On the vanishing dissipation limit for the incompressible MHD equations on bounded domains
aa r X i v : . [ m a t h . A P ] J u l On the vanishing dissipation limit for the incompressibleMHD equations on bounded domains
Qin Duan , Yuelong Xiao ∗ and Zhouping Xin Hunan Key Laboratory for Computation and Simulation in Science and Engineering,School of Mathematics and Computational Science, Xiangtan University College of Mathematics and Statistics, Shenzhen University The Institute of Mathematical Sciences, The Chinese University of Hong Kong
Abstract
In this paper, we investigate the solvability, regularity and the vanishing dissipa-tion limit of solutions to the three-dimensional viscous magneto-hydrodynamic (MHD)equations in bounded domains. On the boundary, the velocity field fulfills a Navier-slipcondition, while the magnetic field satisfies the insulating condition. It is shown thatthe initial-boundary problem has a global weak solution for a general smooth domain.More importantly, for a flat domain, we establish the uniform local well-posedness ofthe strong solution with higher order uniform regularity and the asymptotic conver-gence with a rate to the solution of the ideal MHD as the dissipation tends to zero.
Keywords:
MHD equations, Initial-boundary-value problem, Vanishing dissipation limit.
Let Ω ⊂ R be a bounded smooth domain with boundary ∂ Ω and n be the outwardnormal vector on ∂ Ω. The 3-dimentional (3-D) viscous magnetic-hydrodynamic equations ∗ Corresponding author: [email protected]. ∂ t u − ν ∆ u + ( ∇ × u ) × u + B × ( ∇ × B ) + ∇ P = 0 in Ω , ∇ · u = 0 in Ω ,∂ t B − µ ∆ B = ∇ × ( u × B ) in Ω , ∇ · B = 0 in Ω , (1.1)while u and B are the velocity field and magnetic field respectively, P is the pressure, ν isthe viscosity coefficient and µ is the magnetic diffusion coefficient, ∇· and ∇× denote thediv and curl operators respectively. (1.1) is supplemented with the following initial data( u, B )( x, t = 0) = ( u , B )( x ) , x ∈ Ω . (1.2)On the boundary, the velocity field is assumed to satisfy the following Navier-slip condition([41, 44]): u · n = 0 , ∇ × u × n = 0 on ∂ Ω , (1.3)and the magnetic field satisfies the insulating boundary condition ([13, 19]) B × n = 0 on ∂ Ω , (1.4)where n is the unit outnormal of ∂ Ω.The aim of this paper is to study the solvability, regularity and the asymptotic behavioras the dissipations vanish ( ν → + and µ → + ) of the solution to the initial-boundary-valueproblem (IBVP) (1.1) − (1.4). In particular, we are concerned with the uniform (with respectto the dissipation ν and µ ) well-posedness of the strong solution to the IBVP (1.1)-(1.4) andwhether there are strong boundary layers near the physical boundary.This study is motivated strongly by the vanishing viscosity limit problem for the incom-pressible Navier-Stokes equations, which is a classical problem in the mathematical theory offluid dynamics and has been studied extensively in the case without physical boundaries (see[10, 11, 20, 21, 28] for instance). Yet the vanishing viscosity problem for the Navier-Stokesequation becomes more complicated and challenging in the presence of physical boundariesdespite its fundamental importance both physically and theoretically in understanding theboundary layer behavior of viscous flows for large Renolds number. Substantial difficultiesarise due to the appearance of strong boundary layers in the case that the commonly usedno-slip condition is imposed on the boundary, which makes it extremely difficult to justify thewell-known Prandtl’s boundary layer theory with few notable exception ([21, 27, 32, 33, 36]).2n the other hand, if a Navier-slip condition, such as (1.3), is imposed on the boundary, thenvorticities created near the boundary can be controlled so the possible viscous boundary lay-ers are weak, which makes it possible to obtain strong convergence of the viscous fluid to theideal one, as shown in many recent works, see ([1]-[9], [16, 17, 24, 25, 29, 37, 38, 39, 40, 42, 43])and the references therein. In particular, for a flat domain Ω ⊂ R with the slip boundarycondition (1.3), Xiao and Xin ([39]) introduced an argument to obtain the uniform H (Ω)estimates on the solution to the 3-D Navier-Stokes equations and further the convergencewith an optimal rate to the solution to the incompressible Euler system as the viscosity tendsto zero. One is also referred to ([1, 2, 38, 39, 40]) for further studies in these directions.As for the hydrodynamic case, the vanishing dissipation problem for the MHD equationsin the presence of physical boundaries is important both in theory and applications, andis also a challenging topic. Due to the great complexities of the MHD system, there havebeen less results in the rigorous mathematical treatment of boundary layers for the MHDcompared to the hydrodynamics until recently. Even for fixed positive dissipation, i.e., ν > µ >
0, the well-posedness of initial boundary value problem (IBVP) for MHD hasbeen developed mostly for the case that the velocity field u satisfies the no-slip conditionand while the magnetic field satisfies the perfect conducting condition, i.e., u = 0 , on ∂ Ω , (1.5) B · n = 0 , ∇ × B × n = 0 on ∂ Ω , (1.6)see [12, 19, 34] and the references therein. Indeed, it is a subtle problem to prescribeboundary conditions for the magnetic field B mathematically since the magnetic field B satisfies a system of parabolic equation up to leading order with the additional divergencefree constraint, i.e., ∂ t B − µ ∆ B = ∇ × ( u × B ) in Ω , ∇ · B = 0 in Ω , (1.7)so the standard parabolic boundary condition for B on ∂ Ω, such as the homogeneous Dirichletboundary condition, may lead to an overdetermined problem. In fact, as far as we are aware,all the known well-posedness theory for the initial boundary value problem for the unsteadyMHD, except [14], deals with boundary conditions as either (1.5), (1.6) ([12, 19, 34]), or(1.3) and (1.6), ([44]), or B · n = 0 , ( ∇ × B ) · n = 0 , ∆ B · n = 0 , on ∂ Ω , (1.8)3ee [35]. In all these three cases, the boundary conditions for the magnetic field are character-ized as that with these boundary conditions for B , the corresponding Stokes and Laplacianoperators are identical, which is not true in general for the Dirichlet boundary conditions,see ([11, 30]). Thus we do not understand the argument in [14]. For the stationary MHD,the well-posedness has been well established in the case that the velocity field satisfies theno-slip (Dirichlet) condition and the magnetic field satisfies either the perfectly conductingcondition (1.6), or the perfect insulating condition (1.4), see [13] and [19] and referencestherein for generalizations. For more discussions on boundary conditions for MHD eithermathematically or physically, we refer to ([15, 18, 19, 31, 35]) and references therein. To ourknowledge, the well-posedness of the IBVP, (1.1)-(1.4) has not been considered before. Thiswill be one of the consequences of the studies in this paper.In the case that the dissipation coefficients are positive but can be arbitrarily small,i.e. 0 < ν, µ ≪
