Lifespan of Solution to MHD Boundary Layer Equations with Analytic Perturbation of General Shear Flow
aa r X i v : . [ m a t h . A P ] J un Lifespan of Solution to MHD Boundary Layer Equations withAnalytic Perturbation of General Shear Flow
Feng Xie
School of Mathematical Sciences, and LSC-MOE,Shanghai Jiao Tong University, Shanghai 200240, P.R.China
Tong Yang
Department of Mathematics, City University of Hong Kong,Tat Chee Avenue, Kowloon, Hong Kong
Dedicated to Professor Philippe G. Ciarlet on the Occasion of his 80th Birthday
Abstract
In this paper, we consider the lifespan of solution to the MHD boundary layer system asan analytic perturbation of general shear flow. By using the cancellation mechanism in thesystem observed in [12], the lifespan of solution is shown to have a lower bound in the orderof ε − if the strength of the perturbation is of the order of ε . Since there is no restrictionon the strength of the shear flow and the lifespan estimate is larger than the one obtained forthe classical Prandtl system in this setting, it reveals the stabilizing effect of the magneticfield on the electrically conducting fluid near the boundary. : 76N20, 35Q35, 76N10, 35M33. Keywords : MHD boundary layer, analytic perturbation, lifespan estimate, shear flow.
Consider the high Reynolds number limit to the MHD system near a no-slip boundary, thefollowing MHD boundary layer system was derived in [12] when both of the Reynolds numberand the magnetic Reynolds number have the same order in two space dimensions. Precisely,consider the MHD system in the domain { ( x, Y ) | x ∈ R , Y ∈ R + } with Y = 0 being the boundary, ∂ t u ǫ + ( u ǫ ∂ x + v ǫ ∂ Y ) u ǫ + ∂ x p ǫ − ( h ǫ ∂ x + g ǫ ∂ Y ) h ǫ = ǫ ( ∂ x u ǫ + ∂ Y u ǫ ) ,∂ t v ǫ + ( u ǫ ∂ x + v ǫ ∂ Y ) v ǫ + ∂ Y p ǫ − ( h ǫ ∂ x + g ǫ ∂ Y ) g ǫ = ǫ ( ∂ x v ǫ + ∂ Y v ǫ ) ,∂ t h ǫ + ( u ǫ ∂ x + v ǫ ∂ Y ) h ǫ − ( h ǫ ∂ x + g ǫ ∂ Y ) u ǫ = κǫ ( ∂ x h ǫ + ∂ Y h ǫ ) ,∂ t g ǫ + ( u ǫ ∂ x + v ǫ ∂ Y ) g ǫ − ( h ǫ ∂ x + g ǫ ∂ Y ) v ǫ = κǫ ( ∂ x g ǫ + ∂ Y g ǫ ) ,∂ x u ǫ + ∂ Y v ǫ = 0 , ∂ x h ǫ + ∂ Y g ǫ = 0 , (1.1) E-mail address: [email protected] (F. Xie) E-mail address: [email protected](T. Yang) ǫ , ( u ǫ , v ǫ ) and ( h ǫ , g ǫ ) represent the velocity and the magnetic field respectively. The no-slipboundary condition is imposed on the velocity field( u ǫ , v ǫ ) | Y =0 = , (1.2)and the perfectly conducting boundary condition is given for the magnetic field( ∂ Y h ǫ , g ǫ ) | Y =0 = . (1.3)Formally, when ǫ = 0, (1.1) is reduced into the following incompressible ideal MHD system ∂ t u e + ( u e ∂ x + v e ∂ Y ) u e + ∂ x p e − ( h e ∂ x + g e ∂ Y ) h e = 0 ,∂ t v e + ( u e ∂ x + v e ∂ Y ) v e + ∂ Y p e − ( h e ∂ x + g e ∂ Y ) g e = 0 ,∂ t h e + ( u e ∂ x + v e ∂ Y ) h e − ( h e ∂ x + g e ∂ Y ) u e = 0 ,∂ t g e + ( u e ∂ x + v e ∂ Y ) g e − ( h e ∂ x + g e ∂ Y ) v e = 0 ,∂ x u e + ∂ Y v e = 0 , ∂ x h e + ∂ Y g e = 0 . (1.4)Since the sovability of the system (1.4) requires only the normal components of the velocity andmagnetic fields ( v e , g e ) on the boundary( v e , g e ) | Y =0 = , (1.5)in the limit from (1.1) to (1.4), a Prandtl-type boundary layer can be derived to resolve the mis-match of the tangential components between the viscous flow ( u ǫ , h ǫ ) and invicid flow ( u , h )on the boundary { Y = 0 } . And this system governing the fluid behavior in the leading order ofapproximation near the boundary is derived in [7, 12, 13]: ∂ t u + u ∂ x u + u ∂ y u = b ∂ x b + b ∂ y b + ∂ y u ,∂ t b + ∂ y ( u b − u b ) = κ∂ y b ,∂ x u + ∂ y u = 0 , ∂ x b + ∂ y b = 0 (1.6)in H = { ( x, y ) ∈ R | y ≥ } with the fast variable y = Y / √ ǫ . Here, the trace of the horizontalideal MHD flow (1.4) on the boundary { Y = 0 } is assumed to be a constant vector so that thepressure term ∂ x p e ( t, x,
0) vanishes by the Bernoulli’s law.Consider the system (1 .
6) with initial data u ( t, x, y ) | t =0 = u ( x, y ) , b ( t, x, y ) | t =0 = b ( x, y ) , (1.7)and the boundary conditions (cid:26) u | y =0 = 0 ,u | y =0 = 0 , and (cid:26) ∂ y b | y =0 = 0 ,b | y =0 = 0 . (1.8)And the far field state is denoted by (¯ u, ¯ b ):lim y → + ∞ u = u e ( t, x, , ¯ u, lim y → + ∞ b = h e ( t, x, , ¯ b. (1.9)2irst of all, a shear flow ( u s ( t, y ) , , ¯ b,
0) is a trivial solution to the system (1.6) with u s ( t, y )solving ∂ t u s ( t, y ) − ∂ y u s ( t, y ) = 0 , ( t, y ) ∈ R + × R + ,u s ( t, y = 0) = 0 , lim y →∞ u s ( t, y ) = ¯ u,u s ( t = 0 , y ) = u s ( y ) . (1.10)In the following discussion, we assume the shear flow u s ( t, y ) has the following properties:( H ) k ∂ iy u s ( t, · ) k L ∞ y ≤ C h t i i/ ( i = 1 , , Z ∞ | ∂ y u s ( t, y ) | dy < C, k θ α ∂ y u s ( t, · ) k L y ≤ C h t i / , for some generic constant C . Remark 1.1
The assumption (H) on the shear flow holds for a large class of initial data u s .For example, it holds for the initial data u s = χ ( y ) with χ ( y ) ∈ C ∞ ( R ) , χ ( y ) = 0 for y ≤ and χ ( y ) = ¯ u for y ≥ considered in [23] for the Prandtl system. Note that here we do not assumethe smallness of the shear flow. In addition, it also holds when u s ( y ) = 1 √ π Z y exp( − z dz considered in [9] for the Prandtl system where the almost global solution is obtained. Note thatfor the classical Prandtl equations, such shear flow in the form of Guassian error function yieldsa time decay damping term in the time evolution equation of u , however, it does not leads toany damping effect in the MHD boundary layer system (1.6) . To define the function space of the solution considered in this paper, the following Gaussianweighted function θ α will be used: θ α ( t, y ) = exp ( αz ( t, y ) , with z ( t, y ) = y p h t i , h t i = 1 + t and α ∈ [1 / , / . With this and M m = √ m + 1 m ! , define the Sobolev weighted semi-norms by X m = X m ( f, τ ) = k θ α ∂ mx f k L τ m M m , D m = D m ( f, τ ) = k θ α ∂ y ∂ mx f k L τ m M m ,Z m = Z m ( f, τ ) = k zθ α ∂ mx f k L τ m M m , Y m = Y m ( f, τ ) = k θ α ∂ mx f k L τ m − mM m . (1.11)Then the following space of analytic functions in the tangential variable x and Sobolev weightedin the normal variable y is defined by X τ,α = { f ( t, x, y ) ∈ L ( H ; θ α dxdy ) : k f k X τ,α < ∞} with τ > k f k X τ,α = X m ≥ X m ( f, τ ) .
