On the local well-posedness and a Prodi-Serrin type regularity criterion of the three-dimensional MHD-Boussinesq system without thermal diffusion
aa r X i v : . [ m a t h . A P ] S e p On the local well-posedness and a Prodi-Serrin type regularitycriterion of the three-dimensional MHD-Boussinesq systemwithout thermal diffusion
Adam Larios, Yuan PeiTuesday 18 th September, 2018
Department of MathematicsUniversity of Nebraska-LincolnLincoln, NE 68588e-mail: [email protected] e-mail: [email protected]
Abstract
We prove a Prodi-Serrin-type global regularity condition for the three-dimensional Magnetohydrodynamic-Boussinesq system (3D MHD-Boussinesq) without thermal diffusion, in terms of only two velocity and twomagnetic components. This is the first Prodi-Serrin-type criterion for a hydrodynamic system which is not fullydissipative, and indicates that such an approach may be successful on other systems. In addition, we provide aconstructive proof of the local well-posedness of solutions to the fully dissipative 3D MHD-Boussinesq system,and also the fully inviscid, irresistive, non-diffusive MHD-Boussinesq equations. We note that, as a special case,these results include the 3D non-diffusive Boussinesq system and the 3D MHD equations. Moreover, they can beextended without difficulty to include the case of a Coriolis rotational term.
Keywords:
Magnetohydrodynamic equations, Boussinesq equations, B´enard convection, Prodi-Serrin, weak solu-tions, partial viscosity, inviscid, global existence, regularity
Mathematics Subject Classification : 35A01, 35K51, 35Q35, 35Q86, 76B03, 76D03, 76W05.
In this paper, we address global regularity criteria for the solutions to the non-diffusive three-dimensional MHD-Boussinesq system of equations. The MHD-Boussinesq system models the convection of an incompressible flowdriven by the buoyant effect of a thermal or density field, and the Lorenz force, generated by the magnetic fieldof the fluid. Specifically, it closely relates to a natural type of the Rayleigh-B´enard convection, which occurs in ahorizontal layer of conductive fluid heated from below, with the presence of a magnetic field (c.f. [30, 31]). Variousphysical theories and numerical experiments such as in [46] have been developed to study the Rayleigh-B´enardas well as the magnetic Rayleigh-B´enard convection and related equations. We observe that by formally settingthe magnetic field b to zero, system (1.1) below reduces to the Boussinesq equations while by formally settingthe thermal fluctuation θ = 0 we obtain the magnetohydrodynamic equations. One also formally recovers theincompressible Navier-Stokes equations if we set b = 0 and θ = 0 simultaneously.1enote by Ω = T the three-dimensional periodic space R / Z = [0 , , and for T > , the 3D MHD-Boussinesq system with full fluid viscosity, magnetic resistivity, and thermal diffusion over Ω × [0 , T ) is givenby ∂u∂t − ν ∆ u + ( u · ∇ ) u + ∇ p = ( b · ∇ ) b + gθe ,∂b∂t − η ∆ b + ( u · ∇ ) b = ( b · ∇ ) u,∂θ∂t − κ ∆ θ + ( u · ∇ ) θ = 0 , ∇ · u = 0 = ∇ · b, (1.1)where ν ≥ , η ≥ , and κ ≥ stand for the constant kinematic viscosity, magnetic diffusivity, and thermaldiffusivity, respectively. The constant g > has unit of force, and is proportional to the constant of gravitationalacceleration. We denote x = ( x , x , x ) , and e to be the unit vector in the x direction, i.e., e = (0 , , T .Here and henceforth, u = u ( x, t ) = ( u ( x, t ) , u ( x, t ) , u ( x, t )) is the unknown velocity field of a viscous in-compressible fluid, with divergence-free initial data u ( x,
0) = u ; b = b ( x, t ) = ( b ( x, t ) , b ( x, t ) , b ( x, t )) is theunknown magnetic field, with divergence-free initial data b ( x,
