olar co ordinates). 2020 Mathematics Sub ject Classification. Primary 35B44, 35Q86, 76B03; Sec- ondary 35B35, 35Q35. Keyw ords. 2D inviscid Boussinesq equations; Constantin–Lax–Ma jda t yp e mo del; ex- plicit blow-up solution; ridge reduction; b ounded energy; linear and nonlinear stabilit y; w eigh ted Sob olev energies. 1. Introduction The question of whether smo oth solutions can develop a singularit y in finite time is a cen tral theme in nonlinear PDE and fluid mechanics. In this pap er we study a (1 + 2)- dimensional closed subsystem (E1), rigorously derived from the inviscid 2D Boussinesq equations: it retains a stretc hing-lik e in teraction betw een t w o scalar fields while sup- pressing geometric and nonlo cal complications so that closed-form analysis is p ossible. Here we provide a rigorous, fully explicit setting in which a Constan tin–Lax–Ma jda type blo w-up mechanism can b e pro v ed and shown stable under p erturbations. Date : March 18, 2026. 2020 Mathematics Subje ct Classific ation. 35B44, 35B40, 35Q86, 76B03, 76D05. Key wor ds and phr ases. 2D in viscid Boussinesq, Constan tin–Lax–Ma jda t ype mo del, finite-time blo w- up, explicit solutions, nonlinear stability , ridge reduction, weigh ted Sob olev energy , divergence-form closure. 1 2 Y AOMING SHI W e will call the Hou-Li t yp e new v ariables { u, v , g } of (2.2) the building blo c ks of v orticit y , b ecause their units are equal to the unit of ω = ∇ × u . Also the quadratic v ortex stretching terms are greatly simplified: ( uv , v 2 − u 2 , − g 2 ). Related w ork and con text. As concisely emphasized b y Elgindi–Jeong [16] and Elgindi– P asqualotto [29], singularit y formation for the in viscid 2D Boussinesq and 3D Euler systems has a long history; w e recall only a small selection of represen tativ e p oin ters here, emphasizing w orks most closely aligned with the closed-form mec hanism and sta- bilit y framew ork dev elop ed below. F or the in viscid 2D Boussinesq system, classical lo cal theory and conditional breakdown criteria go back to Cannon–DiBenedetto [12], Chae– Nam [3] (see also Chae–Kim–Nam [7]), and T aniuchi [26]; see also W u’s lecture notes [27] and the Euler p ersp ectiv e of Beale–Kato–Ma jda [1] and Constantin [10, 11]. On the mo deling side, explicit/didactic mechanisms include Chae–Constan tin–W u [2] and the CLM/De Gregorio lineage [9, 15], as w ell as 1D reductions for Boussinesq-type dynamics suc h as Choi–Kiselev–Y ao [8]. F or rigorous singularity constructions and stability sce- narios in PDE settings, see Elgindi–Jeong [16, 17], Chen–Hou [4, 5], and the synthesis of Driv as–Elgindi [14]. W edge co verage. W e can similarly solv e the problem in w edge 2: x ∈ [0 , ∞ ), θ ∈ [ π 4 , 3 π 4 ] and wedge 3: x ∈ [0 , ∞ ), θ ∈ [ − 3 π 4 , − π 4 ]. Thus, by symmetry , w e cov er the righ t half-plane in Cartesian v ariables (equiv alently , x ∈ [0 , ∞ ) and θ ∈ [ − π 2 , π 2 ] in p olar co ordinates). Main ac hiev emen ts. (0) W e construct smooth, closed-form finite-time blow-up solutions to a (1 + 2)D system (E1) with b ounded energy on [0 , T ]. (1) W e sho w existence of ridge rays θ 0 = ± π / 4 on whic h (E1) reduces to a (1 + 1)D con v ection-free reaction system. (2) W e extend these ridge solutions to wedge backgrounds b y choosing suitable θ - dep enden t seed profiles. (3) W e derive p erturbation equations in w eigh ted divergence form and pro v e lin- ear/nonlinear stabilit y up to the blow-up time. Organization. Section 2 giv es a self-contained deriv ation of the closed subsystem (2.5) from the in viscid 2D Boussinesq equations on R 2 (in p olar co ordinates) and constructs the explicit blo w-up bac kground on the wedge b et w een the ridge rays. Section 3 describes emm b eding strategy and derives p erturbation equation. Section 4 prov es b oundedness of the natural energy up to the blo w-up time. Section 5 prov es linear and nonlinear stabilit y of the background on [0 , T ). Section 6 summarizes the results and discusses extensions (including the other wedge sectors). 2. The deriv a tion of system (E1) fr om 2D inviscid Boussinesq equa tions 2.1. The deriv ation of system (E1) from 2D in viscid Boussinesq equations. In this section w e con v ert the v elocity-pressure form of 2D in viscid Boussinesq equations (see for example W u [27], Elgindi–Jeong [17], Kiselev–Pu–Y ao [22]) into a new form (vorticit y building blo c ks). The structure of the v ortex stretc hing term and conv ection term are clearly rev ealed. Th us studying v ortex strething in might be easier than b efore. In the velocity-pressure form, the 2D inviscid Boussinesq equations for v elo cit y u = u 2 e 2 + u 3 e 3 , pressure P and densit y ϑ (assuming the gravit y in − e 2 direction) in R 2 are giv en by ST ABLE FINITE-TIME BLO W-UP FOR SYSTEM (E1) 3                  ˜ D D t ϑ = 0 , t ∈ [0 , T ) , ( x 2 , x 3 ) ∈ R 2 ˜ D D t u 2 = − ∂ 2 P + ϑ, ˜ D D t u 3 = − ∂ 3 P , ∂ 2 u 2 + ∂ 3 u 3 = 0 , ˜ D D t = ∂ t + u 2 ∂ 2 + u 3 ∂ 3 , (2.1) No w assuming that u 2 is odd in x 2 and ev en in x 3 ; u 3 is ev en in x 2 and odd in x 3 , p is ev en in ( x 2 , x 3 ), ϑ is o dd in x 2 and ev en in x 3 . Define Hou–Li [21] t yp e new v ariables { v , g , u 2 , p } :=  − u 2 x 2 , u 3 x 3 , ϑ x 2 , P  (2.2) w e can con v ert (2.1) to a new set of equations (with ( ρ, z ) = ( x 2 , x 3 ) to simplify the notation) : V erification of the reduction. Under the stated parity assumptions, the whole plane system (2.1) can b e extended to R 2 b y o dd/even reflection. In particular, along the symmetry axes x 2 = 0 and x 3 = 0 the quotients v := − u 2 x 2 , g := u 3 x 3 , u 2 := ϑ x 2 (2.3) extend smo othly (the numerator has the matc hing o dd symmetry), and we ma y write u 2 = − ρ v , u 3 = z g , ϑ = ρ u 2 . With this con ven tion the material deriv ative becomes D D t = ∂ t + u 2 ∂ ρ + u 3 ∂ z = ∂ t − ρv ∂ ρ + z g ∂ z , (2.4) whic h matches the last line in (2.5). Now: • F rom ˜ D D t ϑ = 0 and ϑ = ρu 2 , 0 = D D t ( ρu 2 ) =  D ρ D t  u 2 + ρ D D t ( u 2 ) = u 2 u 2 + ρ D D t ( u 2 ) = − ρv u 2 + ρ D D t ( u 2 ) , hence D D t ( u 2 ) = v u 2 , equiv alently D D t u = 1 2 uv . • F rom ˜ D D t u 2 = − ∂ ρ P + ϑ and u 2 = − ρv , D D t u 2 = D D t ( − ρv ) = − ( D ρ D t ) v − ρ D v D t = − u 2 v − ρ D v D t = ρv 2 − ρ D v D t , so dividing b y ρ > 0 giv es D v D t = v 2 − u 2 + 1 ρ P ρ . • F rom ˜ D D t u 3 = − ∂ z P and u 3 = z g , D D t u 3 = D D t ( z g ) = ( D z D t ) g + z D g D t = u 3 g + z D g D t = z g 2 + z D g D t , hence D g D t = − g 2 − 1 z P z . • Finally , incompressibility ∂ ρ u 2 + ∂ z u 3 = 0 with u 2 = − ρv , u 3 = z g yields 0 = ∂ ρ ( − ρv ) + ∂ z ( z g ) = − ( v + ρv ρ ) + ( g + z g z ) , i.e. z g z − ρv ρ + g − v = 0. 4 Y AOMING SHI This pro ves that the c hange of v ariables reduces (2.1) exactly to (2.5).                  