3n addition, the following two semi-norms will also be used: k f k D τ,α = X m ≥ D m ( f, τ ) = k ∂ y f k X τ,α , k f k Y τ,α = X m ≥ Y m ( f, τ ) . Here, the summation over m is considered in the l sense that is similar to the definition usedin [9, 23] rather than in the l sense used in [10]. With the above notations, we are now readyto state the main Theorem as follows. Theorem 1.1
For any λ ∈ [3 / , , there exists a small positive constant ε ∗ depending on − λ .Under the assumption (H) on the backgroud shear flow ( u s ( t, y ) , , ¯ b, with ¯ b = 0 , assume theinitial data u and b satisfy k u − u s (0 , y ) k X τ , / ≤ ε, k b − ¯ b k X τ , / ≤ ε, (1.12) for some given ε ∈ (0 , ε ∗ ] . Then there exists a unique solution ( u , u , b , b ) to the MHDboundary layer equations (1.6)-(1.9) such that ( u − u s ( t, y ) , b − ¯ b ) ∈ X τ,α , α ∈ [1 / , / , with analyticity radius τ larger than τ / in the time interval [0 , T ε ] . And the lifespan T ε hasthe following low bound estimate T ε ≥ Cε − λ , (1.13) where the constant C is independent of ε . As is well-known that the leading order characteristic boundary layer for the incompressibleNavier-Stokes equations with no-slip boundary condition is described by the classical Prandtlequations derived by Prandtl [19] in 1904. In the two space dimensions, under the monotonic-ity assumption on the tangential velocity in the normal direction, Oleinik firstly obtained thelocal existence of classical solutions by using the Crocco transformation, cf. [17] and Oleinik-Samokhin’s classical book [18]. Recently, this well-posedness result was re-proved by using anenergy method in the framework of Sobolev spaces in [1] and [16] independently by observingthe cancellation mechanism in the convection terms. And by imposing an additional favorablecondition on the pressure, a global in time weak solution was obtained in [22].When the monotonicity condition is violated, singularity formation or separation of theboundary layer is well expected and observed. For this, E-Engquist constructed a finite timeblowup solution to the Prandtl equations in [3]. Recently, when the background shear flowhas a non-degenerate critical point, some interesting ill-posedness (or instability) phenomenaof solutions to both linear and nonlinear classical Prandtl equations around shear flows arestudied, cf. [4, 5, 6]. All these results show that the monotonicity assumption on the tangentialvelocity plays a key role for well-posedness theory except in the frameworks of analytic functionsand Gevrey regularity classes. Indeed, in the framework of analytic functions, Sammartino andCaflisch [20, 2] established the local well-posedness theory of the Prandtl system in three spacedimensions and also justified the Prandtl ansatz in this setting by applying the abstract Cauchy-Kowalewskaya (CK) theorem initated by Asano’s unpublished work. Later, the analyticity4equirement in the normal variable y was removed by Lombardo, Cannone and Sammartino in[15] because of the viscous effect in the normal direction.Recently, Zhang and Zhang obtained the lifespan of small analytic solution to the classicalPrandtl equations with small analytic initial data in [23]. Precisely, when the strength of back-ground shear flow is of the order of ε / and the perturbation is of the order of ε , they showedthat the classical Prandtl system has a unique solution with a lower bound estimate on thelifespan in the order of ε − / . Furthermore, if the initial data is a small analytic perturbationof the Guassian error function (1.10), an almost global existence for the Prandtl boundary layerequations is proved in [9].On the other hand, to study the high Reynolds number limits for the MHD equations(1.1) with no-slip boundary condition on the velocity (1.2) and perfect conducting boundarycondition (1.3) on the magnetic field, one can apply the Prandtl ansatz to derive the boundarylayer system (1.6) as the leading order description on the flow near the boundary. For this,readers can refer to [7, 12, 13, 14, 21] about the formal derivation of (1.6), the well-posednesstheory of the system and the justification of the Prandtl ansatz locally in time.This paper is about long time existence of solutions to (1.6)-(1.9). Precisely, we will showthat if the initial data is a small perturbation of a shear flow analytically in the order of ε ,then there exists a unique solution to (1.6)-(1.9) with the lifespan T ε of the order of ε − .Compared with the estimate on the lifespan of solutions to the classical Prandtl system studiedin [23], the lower bound estimate is larger and there is no requirement on the smallness of thebackground shear flow because the