0) = b ; and the scalar p = p ( x, t ) represents theunknown pressure, while θ = θ ( x, t ) can be thought of as the unknown temperature fluctuation, with initial value θ = θ ( x, . Setting κ = 0 , we obtain the non-diffusive MHD-Boussinesq system ∂u∂t − ν ∆ u + ( u · ∇ ) u + ∇ p = ( b · ∇ ) b + gθe ,∂b∂t − η ∆ b + ( u · ∇ ) b = ( b · ∇ ) u,∂θ∂t + ( u · ∇ ) θ = 0 , ∇ · u = 0 = ∇ · b, (1.2)which we study extensively in this paper. We also provide a proof for the local existence and uniqueness ofsolutions to the fully inviscid MHD-Boussinesq system with ν = η = κ = 0 , namely, ∂u∂t + ( u · ∇ ) u + ∇ p = ( b · ∇ ) b + gθe ,∂b∂t + ( u · ∇ ) b = ( b · ∇ ) u,∂θ∂t + ( u · ∇ ) θ = 0 , ∇ · u = 0 = ∇ · b, (1.3)with the initial condition u , b , and θ in H . We note that the proof of this result differs sharply from the proofof local existence for solutions of (1.1), due to a lack of compactness. Therefore, we include the proof for the sakeof completeness.In recent years, from the perspective of mathematical fluid dynamics, much progress have been made in thestudy of solutions of the Boussinesq and MHD equations. For instance, in [10, 11], Chae et al. obtained the localwell-posedness of the fully inviscid 2D Boussinesq equations with smooth initial data. A major breakthrough camein [9] and [19], where the authors independently proved global well-posedness for the two-dimensional Boussinesqequations with the case ν > and κ = 0 and the case ν = 0 and κ > , On the other hand, Wu et al. proved2n [4, 7, 8, 28, 49] the global well-posedness of the MHD equations, for a variety of combinations of dissipationand diffusion in two dimensional space. Furthermore, a series of results concerning the global regularity of the 2DBoussinesq equations with anisotropic viscosity were obtained in [27, 1, 8, 15]. For the 2D Boussinesq equations,the requirements on the initial data were significantly weakened in [14, 20, 21]. Regarding the MHD-B´enardsystem, some progress has been made in 2D case under various contexts, see, e.g., [12, 16]. However, there haslittle work in the 3D case. Specifically, outstanding open problems such as global regularity of classic solutions forthe fully dissipative system and whether the solutions blow up in finite time for the fully inviscid system remainunresolved.The main purpose of our paper is to obtain a Prodi-Serrin type regularity criterion for the non-diffusive 3DMHD-Boussinesq system. Unlike the case of the 3D Navier-Stokes equations, Prodi-Serrin type regularity criteriaare not available for Euler equations in three-dimensional space. Thus, it is difficult to obtain global regularityfor u , b , and θ simultaneously since there is no thermal diffusivity in the equation for θ . However, we are able tohandle this by proving the higher order regularity for u and b first, before bounding k∇ θ k L x . We emphasize thatthis is the first work, to the best of our knowledge, that proves a Prodi-Serrin-type criterion in the case where thesystem is not fully dissipative.The pioneering work of Serrin, Prodi, et al. (c.f. [17, 29, 43, 44, 37, 39, 40]) for the 3D Navier-Stokes equationsproved that, for any T > , if u ∈ L rt ([0 , T ]; L sx ) with /r + 3 /s < and < s < ∞ , then the solution for the3D Navier-Stokes equations remains regular on the interval [0 , T ] . Proof for the borderline case in various settingswas obtained in [17, 29, 43, 44]. Similar results concerning the 3D Navier-Stokes, Boussinesq and MHD equationswere obtain in [2, 3, 6, 5, 24, 25, 32, 36, 34, 35, 55, 56, 41]. In particular, in [50, 51], regularity criteria for MHDequations involving only two velocity components was proved but in a smaller Lebesgue space. However, thereis no literature on the regularity criteria for the solutions of systems (1.1) and (1.2). In this paper, we obtain aregularity condition using several velocity and magnetic field components in much larger space that is closer to thecritical space of the equations. A central message of the present work is that with optimal and delicate applicationof our method, as well as potential new techniques such as in [22, 38, 42, 52, 53, 54], one might further improvethe criterion on the global regularity for system (1.2).Moreover, we prove the local-in-time existence and uniqueness of the solutions to the system (1.2) with H initial datum. We obtain the necessary a priori estimates and construct the solution via Galerkin methods for boththe full and the non-diffusive systems. In particular, we