D D t u = 1 2 uv , t ∈ [0 , T ) , ( ρ, z ) ∈ R 2 D D t v = v 2 − u 2 + 1 ρ p ρ D D t g = − g 2 − 1 z p z z ∂ z g − ρ∂ ρ v + g − v = 0 . D D t := ∂ t + g z ∂ z − v ρ∂ ρ , (2.5) Remark 2.1. F r om the insp e ction of (2.5) , we notic e that if the intitial c onditions for { u, v , g , p } ar e b oth symmetric in r and z , then the PDEs wil l pr eserve this symmetric pr op erties. In this sense, we c onsider (2.5) is define d on R 2 . Remark 2.2. We wil l c al l { u, v , g } the building blo cks of vorticity, b e c ause their units ar e e qual to the unit of ω = ∇ × u . Also the quadr atic vortex str etching terms ar e gr e atly simplifie d: ( uv , v 2 − u 2 , − g 2 ) . 2.2. System (E 1 ). W e in tro duce a stream function ¯ ψ and app end tw o more equations to (2.5) and put everything together in (2.6). W e also wan t to sav e symbols ( u, v , g , p ) for future con v enience, so we add ov er bar to the dep edent v ariables. Th us system (E1) in (2.6) con tains 5 dep endent v ariables ( ¯ u, ¯ v , ¯ g , ¯ p, ¯ ψ ) as functions of t and ev en functions of ( ρ, z ) (the meridian plane) in 6 equations. The last one is definition for D D t . (i) 0 = D D t ¯ u − 1 2 ¯ u ¯ v , ( t, ρ, z ) ∈ [0 , T ) × R 2 (2.6a) (ii) 0 = D D t ¯ v − ¯ v 2 + ¯ u 2 − 1 ρ ¯ p ρ , (2.6b) (iii) 0 = D D t ¯ g + ¯ g 2 + µ 2 z ¯ p z , (2.6c) (iv) 0 = z ∂ z ¯ g − ρ∂ ρ ¯ v + ¯ g − ¯ v , (2.6d) (v) 0 = ¯ v − ¯ ψ − z ∂ z ¯ ψ , (2.6e) (vi) 0 = ¯ g − ¯ ψ − ρ∂ ρ ¯ ψ , (2.6f ) (vii) D D t : = λ ∂ t + ¯ g z ∂ z − ¯ v ρ∂ ρ , . (2.6g) Remark 2.3 ( t -scaling factor λ and z -scaling factor µ ) . A t -sc aling factor λ and z -sc aling factor µ ar e include d for futur e c onvenienc e. They only app e ar e d in the c ombinations λ ∂ t , µ 2 z ∂ z and µ 2 ∂ 2 z . On the other hand substition of (v) and (vi) into (iv) le ads to in identity. So ther e is no r e daudency. F or conv enience, w e will study the system in p olar co ordinates. 2.3. P olar co ordinates ( x = r = p ρ 2 + z 2 , θ = arctan( z /ρ )) . W e use the p olar co ordinates in Meridian plane: ρ = x cos ( θ ) , z = x sin ( θ ) . (2.7) Remark 2.4. The p olar c o or dinates ( x, θ ) in Meridian plane ( ρ, z ) is also a sph eric al c o or dinates ( x, θ , ϕ ) (with north-p ole at θ = π / 2 ) for 3D axisyymetric functions in R 3 . 2.4. Bac kground and closure-induced cancellations. W e write the bac kground so- lutions as ¯ u = U ( t, x, θ ) , ¯ v = V ( t, x, θ ) , ¯ g = G ( t, x, θ ) , ¯ p = P ( t, x, θ ) . (2.8) W e wan t to c hec k how the equations lo ok lik e under the following Ansartz A: G ( t, x, θ ) = V ( t, x, θ ) . (2.9) ST ABLE FINITE-TIME BLO W-UP FOR SYSTEM (E1) 5 After substitution of (3.4) in to (2.6) and using (2.9) to simplify the results, we end-up 4 equations with an interesting pattern: λU t = cos(2 θ ) V xU x + 1 2 V U − sin(2 θ ) V U θ , λV t = cos(2 θ ) V xV x + V 2 − U 2 + 1 x P x − tan( θ ) 1 x 2 P θ − sin(2 θ ) V V θ , λV t = cos(2 θ ) xV x V − V 2 − µ 2 x P x − cot( θ ) µ 2 x 2 P θ − sin(2 θ ) V V θ , 0 = cos(2 θ ) xV x − sin(2 θ ) V θ . (2.10) 2.5. Ridge ra y and Ridge functions. Theorem 2.5. System (2.6) (including the diver genc e-fr e e c ondition (2.6) (4)) r estricte d to the r ays determine d by (tan θ ) 2 = (tan θ 0 ) 2 = 1 Assuming the ansartz (2.9) and ridge flatness (in dir e ctions normal to the ridge) ansatz: ( V θ , U θ , P θ ) | θ 0 = 0 , (2.11) Then we have: (A): Diver genc e c onstr aint fixes the ridge r ays. Under (2.9) , the diver genc e identity (2.6d) implies cos(2 θ ) = cos(2 θ 0 ) = cos( π 2 ) = 0 . Equivalently, θ 0 ∈ n ± π 4 , π ± π 4 o . In the ( r , z ) –plane this c orr esp onds to the two str aight lines thr ough the origin z = ± r , i.e. four r ays (two r ays on e ach diagonal line). When working in the first quadr ant r ≥ 0 , z ≥ 0 , we take the princip al choic e θ 0 = π 4 . (B). Conve ction-fr e e on the principle ridge r ay. On θ = θ 0 , D D t   θ = θ 0 = ∂ t . (C) With the sp e cific choic e for sc aling pr ameters ( λ, µ ) : λ = 3 2 , µ =  5 3  1 / 2 , (2.12) the dynamics of al l ridge functions { U, V , G, P , } ( t, x, θ 0 ) ar e c ompletely determine d by the 1+1D c onvention-fr e e r e action system for { U ( t, x, θ 0 ) , V ( t, x, θ 0 ) } . ( U t = 1 3 V U, V t = 1 6 V 2 − 5 12 U 2 . (2.13) After setting τ = 6 t,  2 5  1 / 2 U ( t, x ) = 1 2 ω ( τ , x ) , V ( t, x ) = 1 2 H ( ω ( τ , x )) , we c an se e that this system r e duc e d to the famous Constantin-L ax-Majda system, that is pr op ose d to mo del the vorticity str etching dynamics. ( ∂ τ ω = ω H ( ω ) ∂ τ H ( ω ) = 1 2  H ( ω ) 2 − ω 2  . (2.14) 6 Y AOMING SHI Pr o of of the or em 2.5. After insp ection of (2.10), Claim (A) is ob viously true. W e notice that because cos(2 θ ) = 0, the con v ection terms U ∂ x and V ∂ x v anished in (2.10). Thus they are con v ection-free on the ridge rays. Claim (B) then follows. Th us our ridge ra y system is comp osed of 3 PDEs, as shown b elo w          3 2 U t = 1 2 V U, 3 2 V t = V 2 − U 2 + 1 x P x , 3 2 V t = − V 2 − 5 3 x P x . (2.15) Separation P x and V t leads to (2.13) and a simplified Poisson equation for pressure P x = 3 x 8  U 2 − 2 V 2  . (2.16) This pro ved Claim (C) and ends the Pro of of theorem 2.5. □ It is well known that system (2.13), lik e Constantin-Lax-Ma jda system, admits the fllo wing closed-form solutions        V ( t, a, b ) = − 6 (5 tb 2 + 2 a ( ta − 6)) 2( ta − 6) 2 + 5 t 2 b 2 , U ( t, a, b ) = ± 72 b 2( ta − 6) 2 + 5 t 2 b 2 . (2.17) If w e provide the follo wing initial constions        a = a ( x ) = A (1 + x 4 ) , A > 0 , b = b ( x ) = B x 4 (1 + x 4 ) 2 , B > 0 . (2.18) Then this solution will blowup at the origin in finite time T = 6 A . V ( t, x ) → ∞ , U ( t,x ) U (0 ,x ) → ∞ , as t → T = 6 A , x → 0 . (2.19) 3. Stra tegy to extend the ridge back ground to the meridian sector x ∈ R , θ ∈ [ − π / 4 , π / 4] W e use the ridge solutions (2.17) but with θ -dep endent inintial data a = a ( x, θ ) , b = b ( x, θ ) as the θ -dep endent bac kground solution.                ¯ u = U ( t, x, θ ) + u ( t, x, θ ) , ¯ v = V ( t, x, θ ) + v ( t, x, θ ) , ¯ g = G ( t, x, θ ) + g ( t, x, θ ) , ¯ p = P ( t, x, θ ) + p ( t, x, θ ) , G ( t, x, θ ) = V ( t, x, θ ) . (3.1) ST ABLE FINITE-TIME BLO W-UP FOR SYSTEM (E1) 7 The bac kground solutions are the same as in (2.17). But the x -dep endent initial condi- tions in (2.18) are replaced with ( x, θ )-dep endent ones:              r = x 2 + A 1 cos 2 (2 θ ) , A 1 > 0 a = a ( x, θ ) = A (1 + r 4 ) , A > 0 , b = b ( x, θ ) = B r 4 (1 + r 4 ) 2 , B > 0 . (3.2) This solution will blowup in finite time T = 6 A . V ( t, x, θ ) → ∞ , U ( t,x ) U (0 ,x ) → ∞ , as t → T = 6 A , x → 0 , θ = ± π 4 ± π . (3.3) W e remark that inside the w edge θ ∈  − π 4 , π 4  , | V ( t, x, θ ) | and | U ( t, x, θ ) | are bounded from ab ov e. 3.1. P erturbation PDEs. W e derive the p erturbation PDEs. The full full solutions are giv en by:                ¯ u = U ( t, x, θ ) + u ( t, x, θ ) , ¯ v = V ( t, x, θ ) + v ( t, x, θ ) , ¯ g = G ( t, x, θ ) + g ( t, x, θ ) , ¯ p = P ( t, x, θ ) + p ( t, x, θ ) , G ( t, x, θ ) = V ( t, x, θ ) . (3.4) After substitution of (3.4) into (2.6) and using (3.4)(4) to simplify the results, w e end-up 5 equations with the following pattern: (i) 3 2 U t = 1 2 V U + cos(2 θ ) V xU x − sin(2 θ ) U θ V − U 1 , 3 2 u t = cos(2 θ ) V xu x − sin(2 θ ) V u θ + U 1 , + ( v cos 2 θ − g sin 2 θ ) xU x − 1 2 sin(2 θ )( v + g ) U θ + 1 2 ( V u + v U ) + N 1 ( u, v , g ) N 1 : = 1 2 uv + ( v cos 2 θ − g sin 2 θ ) xu x − 1 2 sin(2 θ )( v + g ) u θ , (3.5) (ii) 3 2 V t = V 2 − U 2 + 1 x P x + cos(2 θ ) V xV x − sin(2 θ ) V V θ − tan( θ ) 1 x 2 P θ − V 1 , 3 2 v t = cos(2 θ ) V xv x − sin(2 θ ) V v θ + 1 x p x − tan( θ ) 1 x 2 p θ + V 1 , + ( v cos 2 θ − g sin 2 θ ) xV x − 1 2 sin(2 θ )( v + g ) V θ + 2( v V − uU ) + N 2 ( u, v , g ) N 2 : = v 2 − u 2 + ( v cos 2 θ − g sin 2 θ ) xv x − 1 2 sin(2 θ )( v + g ) v θ , (3.6) 8 Y AOMING SHI (iii) 3 2 V t = − V 2 − µ 2 x P x + cos(2 θ ) xV x V − sin(2 θ ) V V θ − cot( θ ) 5 3 x 2 P θ − G 1 , 3 2 g t = cos(2 θ ) V xg x − sin(2 θ ) V g θ − 5 3 x p x − cot( θ ) 5 3 x 2 p θ + G 1 , + ( v cos 2 θ − g sin 2 θ ) xV x − 1 2 sin(2 θ )( v + g ) V θ − 2 g V + N 3 ( u, v , g ) N 3 : = − g 2 + ( v cos 2 θ − g sin 2 θ ) xg x − 1 2 sin(2 θ )( v + g ) g θ , (3.7) (v) V = Ψ + 1 2 x Ψ x − 1 2 cos(2 θ ) x Ψ x + 1 2 sin(2 θ )Ψ θ − V 2 , v = ψ + 1 2 sin(2 θ ) ψ θ + sin 2 ( θ ) xψ x + V 2 . (3.8) (vi) V = Ψ + 1 2 x Ψ x + 1 2 cos(2 θ ) x Ψ x − 1 2 sin(2 θ )Ψ θ − G 2 , g = ψ − 1 2 sin(2 θ ) ψ θ + cos 2 ( θ ) xψ x + G 2 . (3.9) Up till no w, ev erything is exact. The symbols ( U 1 , V 1 , G 1 , V 2 , G 2 ) are just separation functions that allow us to split 5 equations into 10. Now w e make a bra v e assumption : w e define ( U 1 , V 1 , G 1 , V 2 , G 2 ) in suc h a wa y that the red p ortion in the first equation of eac h b ox v anishes. 0 = cos(2 θ ) V xU x − sin(2 θ ) V U θ − U 1 , 0 = cos(2 θ ) V xV x − sin(2 θ ) V V θ − tan( θ ) 1 x 2 P θ − V 1 , 0 = cos(2 θ ) V xV x − sin(2 θ ) V V θ − cot( θ ) 5 3 x 2 P θ − G 1 , 0 = − 1 2 cos(2 θ ) x Ψ x + 1 2 sin(2 θ ) Ψ θ − V 2 , 0 = 1 2 cos(2 θ ) x Ψ x − 1 2 sin(2 θ ) Ψ θ − G 2 , Note: V 2 + G 2 = 0 . (3.10) 3.2. Bac kground solutions for ( U, V , P , Ψ) . Th us we ha v e selected four con v ection-free PDEs for four background v ariables { U, V , P , Ψ } ( t, x ).                3 2 U t = 1 2 V U, 3 2 V t = V 2 − U 2 + 1 x P x , 3 2 V t = − V 2 − 5 3 x P x , V = Ψ + 1 2 x Ψ x . (3.11) whic h can b e simplified as                3 2 U t = 1 2 V U, 3 2 V t = 1 8  2 V 2 − 5 U 2  , P x = 3 8 x ( U 2 − 2 V 2 ) , V = Ψ + 1 2 x Ψ x . (3.12) ST ABLE FINITE-TIME BLO W-UP FOR SYSTEM (E1) 9 3.3. Definition (weigh ted div ergence-form angular closure for Ψ ). Fix the angu- lar w eight w ( θ ) := sin 2 θ cos 2 θ = 1 4 sin 2 (2 θ ) , and define the symmetric (weigh ted) angular op erator L θ f := 1 w ( θ ) ∂ θ  w ( θ ) ∂ θ f  , θ ∈ h − π 4 , π 4 i , (3.13) in terpreted in the w eak sense at θ = 0 (where w (0) = 0) with the natural zero-flux con- dition inherited from the weigh t. F or eac h ( t, x ), since V = Ψ + 1 2 x Ψ x of (3.12) 4 only sp ecifies the c hange of Ψ( t, x, θ ) in x direction, w e provide the closure for c hange in θ -direction b y defining θ -even Ψ( t, x, θ ) as the unique solution of the Newman-type b oundary-v alue problem ( L θ Ψ( t, x, θ ) = 4  Ψ( t, x, θ ) − V ( t, x, θ )  , θ ∈  − π 4 , π 4  , Ψ θ ( t, x, ± π / 4) = 0 , (3.14) Remark 3.1. The closur e (3.14) is a gauge choic e for the auxiliary p otential Ψ . It do es not alter the b ackgr ound fields ( U, V ) in (3.12) nor their blow-up at t = T = 6 / A . Substitution of (3.13) leads to Ψ θθ = 4  Ψ − V − Ψ θ cot(2 θ )  (3.15) F urther substitution of (3.12) 4 leads to Ψ θθ + 4Ψ θ cot(2 θ ) + 2 x Ψ x = 0 . (3.16) No w we define new indep enden t v ariables τ := − log x, ξ := cos( θ ) and dep endent v ariable f ( t, τ , ξ ) := Ψ( t, x, θ ), then                f τ = ν ( ξ )  f ξ ξ + α ( ξ ) ξ f ξ  = ν ( ξ ) ξ α ( ξ ) ∂ ξ  ξ α ( ξ ) f ξ  , τ ∈ R , ξ ∈ h 1 √ 2 , 1 i , f ξ   ξ =1 = f ξ   ξ = 1 √ 2 = 0 , ν ( ξ ) := 1 2  1 − ξ 2  ≥ 0 , α ( ξ ) := 2 − 5 ξ 2 1 − ξ 2 . (3.17) Equation (3.17) is a (degenerate) linear parab olic equation in the log–radius v ariable τ = − log x with diffusion in the angular v ariable ξ = cos θ . W e also sp ecified the Newman b oundary conditions b oth at ridge top ξ = 1 √ 2 (or θ = π 4 ) and at the v alley bottom ξ = 1 (or θ = 0) (i.e. the center line of the w edge). Lemma 3.2 (W ell-p osedness of the w eigh ted angular closure) . Fix t ∈ [0 , T ) and supp ose f ( t, τ , ξ ) satisfies (3.17) with Neumann c onditions at ξ = 1 / √ 2 and ξ = 1 . Given an “initial” pr ofile at some τ 0 ∈ R , f ( t, τ 0 , · ) = f in ( · ) ∈ H 1   1 √ 2 , 1   , ther e exists a unique we ak solution f ( t, · , · ) ∈ C  [ τ 0 , ∞ ); L 2  ∩ L 2 loc  [ τ 0 , ∞ ); H 1  to (3.17) on [ τ 0 , ∞ ) × [1 / √ 2 , 1] , and it satisfies the ener gy identity 1 2 d dτ Z 1 1 / √ 2 | f ( t, τ , ξ ) | 2 dξ + Z 1 1 / √ 2 ν ( ξ ) | f ξ ( t, τ , ξ ) | 2 ξ α ( ξ ) dξ = 0 , interpr ete d with the natur al no-flux b oundary c onditions. 10 Y AOMING SHI Pr o of. W rite (3.17) in divergence form f τ = ∂ ξ  a ( ξ ) f ξ  with a ( ξ ) := ν ( ξ ) ξ α ( ξ ) ≥ 0 on [1 / √ 2 , 1], and imp ose Neumann/no-flux conditions a ( ξ ) f ξ | ∂ = 0, whic h are equiv alen t to f ξ | ∂ = 0 since a > 0 on (1 / √ 2 , 1). On an y truncated interv al [1 / √ 2 , 1 − δ ] the co efficien t a is smo oth and uniformly elliptic, so standard Galerkin or semigroup theory for linear parabolic equations in divergence form yields existence/uniqueness and the energy identit y . Letting δ ↓ 0 giv es the claimed weak solution on [1 / √ 2 , 1]; the degeneracy ν (1) = 0 is weak and pro duces no b oundary flux, consistent with the Neumann condition at ξ = 1. □ 3.4. P erturbation equations. W e are left with 5 p erturbation PDEs (with external force coming from the background) for v ariables { u, v , g , p, ψ } ( t, x, θ ). 3 2 u t = cos(2 θ ) V xu x − sin(2 θ ) V u θ + U 1 , + ( v cos 2 θ − g sin 2 θ ) xU x − 1 2 sin(2 θ )( v + g ) U θ + 1 2 ( V u + v U ) + N 1 ( u, v , g ) 3 2 v t = cos(2 θ ) V xv x − sin(2 θ ) V v θ + 1 x p x − tan( θ ) 1 x 2 p θ + V 1 , + ( v cos 2 θ − g sin 2 θ ) xV x − 1 2 sin(2 θ )( v + g ) V θ + 2( v V − uU ) + N 2 ( u, v , g ) 3 2 g t = cos(2 θ ) V xg x − sin(2 θ ) V g θ − 5 3 x p x − cot( θ ) 5 3 x 2 p θ + G 1 , + ( v cos 2 θ − g sin 2 θ ) xV x − 1 2 sin(2 θ )( v + g ) V θ − 2 g V + N 3 ( u, v , g ) v = ψ + 1 2 sin(2 θ ) ψ θ + sin 2 ( θ ) xψ x + V 2 . g = ψ − 1 2 sin(2 θ ) ψ θ + cos 2 ( θ ) xψ x + G 2 . (3.18) 3.5. Getting of ( p θ , p x ) . Our next step is to get rid of ( p θ , p x ). Define (Ω , ω ) as ω : = − 1 3 x (5 v x + 3 g x ) − 1 3 x 2 (5 v θ cot θ − 3 g θ tan θ ) , Ω : = − 8 3 x V x − 1 3 x 2 (5 cot θ − 3 tan θ ) V θ . (3.19) So w e obtain g x = − xω − 5 3 v x − 1 3 x (5 v θ cot θ − 3 g θ tan θ ) , V x = − 3 8 x Ω − 1 8 x (5 cot θ − 3 tan θ ) V θ . (3.20) No w we rewrite (3.18) 2 , 3 as: ( 3 2 v t = A + N 2 + V 1 + 1 x p x − tan( θ ) 1 x 2 p θ , 3 2 g t = B + N 3 + G 1 − 5 3 x p x − cot( θ ) 5 3 x 2 p θ , (3.21) where ( A, B ) are defined as                      A : = cos(2 θ ) V xv x − sin(2 θ ) V v θ , + ( v cos 2 θ − g sin 2 θ ) xV x − 1 2 sin(2 θ )( v + g ) V θ + 2( v V − uU ) B : = cos(2 θ ) V xg x − sin(2 θ ) V g θ , + ( v cos 2 θ − g sin 2 θ ) xV x − 1 2 sin(2 θ )( v + g ) V θ − 2 g V . (3.22) ST ABLE FINITE-TIME BLO