mechanism in the system is used due to the non-degeneracyof the tangential magnetic field. However, it is not known whether one can obtain a global oralmost global in time solution like the work on the Prandtl system when the background shearvelocity is taken to be a Gaussian error function in [9]. We mention that even though Lin andZhang showed the almost global existence of solution to MHD boundary layer equations withzero Dirichlet boundary condition on the magnetic field in [11] when the components of both thebackground velocity and magnetic fields are Guassian error functions, it is not clear wheatherthe system (1.6) holds with zero Dirichlet boundary condition even in formal derivation.The analysis on the lifespan of the perturbed system in this paper relies on the introductionof some new unknown functions that capture the cancellation of some linear terms. Unlike thework in [9] on the Prandtl system for which the cancellation yields a damping term in the timeevolution of the perturbation of the tangetial velocity field, there is no such damping effectobserved for the MHD boundary layer system.Finally, the rest of the paper is organized as follows. After giving some preliminary esti-mates, a uniform estimate on the solution will be proved in the next section. Based on thisuniform estimate, a low bound of the lifespan of solution is derived in Section 3. The uniquenesspart is done in Section 4. Throughout the paper, constants denoted by C , ¯ C, C , C and C aregeneric and independent of the small parameter ε . We first list the following two priliminary estimates on the functions in the norms defined in theprevious section. The first estimate indeed is from Lemma 3.3 in [9] (also see [8]).5 emma 2.1 (Poincar´e type inequality with Gaussian weight) Let f be a function such that f | y =0 = 0 ( or ∂ y f | y =0 ) and f | y = ∞ = 0 . Then, for α ∈ [1 / , / , m ≥ and t ≥ , it holds that α h t i k θ α ∂ mx f k L y ≤ k θ α ∂ y ∂ mx f k L y . (2.1)The second lemma is used in [9] and we include it here with a short proof for convenienceof readers. Lemma 2.2
Let f be a function such that f | y =0 = 0 ( or ∂ y f | y =0 ) and f | y = ∞ = 0 . Then X m ≥ k θ α ∂ y ∂ mx f k L k θ α ∂ mx f k L τ m M m ≥ α / β h t i / k f k D τ,α + α (1 − β ) h t i k f k X τ,α , (2.2) for β ∈ (0 , / .Proof. In fact, by Lemma 2.1, one has k θ α ∂ y ∂ mx f k L k θ α ∂ mx f k L ≥ β k θ α ∂ y ∂ mx f k L k θ α ∂ mx f k L + 2 − β α / h t i / k θ α ∂ y ∂ mx f k L ≥ β k θ α ∂ y ∂ mx f k L k θ α ∂ mx f k L + βα / h t i / k θ α ∂ y ∂ mx f k L + α (1 − β ) h t i k θ α ∂ mx f k L ≥ βα / h t i / k θ α ∂ y ∂ mx f k L + α (1 − β ) h t i k θ α ∂ mx f k L . Multiplying the above inequality by τ m M m and summing up in m ≥ u, v, b, g ) of the ( u s ( t, y ) , , ¯ b,
0) by denoting (cid:26) u = u s ( t, y ) + u,u = v, (cid:26) b = ¯ b + b,b = g. (2.3)Without loss of generality, take ¯ b = 1 and κ = 1. Then (1.6) yields (cid:26) ∂ t u + ( u s + u ) ∂ x u + v∂ y ( u s + u ) − (1 + b ) ∂ x b − g∂ y b − ∂ y u = 0 ,∂ t b − (1 + b ) ∂ x u − g∂ y ( u s + u ) + ( u s + u ) ∂ x b + v∂ y b − ∂ y b = 0 . (2.4)And the initial and boundary data of ( u, v ) and ( b, g ) are given by u ( t, x, y ) | t =0 = u ( x, y ) − u s (0 , y ) , b ( t, x, y ) | t =0 = b ( x, y ) − , (2.5) (cid:26) u | y =0 = 0 ,v | y =0 = 0 , and (cid:26) ∂ y b | y =0 = 0 ,g | y =0 = 0 , (2.6)with the corresponding far field conditionlim y → + ∞ u = 0 , lim y → + ∞ b = 0 . (2.7)6t suffices to establish the long time existence of solutions to (2.4)-(2.7). In this section,we focus on the uniform a priori estimate on the solution to (2.4) in the analytical frameworkdefined in Section 1.Integrating equation (2 . over [0 , y ] gives that ∂ t Z y bd ˜ y + v (1 + b ) − ( u s + u ) g = ∂ y Z y bd ˜ y, (2.8)where the boundary conditions that ∂ y b | y =0 = v | y =0 = g | y =0 = 0 are used.Define ψ ( t, y ) = Z y bd ˜ y, one has ∂ t ψ + v (1 + b ) − ( u s + u ) g = ∂ y ψ. (2.9)Now introduce new unknown functions by taking care of the cancellation mechamism in thesystem as obseved in [12] as follows˜ u = u − ∂ y u s ψ, ˜ b = b. (2.10)Then (˜ u, ˜ b ) satisfies the following equations. (cid:26) ∂ t ˜ u − ∂ y ˜ u + ( u s + u ) ∂ x ˜ u + v∂ y ˜ u − (1 + b ) ∂ x ˜ b − g∂ y ˜ b − ∂ y u s ˜ b + v∂ y u s ψ = 0 ,∂ t ˜ b − ∂ y ˜ b − (1 + b ) ∂ x ˜ u − g∂ y ˜ u + ( u s + u ) ∂ x ˜ b + v∂ y ˜ b − g∂ y u s ψ = 0 . (2.11)Here we have used the following fact that u s is the solution to the heat equation. That is, ∂ t u s − ∂ y u s = 0 , ∂ t ∂ y u s − ∂ y u s = 0 . By a direct calculation, the boundary conditions of (˜ u, ˜ b ) are given by˜ u | y =0 = 0 , ∂ y ˜ b | y =0 = 0 , (2.12)˜ u | y = ∞ = 0 , ˜ b | y = ∞ = 0 . (2.13)We then turn to show the existence of solution (˜ u, ˜ b ) to (2.11)-(2.13) with the correspondinginitial data.