show that the existence time of solutions to the full systemdoes not depend on κ , which enables us to prove that the solutions to the full system approaches that of the non-diffusive system as κ tends to on their time interval of existence. Regarding the fully inviscid system, we remarkthat the local well-posedness of either of the full system (1.1) or the non-diffusive system (1.2) is not automaticallyimplied by that of the fully inviscid system (1.3), as observed in [26] for multi-dimensional Burgers equation ∂u∂t + ( u · ∇ ) u = ν ∆ u, in two and higher dimensions. One might expect to that adding more diffusion, namely in the form of a hyper-diffusion term − ν ∆ u , might make the equation even easier to handle. However, the question well-posedness ofthe resulting equation, namely ∂u∂t + ( u · ∇ ) u = − ν ∆ u + ν ∆ u, All through this paper we denote ∂ j = ∂/∂x j , ∂ jj = ∂ /∂x j , ∂ t = ∂/∂t , ∂ α = ∂ | α | /∂x α · · · x α n n , where α isa multi-index. We also denote the horizontal gradient ∇ h = ( ∂ , ∂ ) and horizontal Laplacian ∆ h = ∂ + ∂ .Also, we denote the usual Lebesgue and Sobolev spaces by L px and H sx ≡ W s, x , respectively, with the subscript x (or t ) indicating that the underlying variable is spatial (resp. temporal). Let F be the set of all trigonometricpolynomial over T and define the subset of divergence-free, zero-average trigonometric polynomials V := (cid:26) φ ∈ F : ∇ · φ = 0 , and Z T φ dx = 0 (cid:27) . We use the standard convention of denoting by H and V the closures of V in L x and H x , respectively, with innerproducts ( u, v ) = X i =1 Z T u i v i dx and ( ∇ u, ∇ v ) = X i,j =1 Z T ∂ j u i ∂ j v i dx, respectively, associated with the norms | u | = ( u, u ) / and k u k = ( ∇ u, ∇ u ) / . The latter is a norm due to thePoincar´e inequality k φ k L x ≤ C k∇ φ k L x holding for all φ ∈ V . We also have the following compact embeddings (see, e.g., [13, 45]) V ֒ → H ֒ → V ′ , where V ′ denotes the dual space of V .The following interpolation result is frequently used in this paper (see, e.g., [33] for a detailed proof). Assume ≤ q, r ≤ ∞ , and < γ < . For v ∈ L qx ( T n ) , such that ∂ α v ∈ L rx ( T n ) , for | α | = m , then k ∂ s v k L p ≤ C k ∂ α v k γL r k v k − γL q , where p − sn = (cid:18) r − mn (cid:19) γ + 1 q (1 − γ ) . (2.4)The following materials are standard in the study of fluid dynamics, in particular for the Navier-Stokes equa-tions, and we refer to reader to [13, 45] for more details. We define the Stokes operator A , − P σ ∆ with domain4 ( A ) , H x ∩ V , where P σ is the Leray-Helmholtz projection. Note that under periodic boundary conditions, wehave A = − ∆ P σ . Moreover, the Stokes operator can be extended as a linear operator from V to V ′ as h Au, v i = ( ∇ u, ∇ v ) for all v ∈ V. It is well-known that A − : H ֒ → D ( A ) is a positive-definite, self-adjoint, and compact operator from H intoitself, thus, A − possesses an orthonormal basis of positive eigenfunctions { w k } ∞ k =1 in H , corresponding to asequence of non-increasing sequence of eigenvalues. Therefore, A has non-decreasing eigenvalues λ k , i.e., ≤ λ ≤ λ , . . . since { w k } ∞ k =1 are also eigenfunctions of A . Furthermore, for any integer M > , we define H M , span { w , w , . . . , w M } and P M : H → H M be the L x orthogonal projection onto H M . Next, for any u, v, w ∈ V , we introduce the convenient notation for the bilinear term B ( u, v ) := P σ (( u · ∇ ) v ) , which can be extended to a continuous map B : V × V → V ′ such that h B ( u, v ) , w i = Z T ( u · ∇ v ) · w dx. for smooth functions u, v, w ∈ V . Notice that θ is a scalar function so we cannot actually apply P σ on it; hence,the notation P M θ should be understood as projection onto the space spanned by the first M eigenfunctions of − ∆ only. Therefore, in order to avoid abuse of notation, we denote B ( u, θ ) := u · ∇ θ for smooth functions, andextended it to a continuous map B : V × H → H − similarly to B ( · , · ) . We will use the following importantproperties of the map B . Detailed proof can be found in, e.g., [13, 18]. Lemma 2.1.