1, the uniform well-posedness and asymptotic behavior of solutions tothe initial-boundary value problem for the MHD are difficult to study due to the possibleappearance of boundary layers. The first rigorous result along this line is due to Xiao-Xin-Wu [44] in where they treat 3-dimensional flat domains with the slip boundary conditions,(1.3), for the velocity field and the perfect conducting boundary conditions, (1.6), for themagnetic field, get the local uniform well-posedness of the solutions to the IBVP for theMHD with uniform H (Ω) − estimate independent of the dissipations, and finally obtain theasymptotic convergence with an optimal rate as the dissipations tend to zero, just as thecorresponding results for the hydrodynamic case established by Xiao-Xin in [39]. However, itshould be noted that in the theory of [44], there are no strong boundary layers due to the slipboundary condition as (1.3). In the case that the velocity field satisfies the no-slip boundarycondition (1.5), strong boundary layers are expected and it is much harder to study the zerodissipation limit problem. However, recently, for the case that the velocity field satisfies theno-slip conditions (1.5) and the magnetic field satisfies (1.6) for the domain which is a halfplane, Liu-Xie-Yang [26] solved successfully the zero dissipation limit problem for the MHDsystem (locally) provided the magnetic field is non-trivial by solving the Prandtl’s boundarylayer equations for MHD first. Note that the theory in [26] depends crucially on the presenceof the magnetic field so that the results do not hold for the hydro-dynamic equations.The main results in the paper concern with solutions to the MHD equation (1.1) ona smooth bounded domain, subject to the slip boundary condition (1.3) for the velocityfield u and the perfect insulating boundary condition (1.4) for the magnetic field B withsuitable initial condition (1.2). First, for fixed positive viscosity and magnetic diffusion4oefficient and general smooth bounded domains, we show that there exists a global in timeweak solution to the IBVP (1.1)-(1.4) for general initial data, and such a weak solutionbecomes the unique strong solution for short time for general regular initial data. As forthe Navier-Stokes equations, such a local strong solution can be extended globally in timefor suitable small initial data (depending on the dissipation coefficients). Second, for flatdomains, Ω = T × (0 , T being the torus, we will derive the uniform (independentof the dissipation coefficients) H (Ω) estimates on the solutions (and thus the uniform well-posedness), which then enable us to prove the convergence of the solutions of IBVP (1.1)-(1.4)to the solution of the ideal MHD equations as ν and µ tend to zero. Furthermore, the optimalrate of convergence is obtained also, see Theorem 5.4- Theorem5.6. These are major resultsin this paper.We now make some comments on the analysis of this paper. The existence of weakand strong solutions is proved by using the Galerkin method based on eigenvalue problemsfor the corresponding Stokes operators associated with boundary conditions (1.3) and (1.4)respectively, see Lemma 2.3 and Lemma 2.4. For the uniform well-posedness and vanishingdissipation limit, the basic idea is that the vorticity created near the boundary due tothe slip condition (1.3) is weak and can be controlled as in the case for the Navier-Stokesequation investigated by Xiao-Xin in [39] provided that the effects of the magnetic field canbe taken care of. Thus the results and the approach are similar to the case in [44] where thevelocity field and magnetic field satisfy the boundary condition, (1.3) and (1.6) respectively.However, one cannot apply the analysis in [44] into the case here directly. This is due tothat the boundary conditions (1.3) and (1.6) have the same structure so that one can use theargument of [39] to the system satisfied by velocity field and the magnetic field separatelyto derive the uniform H (Ω) estimates in [44]. However, in our case, the perfect insulatingconditions (1.4) for the magnetic field are completely different from the slip condition (1.3)for the velocity field. Thus to obtain the uniform high order estimates on the solutions tothe IBVP (1.1)-(1.4) by energy method, one needs to use the equations for the velocity fieldand the magnetic field and the corresponding boundary condition simultaneously. Indeed,it turns out that to derive the a priori estimates on the H (Ω) norm of the solutions to(1.1)-(1.4) by suitable energy method, it is crucial to have( ∇× ) u × n = 0 , ( ∇× ) ( B × u ) × n = 0 on ∂ Ω , (1.9)which do not come from the boundary conditions (1.3), (1.4) directly, but hold for strongsolutions to the IBVP (1.1)-(1.4). Indeed, ( ∇× ) u × n = 0 follows from the equations for the5elocity field in (1.1) and the boundary conditions (1.3) and (1.4) (see Lemma 5.1) providedthat ∇ × ( B × ( ∇ × B )) × n | ∂ Ω = 0 . (1.10)To show (1.10), one needs to use that the equations for the magnetic field and Lemma 5.1to obtain that △ B × n | ∂ Ω = 0. Then (1.10) follows from this and the boundary condition(1.3) and (1.4), see Lemma 5.2. Thus ( ∇× ) u × n | ∂ Ω = 0 follows, which, together with (1.3)and (1.4), implies ( ∇× ) ( B × u ) × n | ∂ Ω = 0, Lemma 5.3. As consequences of (1.9) and theobservation of some cancellations of nonlinear terms (see (5.14)), one can use the energymethod to derive the desired uniform H (Ω) estimates and the convergence theory.The rest of the paper is organized as follows. In the next section, we introduce thefunctional spaces associated with the boundary conditions (1.3) and (1.4) and study theeigenvalue problem for the corresponding Stokes operators associated with the boundarycondition (1.3) and (1.4) respectively. The global existence of a weak solution to the IBVP,(1.1)-(1.4), is obtained in section 3, while the well-posedness of local (in time) strong solutionsand regularity properties are established in section 4. Finally, we establish the uniform well-posedness and vanishing dissipation limit results in section 5, which are the main results ofthis paper. In this section, we will introduce some basic tools that will be used for later analysis.Let Ω ⊂ R be a smooth domain and denote H s (Ω) with s ≥ H − s (Ω) with s ≥ H s (Ω). It is well known that: Lemma 2.1.
Let s ≥ be an integer. Let u ∈ H s be a vector-valued function. Then k u k s ≤ C ( k∇ × u k s − + k∇ · u k s − + | u · n | s − + k u k s − ) . (2.1) Lemma 2.2.
Let s ≥ be an integer. Let u ∈ H s (Ω) . Then k u k s ≤ C ( k∇ × u k s − + k∇ · u k s − + | u × n | s − + k u k s − ) . (2.2)See [39, 44] and the references therein.Let X = { u ∈ L ; ∇ · u = 0 , u · n = 0 on ∂ Ω } ,
6e the Hilbert space with the L inner product, and let V = H ∩ X ⊂ X,W = { u ∈ X ∩ H ; ( ∇ × u ) × n = 0 on ∂ Ω } ⊂ X. ˆ X = { x ∈ L ; ∇ · B = 0 } , ˆ V = { x ∈ H ; ∇ · B = 0 , B × n = 0 on ∂ Ω } , ˆ W = H ∩ ˆ V ⊂ ˆ V .
It follows from (2.1) and (2.2) that for any u ∈ H s (Ω) ∩ ( V ∪ ˆ V ) , it holds that k u k s ∼ = k u k + k ( ∇× ) s u k . (2.3) Lemma 2.3.