W-UP FOR SYSTEM (E1) 11 The solutions for ( p x , p θ ) then b ecome              p x = 3 2 x  v t cos 2 θ − g t sin 2 θ  + x 10  6( B + N 3 + G 1 ) sin 2 θ − 10( A + N 2 + V 1 ) cos 2 θ  , p θ = − 3 20 x 2 sin(2 θ ) (5 v t + 3 g t ) + x 2 20 (6( B + N 3 + G 1 ) + 10( A + N 2 + V 1 )) . (3.23) Using p xθ = p θ to get rid of p , and using g xt of (3.20) to simplify the result, w e finally obtain:                      3 2 ω t = L 2 + M 2 + Q 2 L 2 = − 1 3 x  5 A x + 3 B x  − 1 3 x 2  5 cot θ A θ − 3 tan θ B θ  M 2 = − 1 3 x  5 N 2 x + 3 N 3 x  − 1 3 x 2  5 cot θ N 2 θ − 3 tan θ N 3 θ  , Q 2 = − 1 3 x  5 V 1 x + 3 G 1 x  − 1 3 x 2  5 cot θ V 1 θ − 3 tan θ G 1 θ  . (3.24) Substituting ( A, B ) of (3.22) into (3.24)(1) and using ( g x , g xx , g xθ , V x , V xx , V xθ ) of (3.20) to simplify the results, we obtain                            L 2 = 10 3 x 2  U ( xu x + u θ cot θ ) + u ( xU x + U θ cot θ )  + cos(2 θ ) V xω x − sin(2 θ ) V ω θ − 1 4 csc(2 θ )(4 + cos(2 θ )) ω V θ + 1 8 x 2 (4 − 7 cos(2 θ )) ω Ω + 1 3 x 2 ( v + g )  sin(2 θ )4 xV xθ + (4 cos(2 θ ) + 1) V θθ − (4 + cos(2 θ )) csc θ sec θ V θ  + ( v cos 2 θ − g sin 2 θ ) x Ω x + 1 3 (4 − cos(2 θ )) Ω xv x + 1 3 Ω  6( v − g ) + v θ sin(2 θ ) − 3 g θ tan θ  (3.25) Substituting ( V 1 , G 1 ) of (3.10) in to (3.24)(2) and using ( V x , V xx , V xθ ) of (3.20) and P xθ of (3.12)(3) to simplify the results, we obtain          Q 2 = 5 8 x 2 ( U 2 ) θ csc θ sec θ − 3 8 x 2 Ω 2 cos(2 θ ) − 1 8 Ω  4 V + V θ (4 + cos(2 θ )) csc θ sec θ  + V  cos(2 θ ) x Ω x − sin(2 θ )Ω θ  (3.26) Substituting N 2 of (3.6), N 3 of (3.7) into (3.24)(3) and using ( g x , g xx , g xθ of (3.20) to simplify the results, we obtain              M 2 = 10 3 x 2 u ( xu x + u θ cot θ ) + 2 ω ( v − g ) + x 2 ω 2 sin 2 θ + ( v cos 2 θ − g sin 2 θ ) xω x − 1 2 sin(2 θ )( g + v ) ω θ + 1 3 ω  − 3 g θ tan θ + v θ sin(2 θ ) + xv x (4 − cos(2 θ ))  . (3.27) 12 Y AOMING SHI The equation for u t in (3.18)(1) is already in desired form:                    3 2 u t = L 1 + M 1 + Q 1 , L 1 = 1 2 ( V u + v U ) + cos(2 θ ) V xu x − sin(2 θ ) V u θ + ( v cos 2 θ − g sin 2 θ ) xU x − 1 2 sin(2 θ )( v + g ) U θ M 1 = N 1 = 1 2 uv + ( v cos 2 θ − g sin 2 θ ) xu x − 1 2 sin(2 θ )( v + g ) u θ , Q 1 = U 1 = cos(2 θ ) V xU x − sin(2 θ ) U θ V . (3.28) 3.6. Get rid of V 2 , G 2 . W e group the iden tities (3.19), (3.8), (3.9), and (3.10) together b elo w:            ω = − 1 3 x (5 v x + 3 g x ) − 1 3 x 2 (5 v θ cot θ − 3 g θ tan θ ) , v = ψ + 1 2 sin(2 θ ) ψ θ + sin 2 ( θ ) xψ x + V 2 , g = ψ − 1 2 sin(2 θ ) ψ θ + cos 2 ( θ ) xψ x − V 2 , V 2 = − 1 2 cos(2 θ ) x Ψ x + 1 2 sin(2 θ )Ψ θ . (3.29) Our goal is to separate V 2 from the lo w er case v ariables suc h that all lo w er case v ariables do not con tain upp er case bac kground fields ( U, V , Ω , Ψ). Th us we define new p erturbation v ariables ( ˜ v , ˜ g , ˜ ω ) via:              ˜ v : = v − V 2 = ψ + 1 2 sin(2 θ ) ψ θ + sin 2 ( θ ) xψ x , ˜ g : = g + V 2 = ψ − 1 2 sin(2 θ ) ψ θ + cos 2 ( θ ) xψ x , Ω 2 : = − 1 3 x (5 V 2 x + 3 G 2 x ) − 1 3 x 2 (5 V 2 θ cot θ − 3 G 2 θ tan θ ) , ˜ ω : = ω − Ω 2 = − 1 3 x (5 ˜ v x + 3 ˜ g x ) − 1 3 x 2 (5 ˜ v θ cot θ − 3 ˜ g θ tan θ ) . (3.30) Substitution of (3.30) 1 , 2 in to (3.30)(4) leads to ˜ ω : = ∆ ψ , (3.31) where                      ∆ = c 1 ( θ ) ∂ xx + c 2 ( θ ) 1 x ∂ x + c 3 ( θ ) 1 x 2 ∂ θ + c 4 ( θ ) 1 x ∂ xθ + c 5 ( θ ) 1 x 2 ∂ θθ c 1 ( θ ) = − 1 3 (4 − cos(2 θ )) , c 2 ( θ ) = − 1 3 (20 + cos(2 θ )) , c 3 ( θ ) = − 1 6 csc( θ ) sec( θ ) [3 + 16 cos(2 θ ) + cos(4 θ )] , c 4 ( θ ) = − 2 3 sin(2 θ ) , c 5 ( θ ) = − 1 3 (4 + cos(2 θ )) , (3.32) W e found out that the follo wing 3 combinations of ( v , g ):                v + g = 2 ψ + xψ x , v − g = ψ θ sin(2 θ ) − xψ x cos(2 θ ) − Ψ θ sin(2 θ ) + x Ψ x cos(2 θ ) , v cos 2 θ − g sin 2 θ = ψ cos(2 θ ) + 1 2 ψ θ sin(2 θ ) + 1 2 x Ψ x cos(2 θ ) − 1 2 Ψ θ sin(2 θ ) . (3.33) ST ABLE FINITE-TIME BLO W-UP FOR SYSTEM (E1) 13 F or u t equation (3.28), w e obtain:                              3 2 u t = L 1 + M 1 + Q 1 , L 1 = 1 2 V u + V cos(2 θ ) xu x − V sin(2 θ ) u θ + 1 2 U  ψ + 1 2 sin(2 θ ) ψ θ + sin 2 ( θ ) xψ x  + xU x  ψ cos(2 θ ) + 1 2 ψ θ sin(2 θ )  − 1 2 U θ sin(2 θ )(2 ψ + xψ x ) , + 1 4 u  sin(2 θ )Ψ θ − cos(2 θ ) x Ψ x  +  ψ + 1 2 sin(2 θ ) ψ θ + sin 2 ( θ ) xψ x  xu x , (3.34)                                M 1 = 1 2 u  ψ + 1 2 sin(2 θ ) ψ θ + sin 2 ( θ ) xψ x  +  ψ cos(2 θ ) + 1 2 ψ θ sin(2 θ )  xu x − 1 2 sin(2 θ )(2 ψ + xψ x ) u θ , Q 1 = V cos(2 θ ) xU x − V sin(2 θ ) U θ , − 1 2 U  cos(2 θ )( V − Ψ) − 1 2 sin(2 θ )Ψ θ  + xU x  cos(2 θ )( V − Ψ) − 1 2 sin(2 θ )Ψ θ  . (3.35) Remark 3.3. The individual term in Q 1 either c ontains a factor cos(2 θ ) or o dd derivative ( V θ , U θ , Ψ θ ) , which wil l vanish on the ridge θ = ± π 4 b e c ause ( V , U, Ψ) ar e flat ther e. Substituting V 2 of (3.10)(4) in to the definition of Ω 2 , and using (3.15) to to get rid of Ψ θθ , Ψ θθ θ , w e obtain:          3 2 Ω 2 = − 2 x 2 cos 2 (2 θ ) csc(2 θ ) V θ − 1 x 2 cos(2 θ )  xV x + 4 csc(2 θ ) V θ − 2 V + 2Ψ  + 1 x 2  csc(2 θ ) V θ − sin(2 θ )Ψ θ  (3.36) And ∂ t Ω 2 b ecomes the forcing term in the PDE for ˜ ω t b ecause ˜ ω t = ω t − ∂ t Ω 2 :                − 3 2 ∂ t Ω 2 = Q 2 d , Q 2 d = 2 x 2 cos 2 (2 θ ) csc(2 θ ) V tθ + 1 x 2 cos(2 θ )  xV tx + 4 csc(2 θ ) V tθ − 2 V t + 2Ψ t  − 1 x 2  csc(2 θ ) V tθ − sin(2 θ )Ψ tθ  . (3.37) Substituting ( ω , v , g , V 2 ) of (3.29) into (3.25), (3.26), (3.27), and using (3.15) to get rid of Ψ θθ , Ψ xθθ , Ψ θθ θ , w e finally obtain:      3 2 ˜ ω t = L 2 a + L 2 b + M 2 + Q 2 a + Q 2 b − 3 2 ∂ t Ω 2 = L 2 a + L 2 b + M 2 + Q 2 , Q 2 : = Q 2 a + Q 2 b + Q 2 d , (3.38) 14 Y AOMING SHI where the quadratic p erturbation term ( M 2 ( f , f ) , f ∈ [ u, ω , ψ ]) is giv en by                    M 2 = 10 2 x 2 u  xu x + cot( θ ) u θ  + ˜ ω  − 2 cos(2 θ ) xψ x + sin(2 θ ) ψ θ  + x ˜ ω x  cos(2 θ ) ψ + 1 2 sin(2 θ ) ψ θ  − ˜ ω θ sin(2 θ )  ψ + 1 2 xψ x  , (3.39) And the linear p erturbation terms ( L 2 a ( f , F ) , L 2 a ( f , F )) ( f ∈ [ u, ˜ ω , ψ ] , F ∈ [ U, V , Ψ]) are giv en by                        L 2 a = 10 3 x 2 u  xU x + cot( θ ) U θ  + 5 6 x 2  4 xu x + 4 cot( θ ) u θ  U + ˜ ω  cos(2 θ )  2 V − 2Ψ + 2 xV x  − sin(2 θ )(2 V + Ψ) θ  + x ˜ ω x  cos(2 θ )  2 V − Ψ  − 1 2 sin(2 θ )Ψ θ  − ˜ ω θ sin(2 θ ) V , (3.40)                                  L 2 b = 1 x 2 ψ  a 1 ( θ ) V + a 2 ( θ ) V θ + a 3 ( θ ) xV x + a 4 ( θ ) xV xθ + a 5 ( θ ) x 2 V xx  , + 1 x 2 ψ  b 1 ( θ )Ψ + b 2 ( θ )Ψ θ  + 1 x ψ x  a 6 ( θ ) V + a 7 ( θ ) V θ + a 8 ( θ ) xV x + a 9 ( θ ) xV xθ  + 1 x ψ x  b 3 ( θ )Ψ + b 4 ( θ )Ψ θ  , + 1 x 2 ψ θ  a 11 ( θ ) V + a 12 ( θ ) V θ + a 13 ( θ ) xV x + + a 14 ( θ ) xV xθ + a 15 ( θ ) x 2 V xx  + 1 x 2 ψ θ  b 5 ( θ )Ψ + b 6 ( θ )Ψ θ  , (3.41)                                    a 1 ( θ ) = − 16 3 cos 2 (2 θ ) , a 2 ( θ ) = − 2 3 csc(2 θ )  24 + 9 cos(2 θ ) + 8 cos(4 θ ) − cos(6 θ )  , a 3 ( θ ) = − 1 3  3 + 24 cos(2 θ ) − cos(4 θ )  , a 4 ( θ ) = − 1 6 csc(2 θ )  8 + 5 cos(2 θ ) + 24 cos(4 θ ) + 3 cos(6 θ )  , a 5 ( θ ) = − 2 3 cos(2 θ )  4 + cos(2 θ )  , b 1 ( θ ) = 16 3 cos 2 (2 θ ) , b 2 ( θ ) = 4 3 sin(4 θ ) . (3.42) ST ABLE FINITE-TIME BLO W-UP FOR SYSTEM (E1) 15                              a 6 ( θ ) = − 4 3 cos 2 (2 θ ) , a 7 ( θ ) = − 1 6  40 cot( θ ) + 24 tan( θ ) − 32 sin(2 θ ) + 2 sin(4 θ )  , a 8 ( θ ) = − 2 3  1 + 4 cos(2 θ )  , a 9 ( θ ) = 1 6  8 sin(2 θ ) + sin(4 θ )  , b 3 ( θ ) = 4 3 cos 2 (2 θ ) , b 4 ( θ ) = 1 3 sin(4 θ ) . (3.43)                                    a 11 ( θ ) = − 4 3 sin(4 θ ) , a 12 ( θ ) = − 2 3 sin 2 (2 θ ) , a 13 ( θ ) = − 1 6  8 sin(2 θ ) − sin(4 θ )  , a 14 ( θ ) = − 2 3 cos(2 θ )  4 + cos(2 θ )  , a 15 ( θ ) = − 1 6  8 sin(2 θ ) + sin(4 θ )  , b 5 ( θ ) = 2 3 sin(4 θ ) , b 6 ( θ ) = 2 3 sin 2 (2 θ ) . (3.44) The quadratic bac kgroud forcing terms ( Q 2 a ( F , F ) , Q 2 b ( F , F )) ( F ∈ [ U, V , Ψ]) are given b y                Q 2 a = 5 4 x 2 csc(2 θ )( U 2 ) θ + 4 3 x 2  1 + 8 cos(2 θ ) + cos(4 θ )  ( V θ ) 2 − 2 3 x 2 sin 2 (2 θ )(Ψ θ ) 2 + 2 x 2 sin 2 (2 θ ) V θ Ψ θ + 1 x 2 V θ  h 1 ( θ )Ψ + h 2 ( θ ) V + h 3 ( θ ) xV x  + 1 x 2 Ψ θ  e 1 ( θ )Ψ + e 2 ( θ ) V + e 3 ( θ ) xV x + e 4 ( θ ) xV xθ + e 5 ( θ ) x 2 V xx  . (3.45)                                            h 1 ( θ ) = 2 3 x 2 sin(4 θ ) , h 2 ( θ ) = − 1 3 x 2 csc(2 θ )  47 + 17 cos(2 θ ) + 16 cos(4 θ ) − 5 cos(6 θ )  , h 3 ( θ ) = − 1 3 csc(2 θ )  8 + 5 cos(2 θ ) + 24 cos(4 θ ) + 3 cos(6 θ )  , e 1 ( θ ) = − 4 3 x 2 sin(4 θ ) , e 2 ( θ ) = 8 3 x 2 sin(4 θ ) , e 3 ( θ ) = 1 6 x  8 sin(2 θ ) − 5 sin(4 θ )  , e 4 ( θ ) = 2 3 x cos(2 θ )  4 + cos(4 θ )  , e 5 ( θ ) = 1 3 sin(2 θ )  4 + cos(2 θ )  . (3.46) 16 Y AOMING SHI                                      Q 2 b = 2 3 x cos 3 (2 θ ) csc(2 θ ) (2Ψ − 5 V ) V xθ + 1 3 x 2 cos 2 (2 θ )  − 30 V 2 + 40 V Ψ − 8Ψ 2  + 1 3 x 2 cos 2 (2 θ ) (15 V − 10Ψ) xV x + 1 3 x 2 cos 2 (2 θ ) csc(2 θ ) ( − 40 V + 16Ψ) xV xθ + 1 3 x 2 cos 2 (2 θ )  − 4( xV x ) 2 − 4 V x 2 V xx + 2Ψ x 2 V xx  + 2 3 x 2 cos(2 θ ) V  − 16 xV x − 8 x 2 V xx + csc(2 θ ) xV xθ  + 2 3 x 2 cos(2 θ )  Ψ  xV x + x 2 V xx  − 8( xV x ) 2  + 8 3 x 2 csc(2 θ ) xV xθ (3.47) Remark 3.4. The individual term in Q 2 := ( Q 2 a (3.45) + Q 2 b (3.47) + Q 2 d (3.37)) ei- ther c ontains a factor cos(2 θ ) or o dd derivative ( V θ , U θ , Ψ θ ) , which wil l vanish on the ridge θ = ± π 4 b e c ause ( V , U, Ψ) ar e flat ther e. 4. Explicit back gr ound, per turba tion PDEs, and energy bounds 4.1. Bac kground Solutions. The bac kground solutions V ( t, x, θ ) , U ( t, x, θ ) are updated with θ -dep endent initial profiles V 0 ( x, θ ) , U 0 ( x, θ ) (in tro ducing A 1 > 0 as a tuning param- eter):              t ∈ [0 , T ) , x ∈ ( −∞ , ∞ ) , θ ∈  − π 4 , π 4  , V ( t, x, θ ) = − 6 (5 tU 2 0 ( x, θ ) + 2 V 0 ( x, θ )( tV 0 ( x, θ ) − 6)) 2( tV 0 ( x, θ ) − 6) 2 + 5 t 2 U 2 0 ( x, θ ) , (2.17) U ( t, x, θ ) = 72 U 0 ( x, θ ) 2( tV 0 ( x, θ ) − 6) 2 + 5 t 2 U 2 0 ( x, θ ) , (4.1) Initial conditions r := x 2 + A 1 (1 − 2 sin 2 θ ) 2 ≥ 0 , V 0 ( x, θ ) = A 1 + r 4 , U 0 ( x, θ ) = B r 4  1 + r 4  2 . (4.2) 4.2. Initial energy and finiteness. W e record the “initial energy” (at t = 0) associated with the bac kground profile: E (0) := Z π / 4 − π / 4 Z ∞ −∞ x 2 h ( V 0 ( x, θ )) 2 + U 2 0 ( x, θ ) i | x | dx dθ . (4.3) Remark 4.1 (Cho osing new seeds to make E ( t ) b ounded up to t = T ) . In the mo difie d b ackgr ound mo del (with U 2 in the V –e quation) we take the ener gy density V 2 + U 2 . T o eliminate the b or derline lo garithmic gr owth as t ↑ T it is not enough to incr e ase the far- field de c ay exp onent: one must impr ove the near-ridge flatness of V 0 and the vanishing or der of U 0 at the ridge p oint. We now fix the se e d data as in (4.2) : r := x 2 + A 1 (1 − 2 sin 2 θ ) 2 , V 0 ( x, θ ) = A 1 + r 4 , U 0 ( x, θ ) = B r 4 (1 + r 4 ) 2 . Then V 0 (0 , ± π / 4) = A and U 0 (0 , ± π / 4) = 0 , so the blow-up time r emains T = 6 / A . A T aylor exp ansion at r = 0 gives V 0 ( x, θ ) = A  1 − r 4 + O ( r 8 )  , U 0 ( x, θ ) = B r 4 + O ( r 8 ) , ST ABLE FINITE-TIME BLO W-UP FOR SYSTEM (E1) 17 so the ridge flatness is now quartic and U 0 vanishes to or der 4 at the ridge. 4.3. Energy along the background flo w and b eha vior as t ↑ T . F or 0 < t < T we define the time–dep endent energy of the bac kground solution E ( t ) := Z π / 4 − π / 4 Z ∞ −∞ x 2 h ( V ( t, x, θ )) 2 + ( U ( t, x, θ )) 2 i | x | dx dθ . (4.4) Using (4.1) one has the p oin twise bounds | V ( t, x, θ ) | ≲ | tV 0 ( x, θ ) − 6 | + t U 0 ( x, θ ) 2 ( tV 0 ( x, θ ) − 6) 2 + t 2 U 0 ( x, θ ) 2 , | U ( t, x, θ ) | ≲ U 0 ( x, θ ) ( tV 0 ( x, θ ) − 6) 2 + t 2 U 0 ( x, θ ) 2 , (4.5) with implicit constan ts dep ending only on universal n umerical factors. In particular, a w ay from the resonan t set { tV 0 = 6 , U 0 = 0 } the denominator is b ounded from b elow and V , U are uniformly b ounded. Where can the bac kground blo w up? With the present choice of V 0 , U 0 and with T := 6 / A , note that V 0 (0 , ± π / 4) = A and U 0 (0 , ± π / 4) = 0 (b ecause (1 − 2 sin 2 ( ± π / 4)) = 0 and the numerator of U 0 con tains x 2 ). Hence tV 0 − 6 v anishes at ( x, θ ) = (0 , ± π / 4) precisely when t = T , and the p oten tial singularity is confined to these tw o “corner” p oin ts. Everywhere else, either tV 0 − 6 sta ys aw a y from 0 or U 0 > 0 k eeps the denominator in (4.1) strictly p ositive. Lo cal asymptotics near ( x, θ ) = (0 , π / 4) . W rite θ ∗ := π / 4 and η := θ − θ ∗ . Then K ( θ ) = A 1 cos 2 (2 θ ) = A 1 cos 2 ( π / 2 + 2 η ) = A 1 sin 2 (2 η ) = 4 A 1 η 2 + O ( η 4 ) , and set r := x 2 + K ( θ ) ∼ x 2 + 4 A 1 η 2 . A T a ylor expansion (using (4.2)) gives V 0 ( x, θ ) = A 1 + r 4 = A  1 − r 4 + O ( r 8 )  , U 0 ( x, θ ) = B r 4 (1 + r 4 ) 2 = B r 4 + O ( r 8 ) . Let τ := T − t ↓ 0 with T = 6 / A . Since tA = 6 − Aτ , we obtain tV 0 ( x, θ ) − 6 = (6 − Aτ )  1 − r 4 + O ( r 8 )  − 6 = − Aτ − 6 r 4 + O ( τ r 4 + r 8 ) . Consequen tly , the main denominator in (4.1), D ( t, x, θ ) := 2( tV 0 − 6) 2 + 5 t 2 U 2 0 , satisfies the near-ridge low er bound D ( t, x, θ ) ≳ ( Aτ + 6 r 4 ) 2 + r 8 ∼ ( τ + r 4 ) 2 on N := { ( x, θ ) : r ≪ 1 , | θ ∓ π / 4 | ≪ 1 } . (4.6) (Here w e used U 2 0 ∼ B 2 r 8 lo cally .) Ev aluating V at r = 0 sho ws the exp ected blow–up rate V ( t, 0 , θ ∗ ) ∼ 6 T − t = 6 τ , but this o ccurs at a single p oint in ( x, θ ) and is strongly suppressed in the energy by the w eigh t x 2 | x | . Uniform b oundedness of E ( t ) up to T . Split the in tegral in (4.4) in to a neigh b orho o d N of (0 , θ ∗ ) (and similarly of (0 , − θ ∗ )) and its complemen t. On N c , (4.5) giv es | V | + | U | ≲ 1, hence Z N c x 2  V 2 + U 2  | x | dx dθ ≲ Z N c | x | 3 dx dθ < ∞ , uniformly for 0 < t < T . 