For the operator B , we have h B ( u, v ) , w i V ′ = − h B ( u, w ) , v i V ′ , ∀ u ∈ V, v ∈ V, w ∈ V, (2.5a) h B ( u, v ) , v i V ′ = 0 , ∀ u ∈ V, v ∈ V, w ∈ V, (2.5b) | h B ( u, v ) , w i V ′ | ≤ C k u k / L x k∇ u k / L x k∇ v k L x k∇ w k L x , ∀ u ∈ V, v ∈ V, w ∈ V, (2.5c) | h B ( u, v ) , w i V ′ | ≤ C k∇ u k L x k∇ v k L x k w k / L x k∇ w k / L x , ∀ u ∈ V, v ∈ V, w ∈ V, (2.5d) | h B ( u, v ) , w i V ′ | ≤ C k u k L x k∇ v k / L x k Av k / L x k∇ w k L x , ∀ u ∈ H, v ∈ D ( A ) , w ∈ V, (2.5e) | h B ( u, v ) , w i V ′ | ≤ C k∇ u k L x k∇ v k / L x k Av k / L x k w k L x , ∀ u ∈ V, v ∈ D ( A ) , w ∈ H, (2.5f) | h B ( u, v ) , w i V ′ | ≤ C k∇ u k / L x k Au k / L x k∇ v k L x k w k L x , ∀ u ∈ D ( A ) , v ∈ V, w ∈ H, (2.5g) | h B ( u, v ) , w i V ′ | ≤ C k u k L x k Av k L x k w k / L x k∇ w k / L x , ∀ u ∈ H, v ∈ D ( A ) , w ∈ V, (2.5h) | h B ( u, v ) , w i D ( A ) ′ | ≤ C k u k / L x k∇ u k / L x k v k L x k Aw k L x , ∀ u ∈ V, v ∈ H, w ∈ D ( A ) . (2.5i) Moreover, essentially identical results hold for B ( u, θ ) , mutatis mutandis . The following lemma is a special case of the Troisi inequality from [47] and is useful for our estimates through-out the paper.
Lemma 2.2.
There exists a constant
C > such that for v ∈ C ∞ ( R ) , we have k v k L ≤ C Y i =1 k ∂ i v k L . f , the equality f = ∇ p holds for somedistribution p if and only if h f, w i = 0 for all w ∈ V . See [48] for details.Next, we list three fundamental lemmas needed in order to prove Theorem 2.6. Their proofs can be found in[25] and [51], respectively. Lemma 2.3.
Assume u = ( u , u , u ) ∈ H ( T ) ∩ V . Then X j,k =1 Z T u j ∂ j u k ∆ h u k dx = 12 X j,k =1 Z T ∂ j u k ∂ j u k ∂ u dx − Z T ∂ u ∂ u ∂ u dx + Z T ∂ u ∂ u ∂ u dx. Lemma 2.4.
For u and b from the solution of (1.2) and i = 1 , , , we have Z T u j ∂ j u k ∂ ii u k dx − Z T b j ∂ j b k ∂ ii u k dx + Z T u j ∂ j b k ∂ ii b k dx − Z T b j ∂ j u k ∂ ii b k dx = X j,k =1 Z T − ∂ i u j ∂ j u k ∂ i u k dx + Z T ∂ i b j ∂ j b k ∂ i u k dx − Z T ∂ i u j ∂ j b k ∂ i b k dx + Z T ∂ i b j ∂ j u k ∂ i b k dx. The following Aubin-Lions Compactness Lemma is needed in order to construct solutions for (1.1).
Lemma 2.5.
Let
T > , p ∈ (1 , ∞ ) and let { f n ( t, · ) } ∞ n =1 be a bounded sequence of function in L pt ([0 , T ]; Y ) where Y is a Banach space. If { f n } ∞ n =1 is also bounded in L pt ([0 , T ]; X ) , where X is compactly imbedded in Y and { ∂f n /∂t } ∞ n =1 is bounded in L pt ([0 , T ]; Z ) uniformly where Y is continuously imbedded in Z , then { f n } ∞ n =1 is relatively compact in L pt ([0 , T ]; Y ) . The following theorem is our main result. It provides a Prodi-Serrin type regularity criterion for system (1.2).
Theorem 2.6.
For m ≥ , u , b ∈ H mx ∩ V , and θ ∈ H x , if we further assume that u , u , b , b ∈ L rt ([0 , T ); L sx ( T )) and r + 3 s = 34 + 12 s , s > / , for a given T > T ∗ where T ∗ is in Theorem 2.9. Then ( u, b, θ ) remains smooth beyond T ∗ . Namely, k u k H x , k b k H x , and k θ k H x remains bounded up to T > T ∗ , and consequently, u, b, θ ∈ C ∞ (Ω × (0 , T )) . The next three theorems provide local well-posedness for systems (1.3), (1.1), and (1.2). First, for the fullyinviscid system (1.3), we have
Theorem 2.7.