The Stokes operator A = I − P △ = I − ∆ with its domain by D ( A ) = W ⊂ V satisfying ( Au, v ) = a ( u, v ) ≡ ( u, v ) + Z Ω ∇ × u · ∇ × vdx is a self-adjoint and positive defined operator, with its inverse being compact. Consequently,its countable eigenvalues can be listed as ≤ λ ≤ λ ≤ · · · −→ ∞ and the corresponding eigenvectors { e j } ⊂ W ∩ C ∞ ,i.e., Ae j = (1 + λ j ) e j , or − ∆ e j = λ j e j , which form an orthogonal complete basis of X . For the details, refer to [41].
Remark . This Stokes operator P ∆ = ∆ with the domain D ( A ) = W follows from∆ u · n | ∂ Ω = 0 for u ∈ W . See [41] for the details.Set ˆ e j = ∇ × e j k∇ × e j k , j ∈ N. It then follows that
Lemma 2.4. { ˆ e j } , j ∈ N form an orthogonal complete basis for ˆ X . The bilinear form ˆ a ( u, v ) = ( u, v ) + Z Ω ( ∇ × u ) · ( ∇ × v ) , u, v ∈ ˆ V with the domain D (ˆ a ) = ˆ V is positive and closed, ˆ a is densely defined in ˆ X . ˆ A = I − ∆ isthe self-adjoint extension of the bilinear form ˆ a ( u, v ) with domain D ( ˆ A ) = ˆ W . Also ˆ e j is theeigenvector of ˆ A respect to the eigenvalue λ j . roof. It is clear that { ˆ e j } , j ∈ N is an orthogonal series in ˆ X . Let x ∈ ˆ X and ( x, ∇ × e j ) =0 , ∀ j ∈ N . Since ∇ · x = 0 and x ∈ L (Ω), one has x = ∇ × u for some u ∈ V . It follows from0 = ( x, ∇ × e j ) = ( ∇ × u, ∇ × e j ) = ( u, − ∆ e j ) = λ j ( u, e j ) , ∀ j ∈ N, that u = 0, and then x = 0. Since ˆ X is a Hilbert space, it follows that { ˆ e j } , j ∈ N forman orthogonal complete basis of ˆ X . Clearly, ˆ a ( u, v ) is positive. Since ˆ e j ⊂ D (ˆ a ), for j ∈ N ,then ˆ a is densely defined in ˆ X . From (2.3), one can define the bilinear form ˆ a ( u, v ) as a innerproduct on ˆ V . By using of the trace theorem, and noting the continuity of the divergenceoperator, it follows that ˆ V is closed in H (Ω), and then is a Hilbert space respect to theinner product. Hence, ˆ a ( u, v ) is closed. It is clear that ˆ e j also satisfies − ∆ˆ e j = λ j ˆ e j and the boundary condition ˆ e j × n = 0on the boundary. We can define the operator ˆ A : span { ˆ e j } ˆ X byˆ A ( X u j ˆ e j ) = X u j (1 − ∆)ˆ e j = X u j (1 + λ j )ˆ e j (2.4)for u = P u j ˆ e j ∈ span { ˆ e j } , where span { ˆ e j } is the closure of { ˆ e j } in H (Ω). It is easy tocheck that span { ˆ e j } = ˆ W . Indeed, let v ∈ ˆ W and(ˆ e j , v ) + (∆ˆ e j , ∆ v ) = 0 , ∀ j. It follows that (1 + λ j )(ˆ e j , v ) = 0 , ∀ j. Since ˆ W is a Hilbert space respect to the inner product( u, v ) ˆ A = ( u, v ) + (∆ u, ∆ v ) , so we have v = 0 and span { ˆ e j } = ˆ W .Now, let f ∈ ˆ X , we can write f = X f j ˆ e j .
8t follows that u = P u j ˆ e j ∈ ˆ W , with (1 + λ j ) u j = f j is a solution of ˆ Au = f in the sensethat ˆ a ( u, v ) = ( f, v ) , ∀ v ∈ ˆ V . If f = 0, it follows that ( u, ˆ e j ) + λ j ( u, ˆ e j ) = 0 , by taking v = ˆ e j . Then, u = 0, and we conclude that ˆ A : ˆ W ˆ X is a isomorphism. Thelemma is proved.For u ∈ W , B ∈ ˆ W , we define H : W × ˆ W −→ X by H ( u, B ) = ( ∇ × u ) × u + B × ( ∇ × B ) + ∇ p, where p satisfies △ p = −∇ · (( ∇ × u ) × u + B × ( ∇ × B )) , ∇ p · n = − ( ∇ × u ) × u + B × ( ∇ × B ) · n on ∂ Ωand H : W × ˆ W −→ ˆ X by H ( u, B ) = ∇ × ( B × u ) . In this section, we will establish the global existence of weak solutions to the systems(1.1) − (1.4) by using of the method of Galerkin approximation. Here we consider a generalsmooth bounded domain in R unless stated otherwise. We denote H ∗ be the dual space of H . Definition 3.1. ( u, B ) is called a weak solution of (1.1) − (1.4) with the initial data u ∈ X , B ∈ ˆ X on the time interval [0 , T ) if u ∈ L (0 , T ; V ) ∩ C w ([0 , T ); X ) and B ∈ L (0 , T ; ˆ V ) ∩ C w ([0 , T ); ˆ X ) satisfy u ′ ∈ L (0 , T ; V ∗ ), B ′ ∈ L (0 , T ; ˆ V ∗ ) and( u ′ , φ ) + ν ( ∇ × u, ∇ × φ ) + ( ∇ × u × u, φ ) + ( B × ∇ × B, φ ) = 0 , ( B ′ , ψ ) + µ ( ∇ × B, ∇ × ψ ) + ( u · ∇ B − B · ∇ u, ψ ) = 0 (3.1)9or all φ ∈ V , ψ ∈ ˆ V and for a.e. t ∈ [0 , T ), and u (0) = u , B (0) = B . Theorem 3.1.
Let u ∈ X , B ∈ ˆ X . Let T > . Then there exists at least one weak solution ( u, B ) of (1.1) − (1.4) on [0 , T ) which satisfies the energy inequality ddt ( k u k + k B k ) + 2( ν k∇ × u k + µ k∇ × B k ) ≤ in the sense of the distribution. Proof.