18 Y AOMING SHI On N , use (4.6). F or the most singular con tribution, note that for r ≲ τ one has ( Aτ + 12 r ) 2 + r 2 ≳ τ 2 , hence | V | ≲ τ − 1 and | U | ≲ τ − 2 U 0 ≲ τ − 2 r , so Z { r ≲ τ } x 2 V 2 | x | dx dθ ≲ τ − 2 Z {| x | ≲ τ 1 / 2 } Z | η | ≲ τ 1 / 2 | x | 3 dx dη ≲ τ − 2 · τ 2 · τ 1 / 2 = O ( τ 1 / 2 ) , and similarly , using U 2 ≲ τ − 4 r 2 ≲ τ − 2 on { r ≲ τ } , Z { r ≲ τ } x 2 U 2 | x | dx dθ ≲ τ − 2 Z {| x | ≲ τ 1 / 2 } Z | η | ≲ τ 1 / 2 | x | 3 dx dη = O ( τ 1 / 2 ) . F or the intermediate region τ ≲ r ≪ 1, the low er bound (4.6) yields ( Aτ + 12 r ) 2 + r 2 ≳ r 2 , so | V | ≲ r − 1 (since the numerator is ≲ r ) and | U | ≲ r − 1 (since U 0 ∼ r ). Consequently x 2 V 2 | x | ≲ | x | 3 r − 2 , x 2 U 2 | x | ≲ | x | 3 r − 2 . In the local co ordinates ( x, η ) with η = θ − π / 4 and r ∼ x 2 + c θ η 2 (for some c θ > 0), the in tegral of | x | 3 r − 2 o v er { τ ≲ r 2 ≪ 1 } is uniformly b ounded (no logarithmic loss). Indeed, setting ( x, η ) = ( ρ cos ϕ, ρ sin ϕ ) yields r ∼ ρ 2 and hence | x | 3 r − 2 dx dη ∼ ρ 3 | cos ϕ | 3 ρ 4 ρ dρ dϕ = | cos ϕ | 3 dρ dϕ, whic h is integrable at ρ = 0 and therefore contributes O (1) uniformly as τ ↓ 0. Putting the pieces together, we conclude that for the mo dified backgr ound with U 2 , E ( t ) < ∞ uniformly for ev ery t ∈ [0 , T ] . The same analysis holds near (0 , − π / 4), completing the claim. 4.4. Up dated Linear PDEs and Nonlinear T erms. The linear-nonlinear PDE sys- tem for p erturbations ( u, ˜ ω , ψ ) is:      3 2 u t = L 1 + L 1 B + M 1 + Q 1 , (3.35) 3 2 ˜ ω t = L 2 a + L 2 b + M 2 + Q 2 , (3.38) ˜ ω = ∆ ψ . (3.31) (4.7) where 4.5. Co efficient F unctions. The Laplacian ∆ in (4.7) 3 is defined as:                      ∆ = c 1 ( θ ) ∂ xx + c 2 ( θ ) 1 x ∂ x + c 3 ( θ ) 1 x 2 ∂ θ + c 4 ( θ ) 1 x ∂ xθ + c 5 ( θ ) 1 x 2 ∂ θθ c 1 ( θ ) = − 1 3 (4 − cos(2 θ )) , c 2 ( θ ) = − 1 3 (20 + cos(2 θ )) , c 3 ( θ ) = − 1 6 csc( θ ) sec( θ ) [3 + 16 cos(2 θ ) + cos(4 θ )] , c 4 ( θ ) = − 2 3 sin(2 θ ) , c 5 ( θ ) = − 1 3 (4 + cos(2 θ )) , (4.8) Remark 4.2. The app ar ent singular factor csc θ sec θ in c 3 ( θ ) is absorb e d by the weighte d derivative D θ = (sin θ cos θ ) − 1 ∂ θ intr o duc e d ab ove: writing c 3 ( θ ) = e c 3 ( θ ) csc θ sec θ, e c 3 ( θ ) := − 1 5  3 + 16 cos(2 θ ) + cos(4 θ )  , we have c 3 ( θ ) ∂ θ = e c 3 ( θ ) D θ with b ounde d smo oth e c 3 on [ − π / 4 , π / 4] . Likewise, terms involving cot θ ar e tr e ate d as b ounde d multipliers in the weighte d ener gy onc e expr esse d in terms of D θ (or plac e d into diver genc e form in θ ). ST ABLE FINITE-TIME BLO W-UP FOR SYSTEM (E1) 19 All c i ( θ ) are b ounded for θ ∈ [ − π / 4 , π / 4] (no singularities in this in terv al, as csc θ sec θ = 1 / (sin θ cos θ ) is b ounded a w a y from 0 here). 4.6. Initial conditions and b oundary conditions. Boundary conditions (for θ -edges and x → ±∞ ) and initial conditions (no w θ -dep endent with fast deca y) are:                u ( t, x, ± π / 4) = 0 , u ( t, x, θ ) → 0 as | x | → ∞ , ψ ( t, x, ± π / 4) = 0 , ψ ( t, x, θ ) → 0 as | x | → ∞ , u (0 , x, θ ) is even in x and θ , ψ (0 , x, θ ) is ev en in x and θ , u (0 , x, θ ) , ψ (0 , x, θ ) deca y sufficiently fast as θ → ± π / 4 . (4.9) The ”sufficien tly fast decay” as θ → ± π / 4 ensures u (0 , x, θ ) , ψ (0 , x, θ ) ∈ L ∞ ∩ L 2 w (w eigh ted L 2 -space, defined later) and their deriv ativ es v anish at the θ -edges, simplifying in tegration by parts. 5. Linear and nonlinear st ability up to blo w-up time Up dated p erturbation system. Throughout this section we w ork with the final p er- turbation equations derived in Section 2. In particular, under the P ath A weigh ted div ergence-form closure for Ψ, the w eighted div ergence-form angular closure for Ψ (Sec- tion 3.3) implies algebraic iden tities that eliminate the previously problematic bac kground con tributions: in particular, the terms lab eled Q 2 c and Q 2 e cancel after substituting the closure relation, so the remaining bac kground forcing is captured entirely by Q 1 and Q 2 . up to the b ackgr ound blow-up time T for the p erturbation system (4.7) together with the coefficient collections (3.28) and the elliptic op erator (4.8), around the explicit back- ground (4.1). Throughout, all Leb esgue/Sob olev norms are tak en with resp ect to the w eigh ted measure dµ w = w ( θ ) | x | dx dθ in tro duced in the Notation and weighte d norms subsection, and w e use the desingularized angular deriv ative D θ := (sin θ cos θ ) − 1 ∂ θ . 5.1. Bo otstrap framew ork and up dated background co efficient b ounds. Fix an in teger k ≥ 6. Define the p erturbation energy E k ( t ) := X j + ℓ ≤ k  ∥ ∂ j x D ℓ θ u ( t ) ∥ 2 L 2 µ w + ∥ ∂ j x D ℓ θ ˜ ω ( t ) ∥ 2 L 2 µ w  + X j + ℓ ≤ k +1 ∥ ∂ j x D ℓ θ ψ ( t ) ∥ 2 L 2 µ w . (5.1) Bac kground co efficien t b ounds actually needed in the energy metho d. The explicit bac kground (4.1) provides ( U, V ) in closed form, and P θ through the relation for ∂ x P . What the stability estimates require is not a uniform b ound on the raw deriv ativ es ∂ m x ∂ ℓ θ V (which can gro w faster than ( T − t ) − 1 near the intermediate scale r 2 ∼ T − t ), but rather uniform con trol of the de gener ate c ombinations that app ear in (4.7)–(3.28) and in the w eighted Sob olev norms. Define the adapted deriv ativ es Z x := x∂ x , D θ := (sin θ cos θ ) − 1 ∂ θ , and extend the quotients V x /x and U x /x contin uously at x = 0 using evenness in x of the seed data. Then for each integer k ≥ 0 there exists C ∗ = C ∗ ( A, B , seeds , k ) suc h that 20 Y AOMING SHI for all t ∈ [0 , T ), X j + ℓ ≤ k  ∥ Z j x D ℓ θ V ( t ) ∥ L ∞ + ∥ Z j x D ℓ θ U ( t ) ∥ L ∞ +    Z j x D ℓ θ  V x ( t ) x     L ∞ +    Z j x D ℓ θ  U x ( t ) x     L ∞  ≤ C ∗ T − t . (5.2) In particular, ∥ V ( t ) ∥ L ∞ = 6 T − t (attained at ( x, θ ) = (0 , ± π / 4)) , ∥ U ( t ) ∥ L ∞ ≲ 1 T − t . Ho w (5.2) controls the quadratic bac kground forcings in Q 1 , Q 2 . In (3.35) the pure-bac kground term Q 1 is          Q 1 = V cos(2 θ ) xU x − V sin(2 θ ) U θ , − 1 2 U  cos(2 θ )( V − Ψ) − 1 2 sin(2 θ )Ψ θ  + xU x  cos(2 θ )( V − Ψ) − 1 2 sin(2 θ )Ψ θ  . (5.3) In (3.38), (3.45), (3.47), (3.37), Q 2 = ( Q 2 a + Q 2 b + Q 2 d ) con tain similar terms. Remark 5.1. The individual term in Q 1 and Q 2 either c ontains a factor cos(2 θ ) or o dd derivative ( V θ , U θ , Ψ θ ) , which wil l vanish on the ridge θ = ± π 4 b e c ause ( V , U, Ψ) ar e flat ther e. The b ound (5.2) is form ulated exactly to con trol these coefficients through the adapted deriv atives xU x = Z x U, U θ = (sin θ cos θ ) D θ U, xV x = Z x V , V θ = (sin θ cos θ ) D θ V , Ψ θ = (sin θ cos θ ) D θ Ψ , (5.4) Na ¨ ıv e pro duct b ound and the refinement actually used. A direct application of (5.2) giv es ∥ V ( t ) ∥ L ∞ ∼ ( T − t ) − 1 and ∥ Z x V ( t ) ∥ L ∞ , ∥ Z x U ( t ) ∥ L ∞ , ≲ ( T − t ) − 1 . ∥ D θ V ( t ) ∥ L ∞ , ∥ D θ U ( t ) ∥ L ∞ ≲ ( T − t ) − 1 . If one then b ounds each pro duct in Q 1 , Q 2 term-b y-term, one only obtains the na ¨ ıve estimate ∥Q 1 ( t ) ∥ L ∞ + ∥Q 2 ( t ) ∥ L ∞ ≲ ( T − t ) − 2 . Ridge structure and the actual gain. This ( T − t ) − 2 b ound is not sharp for the forcing size relev an t to the stabilit y b o otstrap, b ecause the bac kground singularit y o ccurs only on the tw o ridge rays ( x, θ ) = (0 , ± π / 4) and the forcing combinations w ere engineered so that every individual term in Q 1 and in Q 2 carries an additional smal l factor on the ridge: either a trigonometric v anishing factor cos(2 θ ), or an o dd angular deriv ative of an ev en-in- η