For the initial data ( u , b , θ ) ∈ H x ∩ V , there exists a unique solution ( u, b, θ ) ∈ L ∞ t ((0 , e T ); H x ∩ V ) to the fully inviscid MHD-Boussinesq system (1.3) for some e T > , depending on g and the initial data. Regarding system (1.1), we have
Theorem 2.8.
For m ≥ and u , b ∈ H mx ∩ V , and θ ∈ H mx , there exists a solution ( u, b, θ ) with u, b ∈ C w ([0 , T ); H ) ∩ L t ((0 , T ); V ) and θ ∈ C w ([0 , T ); L x ) ∩ L t ((0 , T ); H x ) for any T > for (1.1). Also, the solutionis unique if u, b ∈ L ∞ t ([0 , T ′ ); H mx ∩ V ) ∩ L t ((0 , T ′ ); H m +1 x ∩ V ) and θ ∈ L ∞ t ([0 , T ′ ); H mx ) ∩ L t ((0 , T ′ ); H m +1 x ) with some T ′ depending only on ν , η , and the initial datum. Theorem 2.9.
For m ≥ and u , b ∈ H mx ∩ V , θ ∈ H mx , there exists a unique solution ( u, b, θ ) to the non-diffusive MHD-Boussinesq system (1.2), where u, b ∈ L ∞ t ([0 , T ∗ ); H mx ∩ V ) ∩ L t ((0 , T ∗ ); H m +1 x ∩ V ) divergencefree, and θ ∈ L ∞ t ([0 , T ∗ ); H mx ) , where T ∗ depends on ν , η , and the initial datum. For Theorem 2.8, we use Galerkin approximation to obtain the solution for the full MHD-Boussinesq system (1.1),while for the existence part of Theorem 2.9, the proof is similar with only minor modification so we omit thedetails.
Proof of Theorem 2.8.
Consider the following finite-dimensional ODE system, which we think of as an approxi-mation to system (1.1) after apply the Leray projection P σ . du M dt − νAu M + P M B ( u M , u M ) = P M B ( b M , b M ) + gP σ ( θ M e ) ,db M dt − ηAb M + P M B ( u M , b M ) = P M B ( b M , u M ) ,dθ M dt − κ ∆ θ M + P M B ( u M , θ M ) = 0 , (3.6)with initial datum P M u ( · ,
0) = u M (0) , P M b ( · ,
0) = b M (0) , and P M θ ( · ,
0) = θ M (0) . Notice that all termsbut the time-derivatives of the above ODE systems are at most quadratic, and therefore they are locally Lipschitzcontinuous. Thus, by the Picard-Lindelhoff Theorem, we know that there exists a solution up to some time T M > .Next we take justified inner-products with the above three equations by u M , b M , and θ M , respectively, integrateby parts, and add the results to obtain ddt (cid:16) k u M k L x + k b M k L x + k θ M k L x (cid:17) + ν k∇ u M k L x + η k∇ b M k L x + κ k∇ θ M k L x = Z T ( b M · ∇ ) b M u M dx + Z T gθ M u M e dx + Z T ( b M · ∇ ) u M b M dx = g Z T θ M u M e dx, where we used the divergence free condition, Lemma 2.1, and the orthogonality of P σ and P M . By the Cauchy-Schwarz and Young’s inequalities, we obtain ddt (cid:16) k u M k L x + k b M k L x + k θ M k L x (cid:17) + 2 ν k∇ u M k L x + 2 η k∇ b M k L x + 2 κ k∇ θ M k L x ≤ C g (cid:16) k u M k L x + k θ M k L x (cid:17) . (3.7)Thus, by the differential form of Gr¨onwall’s inequality, u M and b M are uniformly bounded in L ∞ t ([0 , T M ); H ) ,while θ M is uniformly bounded in L ∞ t ([0 , T M ); L x , independently of T M . Namely, k u M ( t ) k L x + k b M ( t ) k L x + k θ M ( t ) k L x ≤ C g,T k u M (0) k L x + k b M (0) k L x + k θ M (0) k L x , for any < t < T M . Thus, for each M , the solutions can be extended uniquely beyond T M to an interval [0 , T ] ,where T > is arbitrary. In particular, the interval of existence and uniqueness is independent of M . Using the7mbedding L ∞ t ֒ → L t , and extracting a subsequence if necessary (which we relabel as ( u M , b M , θ M ) ), we mayinvoke the Banach-Alaoglu Theorem to obtain u, b ∈ L t ([0 , T ]; H ) , and θ ∈ L t ([0 , T ]; L x ) , such that u M ⇀ u and b M ⇀ b weakly in L t ([0 , T ]; H ) ,θ M ⇀ θ weakly in L t ([0 , T ]; L x ) . ( u, b, θ ) is our candidate solution. Next, integrating (3.7) over time from to t < T , and using Gr¨onwall’sinequality, we have that u M and b M are uniformly bounded in L t ([0 , t ); V ) , while θ M is uniformly bounded in L t ([0 , T ); H x ) for any T > . Next, we obtain bounds on du M /dt , db M /dt , and dθ M /dt in certain functionalspace uniformly with respect to M . Note that du M dt = − νAu M − P M B ( u M , u M ) + P M B ( b M , b M ) + gP M ( θ M e ) ,db M dt = − ηAb M − P M B ( u M , b M ) + P M B ( b M , u M ) ,dθ M dt = − κ ∆ θ M − B (( u M , θ M ) . (3.8)Note in the first equation that Au M is bounded in L t ([0 , T ); V ′ ) due to the fact that u M is bounded in L t ([0 , T ); V ) .Also, we have gP M ( θ M e ) is bounded in L t ([0 , T ); H ) . On the other hand, by Lemma 2.1, we have k P M B ( u M , u M ) k V ′ ≤ C k u M k / L x k∇ u M k / L x , as well as k P M B ( b M , b M ) k V ′ ≤ C k b M k / L x k∇ b M k / L x . Since the L -norm of u M is uniformly bounded and the L -norm of ∇ u M are uniformly integrable, we see that du M /dt are bounded in L / t ([0 , T ); V ′ ) . Similarly, from the second and third equations, we have that db M /dt and dθ M /dt are also bounded in L / t ([0 , T ); V ′ ) and L / t ([0 , T ); H − x ) , respectively. Therefore, by Lemma 2.5and the uniform bounds obtained above, there exists a subsequence (which we again relabel as ( u M , b M , θ M ) ifnecessary) such that u M → u and b M → b strongly in L t ([0 , T ]; H ) ,θ M → θ strongly in L t ([0 , T ]; L x ) ,u M ⇀ u and b M → b weakly in L t ([0 , T ]; V ) ,θ M ⇀ θ weakly in L t ([0 , T ]; H x ) ,u M ⇀ u and b M → b weak- ∗ in L ∞ t ([0 , T ]; H ) ,θ M ⇀ θ weak- ∗ in L ∞ t ([0 , T ]; L x ) , for any T > . Thus, by taking inner products of (3.6) with test function ψ ( t, x ) ∈ C t ([0 , T ]; C ∞ x ) with ψ ( T ) = 0 ,and using the standard arguments of strong/weak convergence for Navier-Stokes equations (see, e.g., [13, 45]), wehave that each of the linear and nonlinear terms in (3.6) converges to the appropriate limit in an appropriate weaksense. Namely, we obtain that (1.1) holds in the weak sense, where the pressure term p is recovered by the approachmentioned in Section 2 and we omit the details here. Finally, we take action of (1.1) with an arbitrary v ∈ V . Then,8y integrating in time over [ t , t ] ⊂ [0 , T ] and sending t → t one can prove by standard arguments (c.f. [13, 45])that u, b and θ are in fact weakly continuous in time. Therefore, the initial condition is satisfied in