This will be proved by using a Galerkin approximation based on eigenvalue problemsin Lemma 2.2 and Lemma 2.3. Define u m ( t ) = m X j =1 u j ( t ) e j , B m ( t ) = m X j =1 b j ( t )ˆ e j , where u j and b j for j = 1 , , · · · , m solve the following ordinary differential equations u ′ j ( t ) + νλ j u j ( t ) + g j ( U ) = 0 , (3.3) b ′ j ( t ) + µλ j b j ( t ) + g j ( U ) = 0 , (3.4) u j (0) = ( u , e j ) , b j (0) = ( B , ˆ e j ) , (3.5)with U = ( u , u , · · · , u m , B , B , · · · B m ) and g j ( U ) = ( H ( u m , B m ) , e j ) ,g j ( U ) = ( H ( u m , B m ) , ˆ e j ) . It follows from the Lipschitz continuity of ( g kj ( U )) that the initial value problem (3.3) − (3.5)is locally well-posed on interval [0 , T ) for some positive T . Consequently, for any t ∈ [0 , T ),( u m , B m ) solves the following systems of equations( u m ) ′ − ν △ u m + P m H ( u m , B m ) = 0 , (3.6)( B m ) ′ − µ △ B m + ˆ P m H ( u m , B m ) = 0 , (3.7) u m (0) = P m u , B m (0) = ˆ P m B (3.8)where P m denotes the projection of X onto the space spanned by { e j } m and ˆ P m denotes theprojection of ˆ X onto the space spanned by { ˆ e j } m .10ultiplying u m and B m on both side of (3.6) and (3.7) respectively and integrating byparts, lead to ( P m H ( u m , B m ) , u m ) = Z Ω ( B m × ∇ × B m ) · u m dx, ( ˆ P m H ( u m , B m ) , B m ) = Z Ω ∇ × ( B m × u m ) · B m dx. = − Z Ω ( B m × ∇ × B m ) · u m dx. Adding them up, one obtains by a simple computation that ddt ( k u m k + k B m k ) + 2( ν k∇ × u m k + µ k∇ × B m k ) = 0 , (3.9)which implies that u m is bounded in L ∞ (0 , T ; X ) , B m is bounded in L ∞ (0 , T ; ˆ X ) , uniformly for m . (3.10) u m is bounded in L (0 , T ; V ) , B m is bounded in L (0 , T ; ˆ V ) , uniformly for m . (3.11)Note that for φ ∈ V and ψ ∈ ˆ V , it holds that | ( −△ u m , φ ) | = | ( ∇ × u m , ∇ × φ ) | , | ( −△ B m , ψ ) | = | ( ∇ × B m , ∇ × ψ ) | . Therefore, {−△ u m } is bounded in L (0 , T ; V ∗ ) , and {−△ B m } is bounded in L (0 , T ; ˆ V ∗ ) . On the other hand, it follows from Sobolev inequalities that for any φ ∈ V , | ( P m H ( u m , B m ) , φ ) | = | ( H ( u m , B m ) , φ m ) |≤ C ( k u m k k u m k + k B m k k B m k ) k φ m k , where φ m = P m φ . From the uniform bound of (3.9), we have { H ( u m , B m ) } is bounded in L (0 , T ; V ∗ ) . Similarly, { H ( u m , B m ) } is bounded in L (0 , T ; ˆ V ∗ ) . u m ) ′ is bounded in L (0 , T ; V ∗ ) , ( B m ) ′ is bounded in L (0 , T ; ˆ V ∗ ) . Now we can use the similar arguments in Constantin and Foias in [11] to complete the proofof Theorem 3.1. The details are omitted.
In this section, we will study the local well-posedness of strong solutions of (1.1) − (1.4)with the corresponding initial data u ∈ V and B ∈ ˆ V . Definition 4.1. ( u, B ) is called a strong solution of (1.1) − (1.4) with the initial data u ∈ V , B ∈ ˆ V on the time interval [0 , T ) if u ∈ L (0 , T ; W ) ∩ C ([0 , T ); V ) and B ∈ L (0 , T ; ˆ W ) ∩ C ([0 , T ); ˆ V ) satisfy u ′ ∈ L (0 , T ; X ), B ′ ∈ L (0 , T ; ˆ X ) and( ∂ t u, ϕ ) − ν (∆ u, ϕ ) + (( ∇ × u ) × u + B × ( ∇ × B ) , ϕ ) = 0( ∂ t B, ψ ) − µ (∆ B, ψ ) = ( ∇ × ( u × B ) , ψ ) (4.1)for any ( ϕ, ψ ) ∈ X × ˆ X and for a.e. t ∈ [0 , T ), and u (0) = u , B (0) = B . Theorem 4.1.
Let u ∈ V , B ∈ ˆ V . Then there exists a time T ∗ > depending only on ν, µ and the H − norm of ( u , B ) such that (1.1) − (1.4) has a unique strong solution ( u, B ) on [0 , T ∗ ) satisfying the energy equation ddt ( k∇ × u k + k∇ × B k ) + 2( ν k△ u k + µ k△ B k )+ 2( ∇ × H ( u, B ) , ∇ × u ) + 2( H ( u, B ) , −△ B ) = 0 . (4.2) Proof.
Taking the curl of (3.6) yields( ∇ × u m ) ′ − ν △ ( ∇ × u m ) + m X j =1 g j ∇ × e j = 0 . (4.3)It follows from (3.7) that ( B m ) ′ − µ △ B m + m X j =1 g j ˆ e j = 0 . (4.4)12aking the inner product ((4.3) , ∇ × u m ) + ((4.4) , −△ B m ), noting that( ∇ × e i , ∇ × e j ) = λ j ( e i , e j ) , one can get ddt ( k∇ × u m k + k∇ × B m k ) + 2( ν k ( ∇× ) u m k + µ k ( ∇× ) B m k )+ 2(( ∇ × H ( u m , B m ) , ∇ × u m ) + ( H ( u m , B m ) , −△ B m )) = 0 . (4.5)Since | ( ∇ × H ( u m , B m ) , ∇ × u m ) | ≤ C ( k B m k ∞ k B m k k ( ∇× ) u m k + k u m k ∞ k u m k k ( ∇× ) u m k ) , and | ( H ( u m , B m ) , −△ B m ) | ≤ C ( k u m k ∞ + k B m k ∞ )( k u m k + k B m k ) k ( ∇× ) B m k . By applying the Agmon inequality k f k ∞ ≤ k f k k f k , ∀ f ∈ H , the equivalent of the norms in (2.3), and the standard interpolation inequalities, one can get | ( ∇ × H ( u m , B m ) , ∇ × u m ) | ≤ C ( k u m k + k B m k )( k ∆ u m k + k△ B m k ) k△ u m k , | ( H ( u m , B m ) , −△ B m ) | ≤ C ( k u m k + k B m k )(1 + k ∆ u m k + k△ B m k ) k△ B m k . Combining with the (4.5), it holds ddt ( k∇× u m k + k∇× B m k )+ ν k ( ∇× ) u m k + µ k ( ∇× ) B m k ≤ C (1+ k u m k + k B m k ) , (4.6)where C depends on ν and µ . Combining this with the energy inequality (3.2) shows thatthere is a time T ∗ > T ∈ (0 , T ∗ ),( u m , B m ) is bounded in L ∞ (0 , T ; H ) , ( u m , B m ) is bounded in L (0 , T ; H ) . It then follows from k P m ( u × v ) k ≤ k u × v k ≤ k u k ∞ k v k , (4.7) k ˆ P m B k ≤ k B k , (4.8)13he definition of H and H , and (3.6)-(3.8) that { ( u m ) ′ } , { ( B m ) ′ } is bounded in L (0 , T ; L ) . (4.9)Due to the Aubin-Lions Lemma, one can find a sequence of ( u m , B m ) still denoted by( u m , B m ) and ( u, B ) such that u m −→ u in L ∞ (0 , T ; V ) weak-star , B m −→ B in L ∞ (0 , T ; ˆ V ) weak-star ,u m −→ u in L (0 , T ; W ) weakly , B m −→ B in L (0 , T ; ˆ W ) weakly ,u m −→ u in L (0 , T ; V ) strongly , B m −→ B in L (0 , T ; ˆ V ) strongly . Passing to the limit, we find that u ∈ L ∞ (0 , T ; V ) ∩ L (0 , T ; W ), B ∈ L ∞ (0 , T ; ˆ V ) ∩ L (0 , T ; ˆ W ) satisfy( ∂ t u, ϕ ) − ν (∆ u, ϕ ) + (( ∇ × u ) × u + B × ( ∇ × B ) , ϕ ) = 0( ∂ t B, ψ ) − µ (∆ B, ψ ) = ( ∇ × ( u × B ) , ψ ) (4.10)for any ( ϕ, ψ ) ∈ X × ˆ X .From (4.9), we find that ( u ′ , B ′ ) ∈ L (0 , T ; X ) × L (0 , T ; ˆ X ). Then we deduce that( u, B ) ∈ C ([0 , T ]; V ) × C ([0 , T ]; ˆ V ), u (0) = u , B (0) = B and the energy equation (4.2)holds.Now we will show that the uniqueness of strong solutions by using of the standard pro-cedure. Let ( u, B ) and ( u , B ) be two strong solutions to (1.1)-(1.4). Set ¯ u = u − u , and¯ B = B − B , then ∂ t ¯ u − ν △ ¯ u + H ( u, B ) − H ( u , B ) = 0 , (4.11) ∂ t ¯ B − µ △ ¯ B + H ( u, B ) − H ( u , B ) = 0 . (4.12)Taking the inner products with ¯ u in (4.11), and ¯ B in (4.12), integrating by parts, from theboundary condition ¯ u · n = 0, ¯ B × n = 0, one gets ddt ( k ¯ u k + k ¯ B k ) + 2 ν k∇ × ¯ u k + 2 µ k∇ × ¯ B k + ( H ( u, B ) − H ( u , B ) , ¯ u ) + ( H ( u, B ) − H ( u , B ) , ¯ B ) = 0 .