˜ u (0 , x, y ) = u (0 , x, y ) − ∂ y u s (0 , y ) Z y b (0 , x, ˜ y ) d ˜ y, ˜ b (0 , x, y ) = b (0 , x, y ) . (2.14)Note that k ˜ u (0 , x, y ) k X τ ,α ≤ k u (0 , x, y ) k X τ ,α + C k b (0 , x, y ) k X τ ,α , (2.15)for α ∈ [1 / , / u, ˜ b ) to (2.11)-(2.14) is obtained, one can define( u, b ) by u ( t, x, y ) = ˜ u ( t, x, y ) + ∂ y u s ( t, y ) Z y ˜ b ( t, x, ˜ y ) d ˜ y, b ( t, x, y ) = ˜ b ( t, x, y ) . (2.16)It is straightforward to check that ( u, b ) is a solution to (2.4)-(2.7) with the following estimates k u k X τ,α ≤ k ˜ u k X τ,α + C k ˜ b k X τ,α , k b k X τ,α = k ˜ b k X τ,α . Therefore, we only need to estimate the solution (˜ u, ˜ b ) to (2.11)-(2.14) in the analytic norms asshown in the next two subsections. For m ≥
0, by applying the tangential derivative operator ∂ mx to (2 . and multiplying it by θ α ∂ mx ˜ u , the integration over H yields Z H ∂ mx ( ∂ t ˜ u − ∂ y ˜ u + ( u s + u ) ∂ x ˜ u + v∂ y ˜ u − (1 + b ) ∂ x ˜ b − g∂ y ˜ b − ∂ y u s ˜ b + vψ∂ y u s ) θ α ∂ mx ˜ udxdy = 0 . (2.17)We now estimate each term in (2 .
17) as follows. Firstly, note that Z H ∂ t ∂ mx ˜ uθ α ∂ mx ˜ udxdy = 12 ddt Z H ( ∂ mx ˜ u ) θ α dxdy − Z H ( ∂ mx ˜ u ) θ α ddt θ α dxdy (2.18)= 12 ddt k θ α ∂ mx ˜ u k L + α h t i k θ α z∂ mx ˜ u k L , and − Z H ∂ y ∂ mx ˜ uθ α ∂ mx ˜ udxdy = k θ α ∂ mx ∂ y ˜ u k L + Z H ∂ y ∂ mx ˜ u∂ y ( θ α ) ∂ mx ˜ udxdy. The boundary term vanishes because of the boundary condition ∂ mx ˜ u | y =0 = 0. Furthermore, Z H ∂ y ∂ mx ˜ u∂ y ( θ α ) ∂ mx ˜ udxdy = − Z H ( ∂ mx ˜ u ) ∂ y ( θ α ) dxdy = − α h t i k θ α ∂ mx ˜ u k L − α h t i k θ α z∂ mx ˜ u k L , where we have used ∂ y ( θ α ) = α h t i θ α + α h t i z ( t, y ) θ α . − Z H ∂ y ∂ mx ˜ uθ α ∂ mx ˜ udxdy = k θ α ∂ mx ∂ y ˜ u k L − α h t i k θ α ∂ mx ˜ u k L − α h t i k θ α z∂ mx ˜ u k L . (2.19)For the nonlinear terms in (2.17), we have Z H ∂ mx (( u s + u ) ∂ x ˜ u ) θ α ∂ mx ˜ udxdy = m X j =0 ( mj ) Z H ∂ m − jx u∂ j +1 x ˜ uθ α ∂ mx ˜ udxdy , R and | R | ≤ [ m/ X j =0 ( mj ) k ∂ m − jx u k L x L ∞ y k θ α ∂ j +1 x ˜ u k L ∞ x L y k θ α ∂ mx ˜ u k L + m X j =[ m/ ( mj ) k ∂ m − jx u k L ∞ xy k θ α ∂ j +1 x ˜ u k L k θ α ∂ mx ˜ u k L . For 0 ≤ j ≤ [ m/ k ∂ m − jx u k L x L ∞ y = k ∂ m − jx (˜ u + ∂ y u s ψ ) k L x L ∞ y ≤k ∂ m − jx ˜ u k L x L ∞ y + k ∂ y u s ∂ m − jx ψ k L x L ∞ y ≤ C k θ α ∂ m − jx ˜ u k / L k θ α ∂ m − jx ∂ y ˜ u k / L + C h t i − / k θ α ∂ m − jx ˜ b k L , where in the last inequality, we have used k ∂ y u s k L ∞ y ≤ C p h t i , according to the assumption (H). Moreover, k ∂ m − jx ψ k L x L ∞ y = k Z y ∂ m − jx ˜ bd ˜ y k L x L ∞ y = k Z y θ α ∂ m − jx ˜ b exp( − α z ) d ˜ y k L x L ∞ y ≤ C h t i / k θ α ∂ m − jx ˜ b k L . And k θ α ∂ j +1 x ˜ u k L ∞ x L y ≤ C k θ α ∂ j +1 x ˜ u k / L k θ α ∂ j +2 x ˜ u k / L . For [ m/
2] + 1 ≤ j ≤ m , we have k ∂ m − jx u k L ∞ xy ≤k ∂ m − jx ˜ u k L ∞ xy + k ∂ y u s ∂ m − jx ψ k L ∞ xy ≤ C k θ α ∂ m − jx ˜ u k / L k θ α ∂ m − jx ∂ y ˜ u k / L k θ α ∂ m − j +1 x ˜ u k / L k θ α ∂ m − j +1 x ∂ y ˜ u k / L + C h t i − / k θ α ∂ m − jx ˜ b k / L k θ α ∂ m − j +1 x ˜ b k / L . | R | τ m M m k θ α ∂ mx ˜ u k L ≤ C ( τ ( t )) / [ m/ X j =0 ( X / m − j D / m − j + h t i − / ¯ X m − j ) Y / j +1 Y / j +2 (2.20)+ m X j =[ m/ ( X / m − j X / m − j +1 D / m − j D / m − j +1 + h t i − / ¯ X / m − j ¯ X / m − j +1 ) Y j +1 . From now on, we use X i , D i , Y i to denote the semi-norms of function ˜ u defined in (1.11), and¯ X i , ¯ D i and ¯ Y i for the corresponding semi-norms for ˜ b . Note that Z H ∂ mx ( v∂ y ˜ u ) θ α ∂ mx ˜ udxdy = m X j =0 ( mj ) Z H ∂ m − jx v∂ jx ∂ y ˜ uθ α ∂ mx ˜ udxdy , R and | R | ≤ [ m/ X j =0 ( mj ) k ∂ m − jx v k L x L ∞ y k θ α ∂ jx ∂ y ˜ u k L ∞ x L y k θ α ∂ mx ˜ u k L + m X j =[ m/ ( mj ) k ∂ m − jx v k L ∞ xy k θ α ∂ jx ∂ y ˜ u k L k θ α ∂ mx ˜ u k L . For 0 ≤ j ≤ [ m/ k ∂ m − jx v k L x L ∞ y = k Z y ∂ m − j +1 x ud ˜ y k L x L ∞ y ≤k Z y ∂ m − j +1 x ˜ ud ˜ y k L x L ∞ y + k Z y ∂ y u s ( Z ˜ y ∂ m − j +1 x ˜ bds ) d ˜ y k L x L ∞ y ≤k Z y ∂ m − j +1 x ˜ ud ˜ y k L x L ∞ y + k Z y ∂ y u s d ˜ y k L ∞ y k Z ˜ y ∂ m − j +1 x ˜ bds k L x L ∞ y ≤ C h t i / k θ α ∂ m − j +1 x ˜ u k L + C h t i / k θ α ∂ m − j +1 x ˜ b k L , where we have used Z ∞ | ∂ y u s ( t, y ) | dy < C by the assumption (H). Note that k θ α ∂ jx ∂ y ˜ u k L ∞ x L y ≤ C k θ α ∂ jx ∂ y ˜ u k / L k θ α ∂ j +1 x ∂ y ˜ u k / L . For [ m/