the weak sense.Next we show that the solution is in fact regular at least for short time, provided ( u , b , θ ) ∈ H m ∩ V . Westart by multiplying (1.1) by Au , Ab , and ∆ θ , respectively, integrate over T , and add, to obtain ddt (cid:16) k∇ u k L x + k∇ b k L x + k∇ θ k L x (cid:17) + ν k ∆ u k L x + η k ∆ b k L x + κ k ∆ θ k L x = − Z T ( u · ∇ ) u ∆ u dx + Z T ( b · ∇ ) b ∆ u dx + g Z T θ ∆ ue dx, − Z T ( u · ∇ ) b ∆ b dx + Z T ( b · ∇ ) u ∆ b dx − Z T ( u · ∇ ) θ ∆ θ dx ≤ C k∇ u k / L x k ∆ u k / L x + C k∇ b k / L x k ∆ b k / L x k ∆ u k L x + g k∇ u k L x k∇ θ k L x + C k∇ u k L x k∇ b k / L x k ∆ b k / L x + C k∇ b k L x k∇ u k / L x k ∆ u k / L x k ∆ b k L x + C k θ k L ∞ x k∇ u k L x k ∆ θ k L x ≤ ν k ∆ u k L x + η k ∆ b k L x + κ k ∆ θ k L x + Cν k∇ u k L x + Cνη k∇ b k L x + C k∇ θ k L x + C k∇ u k L x + Cη k∇ u k L x k∇ b k L x + Cνη k∇ b k L x k∇ u k L x + Cκ k∇ u k L x , where we applied the H¨older’s inequality, Sobolev embedding, and Young’s inequality. By denoting K ( t ) = k∇ u ( t ) k L x + k∇ b ( t ) k L x + k∇ θ ( t ) k L x , we have dKdt ≤ CK + CK , which implies that there exists a T ′ > such that K ( t ) ≤ Ce CT ′ / K (0) p − K (0)( e CT ′ −
1) =: K ( T ′ ) , for all t ∈ [0 , T ′ ] . (3.9)After integrating from t = 0 to t = T ′ and the constant C depends on the initial datum, g , ν , η , and κ . This showsthat ( u, b, θ ) ∈ L ∞ t ((0 , T ′ ); H ∩ V ) as M → ∞ , provided T ′ < /K (0) e C .In order to pass to the limit κ → + , we must show that the above existence time T ′ is independent of κ . Wefollow the vanishing viscosity technique for the Navier-Stokes equations, (c.f. [13]) i.e., let τ = κt , and denote e Q ( τ ) = 1 κ (cid:16) k∇ u ( τκ ) k L x + k∇ b ( τκ ) k L x + k∇ θ ( τκ ) k L x (cid:17) . The above H estimates thus imply that d e Qdτ ≤ e C + e C e Q , where e C depends only on g , ν , η , and is independent of κ . Thus, integrating from τ = 0 to τ = e τ , we obtain e Q ( e τ ) ≤ e Q (0)1 − e C e τ e Q (0) . e C e τ e Q (0) ≤ δ < , i.e., e C ( κ e t ) 1 κ (cid:0) k∇ u (0) k L x + k∇ b (0) k L x + k∇ θ (0) k L x (cid:1) ≤ δ < , it follows that e Q ( e τ ) ≤ C δ e Q (0) . Hence, we have proved that, if T ′ < e C (cid:0) k∇ u (0) k L x + k∇ b (0) k L x + k∇ θ (0) k L x (cid:1) , (3.10)then the above H estimates remain valid for any κ > .On the other hand, we showed earlier that ν Z T ′ k ∆ u k L x dt + η Z T ′ k ∆ b k L x dt + κ Z T ′ k ∆ θ k L x dt remains bounded as M → ∞ . Thus, we have ( u, b, θ ) ∈ L t ((0 , T ′ ); H ∩ V ) . In order to obtain the higher-order regularity in H and H , we follow standard arguments (see, e.g., [30]) and apply the following argumentsuccessively. First, for a multi-index α of order | α | = 2 , we apply the partial differential operator ∂ α , to (1.1), andtest the equations for u , b , and θ by ∂ α u , ∂ α b , and ∂ α θ , respectively, and obtain ddt k ∂ α u k L x + ν k∇ ∂ α u k L x = Z T ∂ α (( b · ∇ ) b ) ∂ α u dx − Z T ∂ α (( u · ∇ ) u ) ∂ α u dx + g Z T ∂ α θ∂ α u dx = I + I + I , ddt k ∂ α b k L x + η k∇ ∂ α b k