14n the other hand, simple manipulations show that( H ( u, B ) − H ( u , B ) , ¯ u ) + ( H ( u, B ) − H ( u , B ) , ¯ B )= ( ∇ × ¯ u × u, ¯ u ) + ( B × ∇ × ¯ B, ¯ u )+ ( ¯ B × ∇ × B , ¯ u ) + (cid:0) ∇ × ( B × u ) − ∇ × ( B × u ) , ¯ B (cid:1) = ( ∇ × ¯ u × u, ¯ u ) + ( B × ∇ × ¯ B, ¯ u ) + ( ∇ × (¯ u × ¯ B ) , B )+ (cid:0) ( B × ¯ u, ∇ × ¯ B ) + ( ¯ B × u , ∇ × ¯ B ) (cid:1) , = ( ∇ × ¯ u × u, ¯ u ) + ( ∇ × (¯ u × ¯ B ) , B ) + ( ¯ B × u , ∇ × ¯ B )) , (4.13)here one has used integration by parts, the boundary condition ¯ B × n = 0, B × n = 0 and( ¯ B × ∇ × B , ¯ u ) = ( ∇ × (¯ u × ¯ B ) , B ) . Combining them together, yields ddt ( k ¯ u k + k ¯ B k ) + 2 ν k∇ × ¯ u k + 2 µ k∇ × ¯ B k ≤ ν k∇ × ¯ u k + µ k∇ × ¯ B k + C ( k u k + k B k + k u k + k B k )( k ¯ u k + k ¯ B k )on [0 , T ∗ ).Then u = u , B = B follows from ¯ u (0) = ¯ B (0) = 0 and the Gronwall’s inequality. Remark . As usual, the local strong solution in Theorem 4.1 can be extended globally intime if the initial data is suitably small. Indeed, note first that if T ∗ be the maximal timefor the existence of strong solution in Theorem 4.1 and T ∗ < ∞ , then k ( u, B ) k −→ t −→ T ∗− . Next, it follows easily from (3.2) and (4.2) that for t ∈ (0 , T ∗ ), ddt ( k u k + k B k ) + (cid:0) µ − C ( k u k + k B k ) (cid:1) k u k + (cid:0) ν − C ( k u k + k B k ) (cid:1) k B k ≤ . (4.14)So the standard continuity argument shows that if k ( u , B ) k ≤ δ ( µ, ν ) , then T ∗ = + ∞ , so that the strong solution can be extended globally. In last section, the existence time interval [0 , T ∗ ) may depends on the parameters ν and µ for general domains. To study the uniform well-posedness independent of the viscosity15nd magnetic diffusion, we consider only the cubic domain Ω = Ω T = T × (0 , T is the torus. In this case, we can get the uniform well-posedness and convergence. We startwith the following three elementary lemmas which will play an essential role in the proof ofthe Theorem 5.4. Lemma 5.1.
Let u, B be two smooth vectors satisfying ∇ · u = 0 , ∇ · B = 0 in Ω and u · n = 0 , ( ∇ × u ) × n = 0 ; B × n = 0 on the boundary, that is ∂ u = 0 , ∂ u = 0 , u = 0 on ∂ Ω , (5.1) B = 0 , B = 0 , ∂ B = 0 on ∂ Ω , (5.2) where ∂ B = 0 is from ∇ · B = 0 . Then, it holds that ∇ × ( B × u ) × n = ( u · ∇ B − B · ∇ u ) × n = 0 on ∂ Ω , ∇ × ( ∇ × u × u ) × n = 0 on ∂ Ω . (5.3) Proof.
Indeed, direct calculations show that( u · ∇ B ) = u ∂ B + u ∂ B + u ∂ B = 0 on ∂ Ω , ( B · ∇ u ) = B ∂ u + B ∂ u + B ∂ u = 0 on ∂ Ω . Similarly ( u · ∇ B ) = u ∂ B + u ∂ B + u ∂ B = 0 on ∂ Ω , ( B · ∇ u ) = B ∂ u + B ∂ u + B ∂ u = 0 on ∂ Ω . Let ω = ∇ × u . Then ∇ × ( ∇ × u × u ) × n = (cid:0) ( u · ∇ ω ) − ( ω · ∇ u ) (cid:1) × n. Since ω = ω = 0 on ∂ Ω , one gets easily that ( u · ∇ ω ) = u ∂ ω + u ∂ ω + u ∂ ω = 0 on ∂ Ω , ( u · ∇ ω ) = u ∂ ω + u ∂ ω + u ∂ ω = 0 on ∂ Ω , and ( ω · ∇ u ) = ω ∂ u + ω ∂ u + ω ∂ u = 0 on ∂ Ω , ( ω · ∇ u ) = ω ∂ u + ω ∂ u + ω ∂ u = 0 on ∂ Ω , that is ∇ × ( ∇ × u × u ) × n = 0 on ∂ Ω . emma 5.2. Let u, B be the vectors in Lemma 5.1. Assume further that ∆ B × n = 0 onthe boundary, that is ∂ B = 0 , ∂ B = 0 on ∂ Ω . (5.4) Then, it holds that ∇ × ( ∇ × B × B ) × n = ( B · ∇ ( ∇ × B ) − ( ∇ × B ) · ∇ B ) × n = 0 on ∂ Ω . (5.5) Proof.