2] + 1 ≤ j ≤ m , k ∂ m − jx v k L ∞ xy ≤k Z y ∂ m − j +1 x ˜ ud ˜ y k L ∞ xy + k Z y ∂ y u s ( Z ˜ y ∂ m − j +1 x ˜ bds ) d ˜ y k L ∞ xy ≤k Z y ∂ m − j +1 x ˜ ud ˜ y k L ∞ xy + k Z y ∂ y u s d ˜ y k L ∞ y k Z ˜ y ∂ m − j +1 x ˜ bds k L ∞ xy C h t i / k θ α ∂ m − j +1 x ˜ u k L y L ∞ x + C h t i / k θ α ∂ m − j +1 x ˜ b k L y L ∞ x ≤ C h t i / k θ α ∂ m − j +1 x ˜ u k / L k θ α ∂ m − j +2 x ˜ u k / L + C h t i / k θ α ∂ m − j +1 x ˜ b k / L k θ α ∂ m − j +2 x ˜ b k / L . Hence, | R | τ m M m k θ α ∂ mx ˜ u k L ≤ C ( τ ( t )) / [ m/ X j =0 ( h t i / Y m − j +1 + h t i / ¯ Y m − j +1 ) D / j D / j +1 (2.21)+ m X j =[ m/ ( h t i / Y / m − j +1 Y / m − j +2 + h t i / ¯ Y / m − j +1 ¯ Y / m − j +2 ) D j . Recall b = ˜ b so that R , m X j =0 ( mj ) Z H ∂ m − jx ˜ b∂ j +1 x ˜ bθ α ∂ mx ˜ udxdy, and | R | ≤ [ m/ X j =0 ( mj ) k ∂ m − jx ˜ b k L x L ∞ y k θ α ∂ j +1 x ˜ b k L ∞ x L y k θ α ∂ mx ˜ u k L + m X j =[ m/ ( mj ) k ∂ m − jx ˜ b k L ∞ xy k θ α ∂ j +1 x ˜ b k L k θ α ∂ mx ˜ u k L . For 0 ≤ j ≤ [ m/ k ∂ m − jx ˜ b k L x L ∞ y ≤ C k θ α ∂ m − jx ˜ b k / L k ∂ m − jx ∂ y ˜ b k / L , and k ∂ j +1 x ˜ b k L ∞ x L y ≤ C k ∂ j +1 x ˜ b k / L k ∂ j +2 x ˜ b k / L . For [ m/
2] + 1 ≤ j ≤ m , k ∂ m − jx ˜ b k L ∞ xy ≤ C k ∂ m − jx ˜ b k / L k ∂ m − jx ∂ y ˜ b k / L k ∂ m − j +1 x ˜ b k / L k ∂ m − j +1 x ∂ y ˜ b k / L . Therefore, | R | τ m M m k θ α ∂ mx ˜ u k L ≤ C ( τ ( t )) / [ m/ X j =0 ¯ X / m − j ¯ D / m − j ¯ Y / j +1 ¯ Y / j +2 (2.22)+ m X j =[ m/ ¯ X / m − j ¯ X / m − j +1 ¯ D / m − j ¯ D / m − j +1 ¯ Y j +1 . Z H ∂ mx ( g∂ y ˜ b ) θ α ∂ mx ˜ udxdy = m X j =0 ( mj ) Z H ∂ m − jx g∂ jx ∂ y ˜ bθ α ∂ mx ˜ udxdy , R and | R | ≤ [ m/ X j =0 ( mj ) k ∂ m − jx g k L x L ∞ y k θ α ∂ jx ∂ y ˜ b k L ∞ x L y k θ α ∂ mx ˜ u k L + m X j =[ m/ ( mj ) k ∂ m − jx g k L ∞ xy k θ α ∂ jx ∂ y ˜ b k L k θ α ∂ mx ˜ u k L . For 0 ≤ j ≤ [ m/ k ∂ m − jx g k L x L ∞ y ≤ C h t i / k θ α ∂ m − j +1 x ˜ b k L , and k θ α ∂ jx ∂ y ˜ b k L ∞ x L y ≤ C k θ α ∂ jx ∂ y ˜ b k / L k θ α ∂ j +1 x ∂ y ˜ b k / L . For [ m/
2] + 1 ≤ j ≤ m , k ∂ m − jx g k L ∞ xy ≤ C h t i / k θ α ∂ m − j +1 x ˜ b k L y L ∞ x ≤ C h t i / k θ α ∂ m − j +1 x ˜ b k / L k θ α ∂ m − j +2 x ˜ b k / L . As a consequence, we have | R | τ m M m k θ α ∂ mx ˜ u k L ≤ C ( τ ( t )) / [ m/ X j =0 h t i / ¯ Y m − j +1 ¯ D / j ¯ D / j +1 + m X j =[ m/ h t i / ¯ Y / m − j +1 ¯ Y / m − j +2 ¯ D j . (2.23)And | Z H ∂ y u s ∂ mx ˜ bθ α ∂ mx ˜ udxdy | ≤k ∂ y u s k L ∞ y k θ α ∂ mx ˜ b k L k θ α ∂ mx ˜ u k L ≤ C h t i − k θ α ∂ mx ˜ b k L k θ α ∂ mx ˜ u k L , that is, | Z H ∂ y u s ∂ mx ˜ bθ α ∂ mx ˜ udxdy |k θ α ∂ mx ˜ u k L ≤ C h t i − k θ α ∂ mx ˜ b k L , (2.24)where we have used k ∂ y u s k L ∞ y ≤ C h t i by the assumption (H). We now consider R , m X j =0 ( mj ) Z H ∂ m − jx v∂ y u s ∂ jx ψθ α ∂ mx ˜ udxdy. | R | ≤ [ m/ X j =0 ( mj ) k ∂ m − jx v k L x L ∞ y k θ α ∂ y u s k L y k ∂ jx ψ k L ∞ xy k θ α ∂ mx ˜ u k L + m X j =[ m/ ( mj ) k ∂ m − jx v k L ∞ xy k θ α ∂ y u s k L y k ∂ jx ψ k L x L ∞ y k θ α ∂ mx ˜ u k L . For 0 ≤ j ≤ [ m/ k ∂ m − jx v k L x L ∞ y ≤ C h t i / k θ α ∂ m − j +1 x ˜ u k L + C h t i / k θ α ∂ m − j +1 x ˜ b k L , and k θ α ∂ y u s k L y ≤ C h t i / , provided that α < k ∂ jx ψ k L ∞ xy ≤ C h t i / k θ α ∂ jx ˜ b k / L k θ α ∂ j +1 x ˜ b k / L . For [ m/
2] + 1 ≤ j ≤ m , we have k ∂ m − jx v k L ∞ xy ≤ C h t i / k θ α ∂ m − j +1 x ˜ u k / L k θ α ∂ m − j +2 x ˜ u k / L + C h t i / k θ α ∂ m − j +1 x ˜ b k / L k θ α ∂ m − j +2 x ˜ b k / L . And k ∂ jx ψ k L x L ∞ y ≤ C h t i / k θ α ∂ jx ˜ b k L . Hence, | R | τ m M m k θ α ∂ mx ˜ u k L ≤ C ( τ ( t )) / [ m/ X j =0 ( h t i − / Y m − j +1 + h t i − / ¯ Y m − j +1 ) ¯ X / j ¯ X / j +1 (2.25)+ m X j =[ m/ ( h t i − / Y / m − j +1 Y / m − j +2 + h t i − / ¯ Y / m − j +1 ¯ Y / m − j +2 ) ¯ X j . Combining the estimates (2.18)-(2.25) and summing over m ≥ ddt k ˜ u k X τ,α + X m ≥ τ m M m k θ α ∂ mx ∂ y ˜ u k L k θ α ∂ mx ˜ u k L + α (1 − α )4 h t i X m ≥ τ m M m k θ α z∂ mx ˜ u k L k θ α ∂ mx ˜ u k L − α h t i k ˜ u k X τ,α − C h t i k ˜ b k X τ,α ≤ ˙ τ ( t ) k ˜ u k Y τ,α (2.26)+ C ( τ ( t )) / (cid:16) h t i − / ( k ˜ u k X τ,α + k ˜ b k X τ,α ) + h t i / ( k ˜ u k D τ,α + k ˜ b k D τ,α ) (cid:17) ( k ˜ u k Y τ,α + k ˜ b k Y τ,α ) , { a j } j ≥ and { b j } j ≥ , X m ≥ m X j =0 a j b m − j ≤ X j ≥ a j X j ≥ b j . Choosing α ≤ / ddt k ˜ u k X τ,α + X m ≥ τ m M m k θ α ∂ mx ∂ y ˜ u k L k θ α ∂ mx ˜ u k L − α h t i k ˜ u k X τ,α − C h t i k ˜ b k X τ,α ≤ ˙ τ ( t ) k ˜ u k Y τ,α + C ( τ ( t )) / (cid:16) h t i − / ( k ˜ u k X τ,α + k ˜ b k X τ,α ) + h t i / ( k ˜ u k D τ,α + k ˜ b k D τ,α ) (cid:17) × ( k ˜ u k Y τ,α + k ˜ b k Y τ,α ) . (2.27) Similarly, for m ≥
0, by applying the tangential derivative operator ∂ mx to (2 . and multiplyingit by θ α ∂ mx ˜ b , the integration over H gives Z H ∂ mx ( ∂ t ˜ b − ∂ y ˜ b − (1 + b ) ∂ x ˜ u − g∂ y ˜ u + ( u s + u ) ∂ x ˜ b + v∂ y ˜ b − g∂ y u s ψ ) θ α ∂ mx ˜ bdxdy = 0 . (2.28)We now estimate (2 .