ridge profile (e.g. V θ , U θ , Ψ θ , Ψ θθ θ ). This is exactly the con ten t of the motiv ation remark righ t after (5.3). Concretely , near θ 0 = π / 4 set η := θ − θ 0 and recall cos(2 θ ) = cos  π 2 +2 η  = − sin(2 η ) = O ( η ) , sin θ cos θ = 1 2 sin(2 θ ) = 1 2 cos(2 η ) = 1 2 + O ( η 2 ) . Moreo v er, b y ridge symmetry of the seed data in the η v ariable (ev enness in η ), the bac kground satisfies V θ ( t, x, θ 0 ) = 0 , U θ ( t, x, θ 0 ) = 0 , Ψ θ ( t, x, θ 0 ) = 0 , and similarly an y o dd θ –deriv ative v anishes on θ = θ 0 . Using (5.4), w e may rewrite the angular deriv atives as U θ = (sin θ cos θ ) D θ U, V θ = (sin θ cos θ ) D θ V , Ψ θ = (sin θ cos θ ) D θ Ψ , so the v anishing of U θ , V θ , Ψ θ on the ridge is equiv alent to the v anishing of D θ U, D θ V , D θ Ψ there. Hence, in a ridge neighborho o d, an y factor of the form V U θ , V V θ , V Ψ θ , etc. gains ST ABLE FINITE-TIME BLO W-UP FOR SYSTEM (E1) 21 an additional O ( | η | ), while any term carrying cos(2 θ ) gains an additional O ( | η | ) directly from cos(2 θ ) = O ( η ). Aw a y from the ridge, the denominator in the closed form (4.1) is uniformly b ounded a w ay from 0, so all bac kground fields and their adapted deriv ativ es are uniformly bounded and there is no ( T − t ) singularit y to start with. Combining these ridge/off-ridge consid- erations yields the refined forcing estimate T argeted forcing-size audit for Q 1 and Q 2 . This subsection records the structural reason wh y the pur e b ackgr ound forcings Q 1 and Q 2 satisfy the mild b ound ∥Q 1 ( t ) ∥ L ∞ + ∥Q 2 ( t ) ∥ L ∞ ≲ 1 T − t , 0 ≤ t < T , rather than the na ¨ ıve pro duct bound ≲ ( T − t ) − 2 . The key p oin t is that every individual term in Q 1 and Q 2 carries an additional ridge gain (either a v anishing trigonometric factor or an o dd θ –deriv ativ e), precisely at the ridge rays where | V | is largest. Lemma 5.2 (Ridge flatness and o dd- θ gains) . L et θ 0 = ± π / 4 and set η := θ − θ 0 . Assume the se e d data ar e even in η and that the angular closur e is imp ose d with the Neumann b oundary c ondition Ψ θ ( ± π / 4) = 0 . Then, for the b ackgr ound fields, al l o dd θ –derivatives vanish on the ridge: V θ ( t, x, θ 0 ) = U θ ( t, x, θ 0 ) = Ψ θ ( t, x, θ 0 ) = 0 , V xθ ( t, x, θ 0 ) = V tθ ( t, x, θ 0 ) = Ψ tθ ( t, x, θ 0 ) = 0 , and henc e in a neighb orho o d of θ 0 one has the T aylor gains V θ , U θ , Ψ θ , V xθ , V tθ , Ψ tθ = O ( | η | ) . Mor e over, cos(2 θ ) = cos  π 2 +2 η  = − sin(2 η ) = O ( η ) , cos 2 (2 θ ) = O ( η 2 ) , sin(2 θ ) = 1+ O ( η 2 ) , while csc(2 θ ) r emains b ounde d on the we dge. Pr o of sketch. Evenness of the seeds in η implies ev enness in η of the bac kground, hence all o dd θ –deriv ativ es v anish at η = 0. The Neumann condition Ψ θ ( θ 0 ) = 0 is consisten t with this symmetry and yields the same v anishing for Ψ θ and its t -deriv ative. The trigonometric expansions are elementary T a ylor series ab out θ 0 = ± π / 4. □ Unique forcing t yp es. Up to b ounded smo oth co efficien t multipliers (functions of θ ), ev ery term in Q 1 and Q 2 falls into one of the “unique” structural types in the summaries b elo w b elo w. The third column records the ridge gain supplied either by cos(2 θ ) or by Lemma 5.2. Awa y from the ridge, the denominator in the closed form (4.1) is b ounded b elo w, so all bac kground quantities are uniformly b ounded and there is no ( T − t ) singu- larit y . F orcing-size audit: unique term types in Q 1 . F rom (5.3), up to b ounded smooth co efficient m ultipliers, Q 1 is a linear combination of the follo wing unique term t yp es: Group Represen tativ e term (from (5.3) ) Ridge gain Q 1 –(i) cos(2 θ ) V xU x = cos(2 θ ) V Z x U cos(2 θ ) = O ( η ) Q 1 –(ii) − sin(2 θ ) V U θ U θ = O ( η ) (Lemma 5.2) Q 1 –(iii) − 1 2 U  cos(2 θ )( V − Ψ) − 1 2 sin(2 θ )Ψ θ  either cos(2 θ ) = O ( η ) or Ψ θ = O ( η ) Q 1 –(iv) xU x  cos(2 θ )( V − Ψ) − 1 2 sin(2 θ )Ψ θ  either cos(2 θ ) = O ( η ) or Ψ θ = O ( η ) 22 Y AOMING SHI F orcing-size audit: unique term t yp es in Q 2 = Q 2 a + Q 2 b + Q 2 d . Group Represen tativ e term Ridge gain Q 2 a 1 x 2 csc(2 θ ) ( U 2 ) θ ( U 2 ) θ = 2 U U θ = O ( η ) Q 2 a 1 x 2 ( V θ ) 2 , 1 x 2 (Ψ θ ) 2 , 1 x 2 V θ Ψ θ V θ , Ψ θ = O ( η ) so pro ducts give O ( η 2 ) Q 2 a 1 x 2 Ψ θ (Ψ , V , xV x , xV xθ , x 2 V xx ) Ψ θ = O ( η ) (and V xθ = O ( η ) if presen t) Q 2 b 1 x cos 3 (2 θ ) csc(2 θ ) (2Ψ − 5 V ) V xθ cos(2 θ ) = O ( η ) and V xθ = O ( η ) Q 2 b 1 x 2 cos 2 (2 θ ) (Ψ 2 , V 2 , V Ψ , ( xV x ) 2 , x 2 V xx ) cos 2 (2 θ ) = O ( η 2 ) Q 2 b 1 x 2 cos 2 (2 θ ) csc(2 θ ) (Ψ , V ) xV xθ cos 2 (2 θ ) = O ( η 2 ) and V xθ = O ( η ) Q 2 b 1 x 2 cos(2 θ ) (Ψ( · · · ) − ( · · · )) cos(2 θ ) = O ( η ) Q 2 d 1 x 2 cos 2 (2 θ ) csc(2 θ ) V tθ cos 2 (2 θ ) = O ( η 2 ) and V tθ = O ( η ) Q 2 d 1 x 2 cos(2 θ ) ( xV tx , V t , Ψ t , V tθ ) cos(2 θ ) = O ( η ) (and V tθ = O ( η ) if presen t) Q 2 d − 1 x 2  csc(2 θ ) V tθ − sin(2 θ )Ψ tθ  difference + closure k eeps an odd- θ gain Conclusion of the audit. Each representativ e term has at least one ridge gain (often t w o). At the lo cation of the background p eak, the core widths satisfy | x | ∼ | η | ∼ ( T − t ) 1 / 8 , so the ridge gain offsets the extra ( T − t ) − 1 factor coming from multiplying t w o bac kground quan tities, yielding the effectiv e forcing size ≲ ( T − t ) − 1 rather than ( T − t ) − 2 . This is the mec hanism b ehind the b ound (5.5) used in the b o otstrap. ∥Q 1 ( t ) ∥ L ∞ + ∥Q 2 ( t ) ∥ L ∞ ≲ 1 T − t . (5.5) and similarly , for each fixed k , X j + ℓ ≤ k  ∥ Z j x D ℓ θ Q 1 ( t ) ∥ L ∞ + ∥ Z j x D ℓ θ Q 2 ( t ) ∥ L ∞  ≲ 1 T − t . This is the forcing size used in the high-order energy inequality below (and is consisten t with the ridge/off-ridge structure: off the ridge the denominator in (4.1) is b ounded a w a y from 0, so there is no ( T − t ) singularit y to b egin with). Lemma 5.3 (Ridge cancellations for the bac kground forcing) . L et θ 0 = π / 4 (the c ase θ 0 = − π / 4 is identic al). Set η := θ − θ 0 . Then in a fixe d neighb orho o d | η | ≤ η 0 ≪ 1 one has the T aylor b ounds cos(2 θ ) = cos  π 2 + 2 η  = − sin(2 η ) = O ( η ) , sin θ cos θ = 1 2 sin(2 θ ) = 1 2 sin  π 2 + 2 η  = 1 2 cos(2 η ) = 1 2 + O ( η 2 ) , (5.6) and in p articular | cos(2 θ ) | ≲ | η | , | sin(2 θ ) | ≤ 1 , | (sin θ cos θ ) − 1 | ≲ 1 . (5.7) Assume mor e over that the se e d data ar e symmetric ar ound the ridge in the sense that, in ridge variables ( x, η ) , V 0 ( x, η ) and U 0 ( x, η ) ar e even in η , so that for the explicit b ackgr ound (4.1) one has V θ ( t, x, θ 0 ) = 0 , U θ ( t, x, θ 0 ) = 0 for al l t < T , x ∈ R . (5.8) ST ABLE FINITE-TIME BLO W-UP FOR SYSTEM (E1) 23 Then the for cing c o efficients in Q 1 , Q 2 satisfy the r efine d ridge gains V xU x = cos(2 θ ) V Z x U = O ( η ) 1 ( T − t ) 2 , V xV x = cos(2 θ ) V Z x V = O ( η ) 1 ( T − t ) 2 , (5.9) and V U θ = V (sin θ cos θ ) D θ U = O ( η ) 1 ( T − t ) 2 , V V θ = V (sin θ cos θ ) D θ V = O ( η ) 1 ( T − t ) 2 . (5.10) In p articular, taking the supr emum over ( x, θ ) and using that off the