It follows from (5.4) and (5.5) that ∇ × ( ω B × B ) = ( B · ∇ ω B − ω B · ∇ B ) , ( B · ∇ ω B ) = B ∂ ω B + B ∂ ω B + B ∂ ω B = B ∂ ( ∂ B − ∂ B ) = B ∂ B − B ∂ B = 0( ω B · ∇ B ) = ω B ∂ B + ω B ∂ B + ω B ∂ B = ( ∂ B − ∂ B ) ∂ B = 0 , where ω B = ∇ × B .Similarly, ∇ × ( ω B × B ) = ( B · ∇ ω B − ω B · ∇ B ) = 0 . The lemma is proved.
Lemma 5.3.
Let the assumptions in Lemma 5.2 hold. Furthermore, ω = ∇ × u satisfies ∆ ω × n = 0 on the boundary. Then, it holds that ( ∇× ) ( B × u ) × n = 0 on ∂ Ω . (5.6) Proof.
Direct computations yield( ∇× ) ( B × u ) = − ∆( u · ∇ B − B · ∇ u ) = − ( ∂ + ∂ )( u · ∇ B − B · ∇ u ) − ∂ ( u · ∇ B − B · ∇ u ) = − ∂ ( u · ∇ B − B · ∇ u ) = − ∂ ( u ∂ B + u ∂ B + u ∂ B − B ∂ u − B ∂ u − B ∂ u ) . Note that ∂ ( u ∂ B ) = ∂ ( ∂ u ∂ B + u ∂ B )= ∂ u ∂ B + 2 ∂ u ∂ B + u ∂ ∂ B = 0 , ( u ∂ B ) = ∂ ( ∂ u ∂ B + u ∂ B )= ∂ u ∂ B + 2 ∂ u ∂ B + u ∂ ∂ B = 0 ,∂ ( u ∂ B ) = ∂ ( ∂ u ∂ B + u ∂ B )= ∂ u ∂ B + 2 ∂ u ∂ B + u ∂ B = 0on the boundary, here one has used ∂ u = ∆ u = − ( ∇ × ω ) = ∂ ω − ∂ ω = 0 . Note also that ∂ ( B ∂ u ) = ∂ B ∂ u + 2 ∂ B ∂ ∂ u + B ∂ ∂ u = 0 ,∂ ( B ∂ u ) = ∂ B ∂ u + 2 ∂ B ∂ ∂ u + B ∂ ∂ u = 0 ,∂ ( B ∂ u ) = ∂ B ∂ u + 2 ∂ B ∂ u + B ∂ u = 0on the boundary, here one has used the fact ∂ u = ∂ ∂ u = ( ∂ ω + ∂ ∂ u )= ∆ ω + ∂ (∆ u − ∂ u + ∂ u ) = 0on the boundary, since − ∆ u = − (∆ u ) = ( ∇ × ω ) = ∂ ω − ∂ ω on the boundary. Then, we conclude that( ∇× ) ( B × u ) = 0on the boundary. By symmetry, it holds that ( ∇× ) ( B × u ) = 0. The lemma is proved.It follows from these lemmas that Theorem 5.4.
Let u ∈ W ∩ H (Ω) , B ∈ ˆ W ∩ H (Ω) . Then there is a T > dependingonly on k ( u , B ) k H such that the strong solution u = u ( ν, µ ) , B = B ( ν, µ ) of the MHDsystem (1.1) − (1.4) with the initial data ( u , B ) has the following uniform bound k u ( · , t ) k + k B ( · , t ) k + Z t ν k u ( · , t ) k + µ k B ( · , t ) k ≤ C for t ∈ [0 , T ] , (5.7) where C is a constant independent of ν and µ . ∂ t ( −△ u ) − ν △ ( −△ u ) − △ ( ∇ × u × u + B × ∇ × B ) = 0 in Ω ,∂ t ( △ B ) − µ △ B + △ ( B · ∇ u − u · ∇ B ) = 0 in Ω , ∇ · △ u = 0 , ∇ · △ B = 0 in Ω , △ u · n = 0 , ( ∇ × △ u ) × n = 0 on ∂ Ω , △ B × n = 0 on ∂ Ω . (5.8)Indeed, it follows from the equation of B in (1.1) with the boundary condition B × n = 0and the Lemma 5.1 that △ B × n = 0 on ∂ Ω . (5.9)Taking the curl of the equation of u in (1.1) and using the Lemma 5.2 and the boundarycondition of u , one can get( ∇× ) u × n = − ( ∇ × △ u ) × n = 0 on ∂ Ω . (5.10)Thus ddt ( k∇ × ( −△ u ) k + k∇ × ( −△ B ) k ) + 2( ν k△ u k + µ k△ B k )+ 2( −△ H ( u, B ) , △ u ) + 2( △ H ( u, B ) , −△ B ) = 0 . (5.11)We claim that T ∗ ( ν, µ ) is bounded below for all ν, µ > −△ H ( u, B ) , △ u ) = (( ∇× ) H ( u, B ) , ∇ × ( −△ u )) , (5.12)and ( △ H ( u, B ) , −△ B ) = (( ∇× ) H ( u, B ) , ∇ × ( −△ B )) , (5.13)here Lemma 5.3 has been used. It remains to estimate (5.12) and (5.13).Noting that( −△ H ( u, B ) , △ u ) = (( ∇× ) H ( u, B ) , ∇ × ( −△ u ))= (( ∇× ) ( B × ( ∇ × B )) , ( ∇× ) u ) + (( ∇× ) ( ∇ × u × u ) , ( ∇× ) u )and ( △ H ( u, B ) , −△ B ) = (( ∇× ) H ( u, B ) , ∇ × ( −△ B )) , ∇× ) ( B × ( ∇ × B )) , ( ∇× ) u ) = − ( B · ∇ (( ∇× ) B ) , ( ∇× ) u ) + R where R can be estimated by the H norm so that | R | ≤ C ( k u k + k B k ) , (( ∇× ) ( ∇ × u × u ) , ( ∇× ) u ) = ( u · ∇ ( ∇× ) u, ( ∇× ) u ) + R for some | R | ≤ C k u k . On the other hand,(( ∇× ) ( B × u ) , ( ∇× ) B )= (( ∇× ) ( u · ∇ B − B · ∇ u ) , ( ∇× ) B )= ( u · ∇ ( ∇× ) B − B · ∇ ( ∇× ) u, ( ∇× ) B ) + R = − ( B · ∇ ( ∇× ) u, ( ∇× ) B ) + R for some | R | ≤ C ( k u k + k B k ). Since( B · ∇ ( ∇× ) u, ( ∇× ) B ) + ( B · ∇ ( ∇× ) B, ( ∇× ) u )= ( B, ∇ (( ∇× ) u · ( ∇× ) B ))= Z ∂ Ω (( ∇× ) u · ( ∇× ) B ) B · n − Z Ω (( ∇× ) u · ( ∇× ) B ) ∇ · B = 0 , (5.14)here we have used ( ∇× ) u × n = 0 , ( ∇× ) B · n = 0 , on ∂ Ωfrom (5.9) and (5.10). Then we conclude that ddt ( k ( ∇× ) u k + k ( ∇× ) B k ) + ν k ∆ u k + µ k ∆ B k ≤ C ( k u k + k B k ) , Combining it with the energy inequality (3.2) yields that ddt ( k u k + k B k ) + ν k u k + µ k B k ≤ C ( k u k + k B k ) , where C is independent of ν and µ , and the norm k · k s is the equivalent one in (2.3).Comparing with the ordinary differential equation y ′ ( t ) = Cy ( t ) , (0) = k u (0) k + k B (0) k , and let T be the blow up time, one obtains that T ∗ ( ν, µ ) ≥ T for all ν, µ > , and (5.7) is valid. Proof of Theorem 5.4:
To make the proof rigorous, one can use the Galerkin approxima-tions. Consider the system satisfied by −△ u m ( x, t ) and −△ B m ( x, t ). Let u ∈ W , B ∈ ˆ W .It follows from (3.6) and (3.7) that( −△ u m ) ′ − ν △ ( −△ u m ) + X g j λ j e j = 0 , (5.15)( −△ B m ) ′ − µ △ ( −△ B m ) + X g j λ j ˆ e j = 0 , (5.16)( −△ u m )(0) = P m ( −△ u ) , ( −△ B m )(0) = ˆ P m ( −△ B ) . (5.17)Since ∇ × e i × n = 0 and n × ∇ × H ( u m , B m ) = 0 on the boundary, so integration by partsyields ( −△ P m H ( u m , B m ) , e i ) = ( X g j λ j e j , e i )= ( H ( u m , B m ) , −△ e i )= Z ∂ Ω H ( u m , B m ) · ( ∇ × e i × n ) + ( ∇ × H ( u m , B m ) , ∇ × e i )= Z ∂ Ω ( n × ∇ × H ( u m , B m )) · e i + ( − ∆ H ( u m , B m ) , e i )= ( −△ H ( u m , B m ) , e i ) . Thus the following commutation holds∆ P m H ( u m , B m ) = P m ∆ H ( u m , B m )where P m H ( u m , B m ) = X g j e j with g j = ( H ( u m , B m ) , e j ), and P m ∆ H ( u m , B m ) = X g ,δj e j with g ,δj = (∆ H ( u m , B m ) , e j ). 21imilarly, integration by parts shows that( − ∆ ˆ P m H ( u m , B m ) , ˆ e i ) = ( X g j λ j ˆ e j , ˆ e i )= ( H ( u m , B m ) , −△ ˆ e i )= Z ∂ Ω ( n × H ( u m , B m )) · ( ∇ × ˆ e i ) + ( ∇ × H ( u m , B m ) , ∇ × ˆ e i )= Z ∂ Ω ∇ × H ( u m , B m ) · (ˆ e i × n ) + ( − ∆ H ( u m , B m ) , ˆ e i )= ( −△ H ( u m , B m ) , ˆ e i )due to the fact that ˆ e i × n = 0 and n × H ( u m , B m ) = 0 on the boundary. Thus the followingcommutation holds ∆ ˆ P m H ( u m , B m ) = ˆ P m ∆ H ( u m , B m )where ˆ P m H ( u m , B m ) = X g j ˆ e j with g j = ( H ( u m , B m ) , ˆ e j ), andˆ P m ∆ H ( u m , B m ) = X g ,δj ˆ e j with g ,δj = (∆ H ( u m , B m ) , ˆ e j ). It follows from (5.15), (5.16) and the above commutationsthat ddt ( k∇ × ( −△ u m ) k + k∇ × ( −△ B m ) k ) + 2( ν k△ u m k + µ k△ B m k )+ 2( △ H ( u m , B m ) , −△ u m ) + 2( △ H ( u m , B m ) , −△ B m ) = 0 . (5.18)Note that ( ∇ × ∆ u m ) × n = 0 on the boundary. It follows that( △ H ( u m , B m ) , −△ u m ) = (( ∇× ) H ( u m , B m ) , ( ∇× ) u m ) . By Lemma 5.3, it follows that ( △ H ( u m , B m )) × n = 0 on the boundary, and then( △ H ( u m , B m ) , −△ B m ) = (( ∇× ) H ( u m , B m ) , ( ∇× ) B m ) . Hence, the estimates in the formal analysis above can also be applied to the Galerkin ap-proximations, and the corresponding bounds can also be obtained, which allow one to passthe limit to derive the desired a priori estimates (5.7). The theorem is proved.The above uniform estimates allow us to obtain the zero dissipation limit.22 heorem 5.5.
Assume that u ∈ W ∩ H (Ω) , B ∈ ˆ W ∩ H (Ω) . Let ( u, B ) = ( u ( ν, µ ) , B ( ν, µ )) be the corresponding strong solution to the MHD system (1.1) − (1.4) on [0 , T ] in Theorem5.4. Then as ν, µ −→ , ( u, B ) converges to the unique solution ( u , B ) of the ideal MHDsystem with the same initial data in the sense u ( ν, µ ) , B ( ν, µ ) −→ u , H in L q (0 , T ; H (Ω)) . (5.19) u ( ν, µ ) , B ( ν, µ ) −→ u , H in C (0 , T ; H (Ω)) (5.20) for all ≤ q < ∞ . Proof.
It follows from theorem 5.4 that u ( ν, µ ) , B ( ν, µ ) is uniformly bounded in C ([0 , T ]; H (Ω)) ,u ′ ( ν, µ ) , B ′ ( ν, µ ) is uniformly bounded in L (0 , T ; H (Ω)) , for all ν, µ >
0. From the Aubin-Lions lemma, there is a subsequence ν n , µ n and u , B suchthat ( u ( ν n , µ n ) , B ( ν n , µ n )) −→ ( u , B ) in L ∞ (0 , T ; H (Ω)) weakly , ( u ( ν n , µ n ) , B ( ν n , µ n )) −→ ( u , B ) in L p (0 , T ; H (Ω)) , ( u ( ν n , µ n ) , B ( ν n , µ n )) −→ ( u , B ) in C ([0 , T ); H (Ω))for any 1 ≤ p < ∞ as ν n , µ n −→
0. Passing to the limit shows that ( u , B ) solves thefollowing limit equations ∂ t u + ( ∇ × u ) × u + B × ∇ × B + ∇ p = 0 in Ω , ∇ · u = 0 in Ω ,∂ t B = ∇ × ( u × B ) in Ω , ∇ · B = 0 in Ω (5.21)with the boundary conditions u · n = 0 , ( ∇ × u ) × n = 0 on ∂ Ω ,B × n = 0 on ∂ Ω , △ B × n = 0 on ∂ Ω , (5.22)and p satisfying △ p = − (cid:0) ∇ · (( ∇ × u ) × u ) − ∇ · (( ∇ × B ) × B ) (cid:1) , ∇ p · n = 0 on ∂ Ω . (5.23)23et ( u, B ) and ( u , B ) be two strong solutions to (5.21)-(5.23). Set ¯ u = u − u , and¯ B = B − B . Then ∂ t ¯ u + H ( u, B ) − H ( u , B ) = 0 , (5.24) ∂ t ¯ B + H ( u, B ) − H ( u , B ) = 0 . (5.25)Taking the inner products with ¯ u in (5.24), and ¯ B in (5.25) and integrating by parts lead to ddt ( k ¯ u k + k ¯ B k ) + ( H ( u, B ) − H ( u , B ) , ¯ u ) + ( H ( u, B ) − H ( u , B ) , ¯ B ) = 0 . (4.13) implies that( H ( u, B ) − H ( u , B ) , ¯ u ) + ( H ( u, B ) − H ( u , B ) , ¯ B )= ( ∇ × ¯ u × u, ¯ u ) + ( ∇ × (¯ u × ¯ B ) , B ) + ( ¯ B × u , ∇ × ¯ B )) . (5.26)Since ( ∇ × ¯ u × u, ¯ u ) = ((¯ u · ∇ ) u, ¯ u ) − (( u · ∇ )¯ u, ¯ u ) = ((¯ u · ∇ ) u, ¯ u ) , (5.27)( ∇ × (¯ u × ¯ B ) , B ) = (¯ u × ¯ B, ∇ × B ) , (5.28)and ( ¯ B × u , ∇ × ¯ B ) = (( u · ∇ ) ¯ B, ¯ B ) − (( ¯ B · ∇ ) u , ¯ B ) = (( u · ∇ ) ¯ B, ¯ B ) , (5.29)here the boundary conditions B × n = 0, B × n = 0, u · n = 0 and u · n = 0 have beenused, one can get ddt ( k ¯ u k + k ¯ B k ) ≤ C ( k u k + k B k + k u k )( k ¯ u k + k ¯ B k ) . Note that u , B , u and B are all in L ∞ (0 , T ; H ) and ¯ u (0) = ¯ B (0) = 0. One obtains theuniqueness by Gronwall’s inequality.Finally, we prove the following convergence rate. Theorem 5.6.