28) term by term as follows. Firstly, Z H ∂ t ∂ mx ˜ bθ α ∂ mx ˜ bdxdy = 12 ddt Z H ( ∂ mx ˜ b ) θ α dxdy − Z H ( ∂ mx ˜ b ) θ α ddt θ α dxdy (2.29)= 12 ddt k θ α ∂ mx ˜ b k L + α h t i k θ α z∂ mx ˜ b k L . And − Z H ∂ y ∂ mx ˜ bθ α ∂ mx ˜ bdxdy = k θ α ∂ mx ∂ y ˜ b k L + Z H ∂ y ∂ mx ˜ b∂ y ( θ α ) ∂ mx ˜ bdxdy, where we have used the boundary condition ∂ y ∂ mx ˜ b | y =0 = 0. Moreover, Z H ∂ y ∂ mx ˜ b∂ y ( θ α ) ∂ mx ˜ bdxdy = − Z H ( ∂ mx ˜ b ) ∂ y ( θ α ) dxdy = − α h t i k θ α ∂ mx ˜ b k L − α h t i k θ α z∂ mx ˜ b k L . Hence, − Z H ∂ y ∂ mx ˜ bθ α ∂ mx ˜ bdxdy = k θ α ∂ mx ∂ y ˜ b k L − α h t i k θ α ∂ mx ˜ b k L − α h t i k θ α z∂ mx ˜ b k L . (2.30)Similar to Subsection 2.1, the nonlinear terms can be estimated as follows. Firstly, Z H ∂ mx ((1 + b ) ∂ x ˜ u ) θ α ∂ mx ˜ bdxdy = m X j =0 ( mj ) Z H ∂ m − jx ˜ b∂ j +1 x ˜ uθ α ∂ mx ˜ bdxdy , R , | R | ≤ [ m/ X j =0 ( mj ) k ∂ m − jx ˜ b k L x L ∞ y k θ α ∂ j +1 x ˜ u k L ∞ x L y k θ α ∂ mx ˜ b k L + m X j =[ m/ ( mj ) k ∂ m − jx ˜ b k L ∞ xy k θ α ∂ j +1 x ˜ u k L k θ α ∂ mx ˜ b k L . For 0 ≤ j ≤ [ m/ k ∂ m − jx ˜ b k L x L ∞ y ≤ C k θ α ∂ m − jx ˜ b k / L k θ α ∂ m − jx ∂ y ˜ b k / L , and k θ α ∂ j +1 x ˜ u k L ∞ x L y ≤ C k θ α ∂ j +1 x ˜ u k / L k θ α ∂ j +2 x ˜ u k / L . For [ m/
2] + 1 ≤ j ≤ m , k ∂ m − jx ˜ b k L ∞ xy ≤ C k θ α ∂ m − jx ˜ b k / L k θ α ∂ m − jx ∂ y ˜ b k / L k θ α ∂ m − j +1 x ˜ b k / L k θ α ∂ m − j +1 x ∂ y ˜ b k / L . Hence, | R | τ m M m k θ α ∂ mx ˜ b k L ≤ C ( τ ( t )) / [ m/ X j =0 ¯ X / m − j ¯ D / m − j Y / j +1 Y / j +2 + m X j =[ m/ ¯ X / m − j ¯ X / m − j +1 ¯ D / m − j ¯ D / m − j +1 Y j +1 . (2.31)Moreover, Z H ∂ mx ( g∂ y ˜ u ) θ α ∂ mx ˜ bdxdy = m X j =0 ( mj ) Z H ∂ m − jx g∂ jx ∂ y ˜ uθ α ∂ mx ˜ bdxdy , R , and | R | ≤ [ m/ X j =0 ( mj ) k ∂ m − jx g k L x L ∞ y k θ α ∂ jx ∂ y ˜ u k L ∞ x L y k θ α ∂ mx ˜ b k L + m X j =[ m/ ( mj ) k ∂ m − jx g k L ∞ xy k θ α ∂ jx ∂ y ˜ u k L k θ α ∂ mx ˜ b k L . For 0 ≤ j ≤ [ m/ k ∂ m − jx g k L x L ∞ y ≤ C h t i / k θ α ∂ m − j +1 x ˜ b k L , and k θ α ∂ jx ∂ y ˜ u k L ∞ x L y ≤ C k θ α ∂ jx ∂ y ˜ u k / L k θ α ∂ j +1 x ∂ y ˜ u k / L . m/
2] + 1 ≤ j ≤ m , k ∂ m − jx g k L ∞ xy ≤ C h t i / k θ α ∂ m − j +1 x ˜ b k / L k θ α ∂ m − j +2 x ˜ b k / L . Therefore, | R | τ m M m k θ α ∂ mx ˜ b k L ≤ C ( τ ( t )) / [ m/ X j =0 h t i / ¯ Y m − j +1 D / j D / j +1 + m X j =[ m/ h t i / ¯ Y / m − j +1 ¯ Y / m − j +2 D j . (2.32)Denote R , m X j =0 ( mj ) Z H ∂ m − jx u∂ j +1 x ˜ bθ α ∂ mx ˜ bdxdy, then | R | ≤ [ m/ X j =0 ( mj ) k ∂ m − jx u k L x L ∞ y k θ α ∂ j +1 x ˜ b k L ∞ x L y k θ α ∂ mx ˜ b k L + m X j =[ m/ ( mj ) k ∂ m − jx u k L ∞ xy k θ α ∂ j +1 x ˜ b k L k θ α ∂ mx ˜ b k L . Similar to the estimation on R , we can obtain | R | τ m M m k θ α ∂ mx ˜ b k L ≤ C ( τ ( t )) / [ m/ X j =0 ( X / m − j D / m − j + h t i − / ¯ X m − j ) ¯ Y / j +1 ¯ Y / j +2 (2.33)+ m X j =[ m/ ( X / m − j X / m − j +1 D / m − j D / m − j +1 + h t i − / ¯ X / m − j ¯ X / m − j +1 ) ¯ Y j +1 . And Z H ∂ mx ( v∂ y ˜ b ) θ α ∂ mx ˜ bdxdy = m X j =0 ( mj ) Z H ∂ m − jx v∂ jx ∂ y ˜ bθ α ∂ mx ˜ bdxdy , R . Thus | R | ≤ [ m/ X j =0 ( mj ) k ∂ m − jx v k L x L ∞ y k θ α ∂ jx ∂ y ˜ b k L ∞ x L y k θ α ∂ mx ˜ b k L + m X j =[ m/ ( mj ) k ∂ m − jx v k L ∞ xy k θ α ∂ jx ∂ y ˜ b k L k θ α ∂ mx ˜ b k L . R , we have | R | τ m M m k θ α ∂ mx ˜ b k L ≤ C ( τ ( t )) / [ m/ X j =0 ( h t i / Y m − j +1 + h t i / ¯ Y m − j +1 ) ¯ D / j ¯ D / j +1 (2.34)+ m X j =[ m/ ( h t i / Y / m − j +1 Y / m − j +2 + h t i / ¯ Y / m − j +1 ¯ Y / m − j +2 ) ¯ D j . Denote R , Z H ∂ mx ( g∂ y u s ψ ) θ α ∂ mx ˜ bdxdy = m X j =0 Z H ∂ m − jx g∂ y u s ∂ jx ψθ α ∂ mx ˜ bdxdy. Then | R | ≤ [ m/ X j =0 ( mj ) k ∂ m − jx g k L x L ∞ y k θ α ∂ y u s k L y k ∂ jx ψ k L ∞ xy k θ α ∂ mx ˜ b k L + m X j =[ m/ ( mj ) k ∂ m − jx g k L ∞ xy k θ α ∂ y u s k L y k ∂ jx ψ k L x L ∞ y k θ α ∂ mx ˜ b k L . For 0 ≤ j ≤ [ m/ k ∂ m − jx g k L x L ∞ y ≤ C h t i / k θ α ∂ m − j +1 x ˜ b k L , and k θ α ∂ y u s k L y ≤ C h t i / , provided that α < k ∂ jx ψ k L ∞ xy ≤ C h t i / k θ α ∂ jx ˜ b k / L k θ α ∂ j +1 x ˜ b k / L . For [ m/
2] + 1 ≤ j ≤ m , k ∂ m − jx g k L ∞ xy ≤ C h t i / k θ α ∂ m − j +1 x ˜ b k / L k θ α ∂ m − j +2 x ˜ b k / L , and k ∂ jx ψ k L x L ∞ y ≤ C h t i / k θ α ∂ jx ˜ b k L . Consequently, | R | τ m M m k θ α ∂ mx ˜ b k L ≤ C ( τ ( t )) / [ m/ X j =0 h t i − / ¯ Y m − j +1 ¯ X / j ¯ X / j +1 (2.35)17 m X j =[ m/ h t i − / ¯ Y / m − j +1 ¯ Y / m − j +2 ¯ X j . From the estimates (2.29)-(2.35), summing over m ≥ ddt k ˜ b k X τ,α + X m ≥ τ m M m k θ α ∂ mx ∂ y ˜ b k L k θ α ∂ mx ˜ b k L + α (1 − α )4 h t i X m ≥ τ m M m k θ α z∂ mx ˜ b k L k θ α ∂ mx ˜ b k L − α h t i k ˜ b k X τ,α ≤ ˙ τ ( t ) k ˜ b k Y τ,α + C ( τ ( t )) / (cid:16) h t i − / ( k ˜ u k X τ,α + k ˜ b k X τ,α ) + h t i / ( k ˜ u k D τ,α + k ˜ b k D τ,α ) (cid:17) × ( k ˜ u k Y τ,α + k ˜ b k Y τ,α ) . (2.36)Similarly, by choosing α ≤ /
2, we have ddt k ˜ b k X τ,α + X m ≥ τ m M m k θ α ∂ mx ∂ y ˜ b k L k θ α ∂ mx ˜ b k L − α h t i k ˜ b k X τ,α ≤ ˙ τ ( t ) k ˜ b k Y τ,α + C ( τ ( t )) / (cid:16) h t i − / ( k ˜ u k X τ,α + k ˜ b k X τ,α ) + h t i / ( k ˜ u k D τ,α + k ˜ b k D τ,α ) (cid:17) × ( k ˜ u k Y τ,α + k ˜ b k Y τ,α ) . (2.37) By the uniform a priori estimates obtained in Section 2, we now estimate the low bound onlifespan of the solution. Consider (2 .
27) + K × (2 .
37) with
K > ddt ( k ˜ u k X τ,α + K k ˜ b k X τ,α ) + X m ≥ τ m M m ( k θ α ∂ mx ∂ y ˜ u k L k θ α ∂ mx ˜ u k L + K k θ α ∂ mx ∂ y ˜ b k L k θ α ∂ mx ˜ b k L ) − α h t i k ˜ u k X τ,α − ( C + Kα h t i k ˜ b k X τ,α (3.1) ≤ (cid:18) ˙ τ ( t ) + C ( K + 1)( τ ( t )) / (cid:16) h t i − / ( k ˜ u k X τ,α + k ˜ b k X τ,α ) + h t i / ( k ˜ u k D τ,α + k ˜ b k D τ,α ) (cid:17)(cid:19) × ( k ˜ u k Y τ,α + K k ˜ b k Y τ,α ) . Choose the function τ ( t ) satisfies the following ODE. ddt ( τ ( t )) / + 3 C ( K + 1)2 (cid:16) h t i − / ( k ˜ u k X τ,α + k ˜ b k X τ,α ) + h t i / ( k ˜ u k D τ,α + k ˜ b k D τ,α ) (cid:17) = 0 . (3.2)From (3.1) and (3.2), one has ddt ( k ˜ u k X τ,α + K k ˜ b k X τ,α ) + X m ≥ τ m M m k θ α ∂ mx ∂ y ˜ u k L k θ α ∂ mx ˜ u k L + K k θ α ∂ mx ∂ y ˜ b k L k θ α ∂ mx ˜ b k L ! α h t i k ˜ u k X τ,α − ( C + Kα h t i k ˜ b k X τ,α ≤ . (3.3)By lemma 2.2, we have X m ≥ τ m M m k θ α ∂ mx ∂ y ˜ u k L k θ α ∂ mx ˜ u k L ≥ α / β h t i / k ˜ u k D τ,α + α (1 − β ) h t i k ˜ u k X τ,α , (3.4)and X m ≥ τ m M m k θ α ∂ mx ∂ y ˜ b k L k θ α ∂ mx ˜ b k L ≥ α / β h t i / k ˜ b k D τ,α + α (1 − β ) h t i k ˜ b k X τ,α , (3.5)for β , β ∈ (0 , / ddt ( k ˜ u k X τ,α + K k ˜ b k X τ,α ) + 12 ( α (1 − β )) 1 h t i k ˜ u k X τ,α + 12 ( α (1 − β ) − CK ) 1 h t i K k ˜ b k X τ,α + α / β h t i / k ˜ u k D τ,α + Kα / β h t i / k ˜ b k D τ,α ≤ . (3.6)Choose α = 12 − δ, β = δ , β = δ , K = 4 Cδ , where 0 < δ < / α (1 − β ) = 12 − δ + δ , and α (1 − β ) − CK = 12 − δ + δ . Then, there exist small positive constants η = δ and η = δ ddt ( k ˜ u k X τ,α + K k ˜ b k X τ,α ) + 1 / − η h t i (cid:16) k ˜ u k X τ,α + K k ˜ b k X τ,α (cid:17) + η h t i / ( k ˜ u k D τ,α + K k ˜ b k D τ,α ) ≤ . (3.7)It implies that ddt ( k ˜ u k X τ,α + K k ˜ b k X τ,α ) h t i / − η + 1 / − η h t i / η (cid:16) k ˜ u k X τ,α + K k ˜ b k X τ,α (cid:17) + η h t i / η ( k ˜ u k D τ,α + K k ˜ b k D τ,α ) ≤ . (3.8)19s a consequence,( k ˜ u k X τ,α + K k ˜ b k X τ,α ) h t i / − η + Z t η h s i / η ( k ˜ u ( s ) k D τ,α + K k ˜ b ( s ) k D τ,α ) ds ≤ ( k ˜ u (0) k X τ,α + K k ˜ b (0) k X τ,α ) ≤ C (1 + K ) ε, (3.9)where we have used (2.15). Then, by noting that K = 4 Cδ , one has3 C K + 1) Z t h s i − / ( k ˜ u ( s ) k X τ,α + k ˜ b ( s ) k X τ,α ) ds = 3 C Cδ + 1) Z t h s i − / ( k ˜ u ( s ) k X τ,α + k ˜ b ( s ) k X τ,α ) ds ≤ CC ε Cδ + 1) Z t h s i − / η ds ≤ CC ε ( 4 Cδ + 1) h t i / η , (3.10)and 3 C K + 1) Z t h s i / ( k ˜ u ( s ) k D τ,α + k ˜ b ( s ) k D τ,α ) ds = 3 C Cδ + 1) Z t h s i / ( k ˜ u ( s ) k D τ,α + k ˜ b ( s ) k D τ,α ) ds = 3 C Cδ + 1) 8 δ Z t h s i / η η h s i / η ( k ˜ u ( s ) k D τ,α + K k ˜ b ( s ) k D τ,α ) ds ≤ ( 4 Cδ + 1) CC δ h t i / η ε. (3.11)On the other hand, (3.2) implies that τ ( t ) / = τ (0) / − C ( K + 1)2 Z t ( h s i − / ( k ˜ u k X τ,α + k ˜ b k X τ,α ) + h s i / ( k ˜ u k D τ,α + k ˜ b k D τ,α )) ds. (3.12)From (3.10), (3.11) and (3.12), one has τ ( t ) / ≥ τ / − max { CC ( 4 Cδ + 1) h t i / η ε, ( 4 Cδ + 1) CC δ h t i / η ε } , for all t ≥ δ = 1ln(1 /ε ) . It is straightforward to show that τ ( t ) ≥ τ , in the time interval [0 , T ε ], where T ε satisfies T ε = ¯ C (cid:18) ε (ln(1 /ε )) (cid:19) − / (ln(1 /ε )+2) − . (3.13)This gives the estimate on the lifespan of solution stated in (1.13).20 The Proof of Uniqueness Part in Theorem 1.1
Assume there are two solutions (˜ u , ˜ b ) and (˜ u , ˜ b ) to (2.11) with the same initial data (˜ u , ˜ b ),which satisfies k (˜ u , ˜ b ) k X τ ,α ≤ ε . And the tangential radii of analyticity of (˜ u , ˜ b ) and (˜ u , ˜ b )are τ ( t ) and τ ( t ), respectively.Define τ ( t ) by d ( τ ( t )) / dt + 3 C ( K + 1)2 (cid:16) h t i − / ( k ˜ u k X τ t ) ,α + k ˜ b k X τ t ) ,α )+ h t i / ( k ˜ u k D τ t ) ,α + k ˜ b k D τ t ) ,α ) (cid:17) = 0 , (4.1)with initial data τ (0) = τ . (4.2)By the estimates obtained in Section 2, there exists a time interval [0 , T ] with T ≤ T ε suchthat τ ≤ τ ( t ) ≤ τ ≤ min { τ , τ } t ∈ [0 , T ].Set U = ˜ u − ˜ u and B = ˜ b − ˜ b . Then ∂ t U − ∂ y U + ( u s + u ) ∂ x U + ( v − v ) ∂ y ˜ u − (1 + b ) ∂ x B − ( g − g ) ∂ y ˜ b − ∂ y u s B + ( v − v ) ∂ y u s ψ + R s = 0 , (4.4)and ∂ t B − ∂ y B − (1 + b ) ∂ x U − ( g − g ) ∂ y ˜ u + ( u s + u ) ∂ x B + ( v − v ) ∂ y ˜ b − ( g − g ) ∂ y u s ψ + R s = 0 , (4.5)with the source terms R s and R s given by R s = ( u − u ) ∂ x ˜ u + v ∂ y U − ( b − b ) ∂ x ˜ b − g ∂ y B + v ∂ y u s ( ψ − ψ ) , (4.6)and R s = − ( b − b ) ∂ x ˜ u − g ∂ y U + ( u − u ) ∂ x ˜ b + v ∂ y B − g ∂ y u s ( ψ − ψ ) . (4.7)Note that the initial data and the boundary conditions are U ( t, x, y ) | t =0 = 0 , B ( t, x, y ) | t =0 = 0 , (4.8)and (cid:26) U | y =0 = 0 ,U | y = ∞ = 0 , and (cid:26) ∂ y B | y =0 = 0 ,B | y = ∞ = 0 . (4.9)21imilar to Section 2, we have ddt k U k X τ,α + X m ≥ τ m M m k θ α ∂ y ∂ mx U k L k θ α ∂ mx U k L − α h t i k U k X τ,α − C h t i k B k X τ,α (4.10) ≤ ˙ τ ( t ) k U k Y τ,α + C ( τ ( t )) / (cid:16) h t i − / ( k ˜ u k X τ ( t ) ,α + k ˜ b k X τ ( t ) ,α ) + h t i / ( k ˜ u k D τ ( t ) ,α + k ˜ b k D τ t ) ,α ) (cid:17) × ( k U k Y τ,α + k B k Y τ,α )+ C ( τ ( t )) / ( k ˜ u k Y τ,α + k ˜ b k Y τ,α )( h t i − / ( k U k X τ,α + k B k X τ,α ) + h t i / ( k U k D τ,α + k B k D τ,α )) , and ddt k B k X τ,α + X m ≥ τ m M m k θ α ∂ y ∂ mx U k L k θ α ∂ mx U k L − α h t i k B k X τ,α (4.11) ≤ ˙ τ ( t ) k B k Y τ,α + C ( τ ( t )) / (cid:16) h t i − / ( k ˜ u k X τ ( t ) ,α + k ˜ b k X τ ( t ) ,α ) + h t i / ( k ˜ u k D τ ( t ) ,α + k ˜ b k D τ ( t ) ,α ) (cid:17) × ( k U k Y τ,α + k B k Y τ,α )+ C ( τ ( t )) / ( k ˜ u k Y τ,α + k ˜ b k Y τ,α )( h t i − / ( k U k X τ,α + k B k X τ,α ) + h t i / ( k U k D τ,α + k B k D τ,α )) . Then, we have ddt ( k U k X τ,α + K k B k X τ,α ) + X m ≥ τ m M m (cid:18) k θ α ∂ y ∂ mx U k L k θ α ∂ mx U k L + K k θ α ∂ y ∂ mx B k L k θ α ∂ mx B | L (cid:19) − α h t i k U k X τ,α − C + Kα h t i k B k X τ,α ≤ (cid:18) ˙ τ ( t ) + C (1 + K )( τ ( t )) / (cid:16) h t i − / ( k ˜ u k X τ ( t ) ,α + k ˜ b k X τ ( t ) ,α ) + h t i / ( k ˜ u k D τ ( t ) ,α + k ˜ b k D τ ( t ) ,α ) (cid:17) × ( k U k Y τ,α + K k B k Y τ,α ) (cid:1) (4.12)+ C (1 + K )( τ ( t )) / ( k ˜ u k Y τ,α + k ˜ b k Y τ,α )( h t i − / ( k U k X τ,α + k B k X τ,α ) + h t i / ( k U k D τ,α + k B k D τ,α )) . From (4.1), one has˙ τ ( t ) + C ( K + 1)( τ ( t )) / (cid:16) h t i − / ( k ˜ u k X τ,α + k ˜ b k X τ,α )+ h t i / ( k ˜ u k D τ,α + k ˜ b k D τ,α ) (cid:17) ( k U k Y τ,α + k B k Y τ,α ) ≤ , (4.13)because τ ( t ) ≤ τ ( t ) and the norms X τ,α and D τ,α are increasing in τ .By the inequalities (3.4), (3.5) and (4.13), one has ddt ( k U k X τ,α + K k B k X τ,α ) + α (1 − β )2 h t i k U k X τ,α + α (1 − β ) − CK h t i K k B k X τ,α α / β h t i / k U k D τ,α + α / β h t i / K k B k D τ,α (4.14) ≤ C (1 + K )( τ ( t )) / ( k ˜ u k Y τ,α + k ˜ b k Y τ,α )( h t i − / ( k U k X τ,α + k B k X τ,α ) + h t i / ( k U k D τ,α + k B k D τ,α )) , for β , β ∈ (0 , / k ˜ u k Y τ,α ≤ τ k ˜ u k X τ,α ≤ τ k ˜ u k X τ ,α ≤ C (1 + K ) τ ε h t i − / η (4.15)and k ˜ b k Y τ,α ≤ τ k ˜ b k X τ,α ≤ τ k ˜ b k X τ ,α ≤ C (1 + K ) τ ε h t i − / η , (4.16)we have C (1 + K )( τ ( t )) / ( k ˜ u k Y τ,α + k ˜ b k Y τ,α ) ≤ K ) CC ε ( τ ( t )) / h t i / − η . (4.17)Notice that t ∈ [0 , T ε ] with T ε = ε − δ , where δ is a fixed small positive constant. As inSection 3, we can choose α = 1 / − δ, β = β = δ , K = 4 Cδ and δ = 1 / ln(1 /ε ), then η can bechosen to be δ . Let ε suitably small to have α (1 − β )2 > K ) CC ε h t i / η ( τ ( t )) / , α (1 − β ) − C/K > K ) CC ε h t i / η ( τ ( t )) / K , and α / β > K ) CC ε h t i / η ( τ ( t )) / , α / β > K ) CC ε h t i / η ( τ ( t )) / K . (4.14) and (4.17) imply that ddt ( k U k X τ,α + K k B k X τ,α ) + η ( k U k X τ,α + K k B k X τ,α ) ≤ η > t ∈ [0 , T ε ]. It implies uniqueness of solution to (2.11) in thetime interval [0 , T ε ]. Acknowledgement:
Feng Xie’ research was supported by National Nature Science Foundationof China 11571231, the China Scholarship Council and Shanghai Jiao Tong University SMC(A).Tong Yang’s research was supported by internal research funding of City University of HongKong, 7004847. 23 eferenceseferences