ridge the denominator in (4.1) is b ounde d away fr om 0 , one obtains the glob al for cing b ound (5.5) . Remark 5.4 (Bookkeeping consequence) . With m = 4 , the for cing term in the high- or der ener gy ine quality c ontributes at size ( T − t ) − 1 E 1 / 2 k , which is c omp atible with any p erturb ation gr owth exp onent 0 < 2 γ < 1 in the b o otstr ap (5.11) . Remark 5.5 (Wh y this is “plug-and-play”) . The only inputs ar e: (a) the trigonometric T aylor exp ansions (5.6) , and (b) the ridge symmetry (5.8) . Al l other factors in Q 1 , Q 2 ar e c ontr ol le d by (5.2) . Thus any alternative se e d choic e pr eserving ridge symmetry le ads to the same c anc el lation me chanism. Pr o of ide a (fr om the close d form). W rite V = − 6 N/D with N ( t ) := 5 tU 2 0 + 2 V 0 ( tV 0 − 6) , D ( t ) := 2( tV 0 − 6) 2 + 5 t 2 U 2 0 . The seeds satisfy 0 < V 0 ≤ A and U 0 ≥ 0, and T = 6 / A . Near the ridge p oint (0 , ± π / 4) one has the expansions V 0 = A (1 − r 4 + O ( r 8 )) and U 0 = B r 4 + O ( r 8 ) with r = x 2 + A 1 (1 − 2 sin 2 θ ) 2 . Hence tV 0 − 6 = − A ( T − t ) − 6 r 4 + O ( τ r 4 + r 8 ) and therefore D ( t, x, θ ) ≳ ( A ( T − t ) + r 4 ) 2 in the ridge neigh b orho o d, while aw a y from the ridge | tV 0 − 6 | is b ounded b elo w b y a p ositiv e constan t. This giv es | V ( t, x, θ ) | ≲ ( T − t + r 2 ) − 1 globally and the exact ridge v alue V ( t, 0 , ± π / 4) = 6 / ( T − t ). Finally , each application of Z x = x∂ x pro duces a factor comparable to r ∂ r (since r dep ends on x 2 ), and eac h D θ pro duces a factor comparable to ∂ η in η = θ − π / 4 with the w eigh t neutralizing the csc θ sec θ singularity . Th us Z j x D ℓ θ applied to ( T − t + r 2 ) − 1 remains ≲ ( T − t + r 2 ) − 1 , yielding (5.2). (whic h holds for all the choices used in Section 2 since r = x 2 + · · · ). Then V ( t, · ) and U ( t, · ) remain ev en in x and smo oth for t < T , hence V x ( t, x, θ ) = O ( x ) and U x ( t, x, θ ) = O ( x ) as x → 0. Consequen tly the p otentially singular m ultipliers x − 1 V x and x − 1 U x that app ear in (3.28) are b ounde d . More precisely: (see (5.2)). Bo otstrap assumption. Fix M ≫ 1 and assume on some in terv al [0 , t ∗ ] ⊂ [0 , T ) that E k ( t ) ≤ M 2 ε 2 ( T − t ) − 2 γ ∀ t ∈ [0 , t ∗ ] , (5.11) for some γ ∈ (0 , 1 / 4) and ε > 0 sufficiently small. 5.2. Elliptic control of ψ from ˜ ω = ∆ ψ . The elliptic relation in (4.7) reads ˜ ω = ∆ ψ , where ∆ is given b y (4.8). Rewrite the angular part in div ergence form using D θ (as already indicated in the w eigh ted-norms subsection), so that ∆ becomes a uniformly elliptic op erator in the weigh ted space L 2 ( dµ w ) with symmetric principal part. Standard w eigh ted elliptic theory then yields, for all in tegers m ≥ 0, ∥ ψ ( t ) ∥ H m +2 µ w ≤ C ∆ ,m ∥ ˜ ω ( t ) ∥ H m µ w , (5.12) where the constan t dep ends only on the strip and the co efficients c i ( θ ). 24 Y AOMING SHI In particular, since k ≥ 6, Sob olev em b edding in the ( x, θ ) v ariables (with D θ coun ted as one deriv ative) giv es ∥ u ( t ) ∥ L ∞ + ∥ ˜ ω ( t ) ∥ L ∞ + ∥ ψ ( t ) ∥ W 1 , ∞ ≤ C E k ( t ) 1 / 2 . (5.13) 5.3. Energy inequality for ( u, ˜ ω ) . Differentiate the u -equation and the ˜ ω -equation in (4.7) (with L i , M i , Q i from (3.35)) b y ∂ j x D ℓ θ for j + ℓ ≤ k , tak e the L 2 µ w inner product with ∂ j x D ℓ θ u and ∂ j x D ℓ θ ω , and sum o v er j + ℓ ≤ k . The bac kground transp ort terms cos(2 θ ) V x∂ x and − sin(2 θ ) V ∂ θ are handled b y in tegration by parts in x and the div ergence-form structure in θ (no b oundary flux b ecause w ( θ ) = 0 at θ = 0 and the strip endp oints are fixed b y the mo del conditions). Using the background b ounds (5.2), comm utator estimates, and (5.13), one obtains an inequalit y of the form d dt E k ( t ) ≤ C T − t E k ( t ) + C  ∥M 1 ( t ) ∥ H k µ w + ∥M 2 ( t ) ∥ H k µ w  E k ( t ) 1 / 2 + C  ∥Q 1 ( t ) ∥ H k µ w + ∥Q 2 ( t ) ∥ H k µ w  E k ( t ) 1 / 2 . (5.14) Quadratic p erturbation terms. F rom the explicit forms of M 1 , M 2 in (3.35) and Moser/Sob olev pro duct estimates, ∥M 1 ( t ) ∥ H k µ w + ∥M 2 ( t ) ∥ H k µ w ≤ C E k ( t ) . Pure bac kground forcing. The explicit Q 1 , Q 2 in (3.35) and (3.38) dep end only on ( U, V , Ψ) and their deriv ativ es. By (5.5) together with Lemma 5.3 (and Leibniz/Moser estimates in H k µ w ), ∥Q 1 ( t ) ∥ H k µ w + ∥Q 2 ( t ) ∥ H k µ w ≤ C T − t . Plugging these b ounds into (5.14) yields, on [0 , t ∗ ], d dt E k ( t ) ≤ C T − t E k ( t ) + C E k ( t ) 3 / 2 + C T − t E k ( t ) 1 / 2 . (5.15) 3.4 Bo otstrap impro v emen t and stability conclusion. Assuming (5.11) and c ho os- ing ε small (dep ending on M and C ), the cubic term C E 3 / 2 k is absorb ed, and the last forcing term is dominated by C T − t E k b ecause E 1 / 2 k ≲ M ε ( T − t ) − γ with γ < 1 / 2. Th us (5.15) impro ves to d dt E k ( t ) ≤ C ′ T − t E k ( t ) , whic h integrates to E k ( t ) ≤ E k (0)  T T − t  C ′ . By choosing the b o otstrap exp onent γ with 2 γ < C ′ , we close (5.11) on [0 , t ∗ ] and extend to t ∗ = T b y con tin uit y . In particular, the p erturbation remains controlled up to T , and the bac kground blow-up rate dominates: ∥ V ( t ) ∥ L ∞ ∼ C 1 T − t , ∥ u ( t ) ∥ L ∞ + ∥ ˜ ω ( t ) ∥ L ∞ + ∥ ψ ( t ) ∥ W 1 , ∞ ≤ C 2 ( T − t ) − 2 γ , 0 < 2 γ < 1 . This matc hes the final stability statemen t in the in tro duction and conclusion. ST ABLE FINITE-TIME BLO W-UP FOR SYSTEM (E1) 25 6. Conclusion W e prov ed that the explicit profile (4.1) is stable under small high-regularity p erturba- tions: the linearized dynamics is con trolled by a co erciv e energy , and the nonlinear terms are perturbative in the b o otstrap regime. Natural extensions include allo wing mo dulation of geometric parameters (e.g. blow-up time/p oint) by adding orthogonalit y conditions, as in Merle–Rapha¨ el [23]; Rapha ¨ el–Ro dnianski [24]; Collot–Merle–Rapha ¨ el [13], and lo w- ering the regularit y by w orking in scale-adapted weigh ted spaces (cf. similarity-v ariable framew orks Giga–Kohn [18]; Giga [19]; Khenissy–Zaag [28]). F rom a mo deling viewp oin t, the fact that the background profile is compatible (af- ter rescaling) with the CLM/De Gregorio 1D v orticity family strengthens its relev ance as a tractable pro xy for vortex-stretc hing-driv en singularity formation; see Constantin– Lax–Ma jda [9]; De Gregorio [15]; Schochet [25]; Hou–Luo [20]; Choi–Hou–Kiselev–Luo– ˇ Sv er´ ak–Y ao [6]. 7. A cknowledgements ChatGPT is credited as a substantiv e con tributor to drafting and tec hnical editing; resp onsibilit y for correctness remains with the author. W e would lik e to thank Prof. Zixiang ZHOU, math dept of F udan Universit y and Prof. Jie QIN, math dept of Uni- v ersit y of California at San ta Cruz for their con tinuous sopp ort and encouragement ov er the years. W e also thank colleagues and the broader PDE/fluid dynamics comm unity for stim ulating discussions on inviscid Boussinesq and CLM-type mo dels. (Computational assistance: ChatGPT, GPT–5.2 Thinking, Op enAI; sessions in March 2026.) 8. 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