Under the same assumptions in Theorem 5.5, it holds that k u ( ν, µ ) − u k + k B ( ν, µ ) − B k ≤ C ( T )( ν + µ ) on the interval [0 , T ] . Proof.
Set ¯ u = u ( ν, µ ) − u and ¯ B = B ( ν, µ ) − B . One can get that −△ ¯ u and −△ ¯ B solve ∂ t ( −△ ¯ u ) − △ ( H ( u, B ) − H ( u , B )) = − ν △ u in Ω , (5.30) ∂ t ( −△ ¯ B ) − △ ( H ( u, B ) − H ( u , B )) = − µ △ B in Ω , (5.31)24 · ¯ u = 0 , ∇ · ¯ B = 0 in Ω , (5.32)¯ u · n = 0 , ¯ B × n = 0 , on ∂ Ω , (5.33)with ∇ × u × n = 0, ∇ × u × n = 0, △ B × n = 0, ( ∇× ) u × n = 0, and △ B × n = 0 on theboundary. Taking inner product of (5.30) with −△ ¯ u and (5.31) with −△ ¯ B and integratingby parts, one gets that ddt ( k△ ¯ u k + k△ ¯ B k ) − △ ( H ( u, B ) − H ( u , B )) , −△ ¯ u ) − △ ( H ( u, B ) − H ( u , B )) , −△ ¯ B )= ν (( ∇× ) u, ( ∇× ) ¯ u ) + µ (( ∇× ) B, ( ∇× ) ¯ B ) . A simple computation yields −△ ( H ( u, B ) − H ( u , B )) = ( u · ∇ )( −△ ¯ u ) − ( B · ∇ )( −△ ¯ B )+ (¯ u · ∇ )( −△ u ) − ( ¯ B · ∇ )( −△ B )+ X i,j =1 , ,i + j =3 F i,j ( D i u , D j ¯ u ) − X i,j =1 , ,i + j =3 F i,j ( D i B , D j ¯ B )+ X i,j =1 , ,i + j =3 F i,j ( D i u, D j ¯ u ) − X i,j =1 , ,i + j =3 F i,j ( D i B, D j ¯ B ) , −△ ( H ( u, B ) − H ( u , B )) = ( u · ∇ )( −△ ¯ B ) − ( B · ∇ )( −△ ¯ u )+ (¯ u · ∇ )( −△ B ) − ( ¯ B · ∇ )( −△ u )+ X i,j =1 , ,i + j =3 F i,j ( D i u , D j ¯ B ) − X i,j =1 , ,i + j =3 F i,j ( D i B , D j ¯ u )+ X i,j =1 , ,i + j =3 F i,j ( D i u, D j ¯ B ) − X i,j =1 , ,i + j =3 F i,j ( D i B, D j ¯ u ) , where F i,j ( D i u, D j v ) ′ s are bilinear forms and D i ′ s are the i-th order differential operators.It follows from ∇ · u = 0 and u · n | ∂ Ω = 0 that (cid:0) ( u · ∇ )( −△ ¯ u ) , −△ ¯ u (cid:1) = 0 , (cid:0) ( u · ∇ )( −△ ¯ B ) , −△ ¯ B (cid:1) = 0 . On the other hand, (cid:0) ( B · ∇ )( −△ ¯ B ) , −△ ¯ u (cid:1) + (cid:0) ( B · ∇ )( −△ ¯ u ) , −△ ¯ B (cid:1) = ( B, ∇ ( △ ¯ B · △ ¯ u ))= Z ∂ Ω ( △ ¯ B · △ ¯ u ) B · n − Z Ω ( △ ¯ B · △ ¯ u ) ∇ · B = 0 , (5.34)25here one has used △ ¯ B × n = 0 , △ ¯ u · n = 0 , on ∂ Ω . Therefore, | ( △ ( H ( u, B ) − H ( u , B )) , −△ ¯ u ) + ( △ ( H ( u, B ) − H ( u , B )) , −△ ¯ B ) |≤ C ( k u k + k B k + k u k + k B k )( k△ ¯ u k + k△ ¯ B k ) . Also, one has that | (( ∇× ) u, ( ∇× ) ¯ u ) | ≤ C k ( ∇× ) u k ( k ( ∇× ) u k + k ( ∇× ) u k )and | (( ∇× ) B, ( ∇× ) ¯ B ) | ≤ C k ( ∇× ) B k ( k ( ∇× ) B k + k ( ∇× ) B k ) . These estimates are uniform respect to ν , µ and thus ddt ( k△ ¯ u k + △ ¯ B k ) ≤ C ( T )( k△ ¯ u k + k△ ¯ B k + ν + µ ) . Due to ¯ u (0) = 0, ¯ B (0) = 0 and Gronwall’s inequality, we deduce that k△ ¯ u k + k△ ¯ B k ≤ C ( T )( ν + µ ) . (5.35)The theorem is proved. Acknowledge.
Yuelong Xiao is partially supported by NSFC Nos. 11871412, 11771300.And Qin Duan is partially supported by NSFC Nos. 11771300. The research of Zhouping Xinwas supported in part by Zheng Ge Ru Foundation, HongKong RGC Earmarked ResearchGrants: CUHK14302819, CUHK14300917, CUHK14302917, and Basic and Applied BasicResearch Foundation of Guangdong Province 2020B1515310002.
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