L x = Z T ∂ α (( b · ∇ ) u ) ∂ α b dx − Z T ∂ α (( u · ∇ ) b ) ∂ α b dx = I + I , ddt k ∂ α θ k L x + κ k∇ ∂ α θ k L x = − Z T ∂ α (( u · ∇ ) θ ) ∂ α θ dx = I . In order to estimate I , we use Lemma 2.1 and get I = X ζ ≤ α (cid:18) αζ (cid:19) Z T (( ∂ ζ b · ∇ ) ∂ α − ζ b ) ∂ α u dx ≤ C k∇ b k L x k ∂ α u k / L x k∇ ∂ α u k / L x k∇ ∂ α b k L x + C k∇ b k L x k ∂ α u k / L x k∇ ∂ α u k / L x k∇ ∂ α b k L x + C k∇ b k L x k ∂ α b k / L x k∇ ∂ α b k / L x k∇ ∂ α u k L x where we used Young’s inequality in the last step. Similarly, I is estimated as I ≤ Cν k ∂ α u k L x + Cν k ∂ α u k L x + ν k∇ ∂ α u k L x . By Cauchy-Schwarz inequality, we obtain, I ≤ g k ∂ α u k L x + g k ∂ α b k L x . I and I , we proceed similarly to the estimates of I . Namely, we have I + I ≤ C (cid:18) Cνη + Cν + Cη + Cη + Cν (cid:19) (cid:16) k ∂ α b k L x + k ∂ α u k L x (cid:17) + (cid:18) Cη + Cν (cid:19) (cid:0) k ∂ α u k L x + k ∂ α b k L x (cid:1) + ν k∇ ∂ α u k L x + η k∇ ∂ α b k L x . Finally, the term I is bounded as I ≤ (cid:18) Cκ + Cκ (cid:19) k ∂ α θ k L x + Cκ k ∂ α θ k L x + Cν k ∂ α u k L x + ν k∇ ∂ α u k L x + κ k∇ ∂ α θ k L x . Summing up the above estimates and denoting ¯ Q = k ∂ α u k L x + k ∂ α b k L x + k ∂ α θ k L x , we arrive at d ¯ Qdt ≤ C + C ¯ Q, (3.11)where C depends on g , ν , η , κ , and K ( T ′ ) defined in (3.9) (i.e., the bounds on the H norms of u , b , and θ ).Hence, by Gr¨onwall inequality, we obtain ( u, b, θ ) ∈ L ∞ t ((0 , T ′ ); H ∩ V ) . Also, we have ν Z T ′ k ∂ α u k L x dt + η Z T ′ k ∂ α b k L x dt + κ Z T ′ k ∂ α θ k L x dt remains finite for | α | = 2 . Next, we apply ∂ α with | α | = 3 to (1.1), and multiply the equations for u , b , and θ by ∂ α u , ∂ α b , and ∂ α θ , respectively, and get ddt k ∂ α u k L x + ν k∇ ∂ α u k L x = Z T ∂ α (( b · ∇ ) b ) ∂ α u dx − Z T ∂ α (( u · ∇ ) u ) ∂ α u dx + g Z T ∂ α θ∂ α u dx = J + J + J , ddt k ∂ α b k L x + η k∇ ∂ α b k L x = Z T ∂ α (( b · ∇ ) u ) ∂ α b dx − Z T ∂ α (( u · ∇ ) b ) ∂ α b dx = J + J , ddt k ∂ α θ k L x + κ k∇ ∂ α θ k L x = − Z T ∂ α (( u · ∇ ) θ ) ∂ α θ dx = J . In order to estimate J , we apply Lemma 2.1 and obtain J ≤ X ≤| ζ |≤| α | (cid:18) αζ (cid:19) Z T | ∂ ζ b ||∇ ∂ α − ζ b || ∂ α u | dx ≤ C k∇ b k L x k ∂ α u k / L x k∇ ∂ α u k / L x k∇ ∂ α b k L x + C X | ζ | =1 k ∂ ζ b k / L x k∇ ∂ ζ b k / L x k ∂ α u k L x k∇ ∂ α − ζ b k L x + C X | ζ | =2 k ∂ ζ b k / L x k ∂ α b k / L x k∇ ∂ α u k L x + C k ∂ α b k L x k ∂ α b k / L x k∇ ∂ α b k / L x k ∂ α u k L x ≤ (cid:18) Cνη + Cη (cid:19) k ∂ α u k L x + (cid:18) Cν + Cη (cid:19) k ∂ α b k L x + ν k∇ ∂ α u k L x + η k∇ ∂ α b k L x , J are similar, i.e., we have J ≤ Cν k ∂ α u k L x + Cν k ∂ α u k L x + ν k∇ ∂ α u k L x . Using Cauchy-Schwarz inequality, we obtain J ≤ g k ∂ α u k L x + g k ∂ α b k L x . Regarding J and J , the estimates are similar to that of J . Namely, we have J + J ≤ C (cid:18) Cνη + Cν + Cη + Cη + Cν (cid:19) (cid:16) k ∂ α b k L x + k ∂ α u k L x (cid:17) + (cid:18) Cη + Cν (cid:19) (cid:0) k ∂ α u k L x + k ∂ α b k L x (cid:1) + ν k∇ ∂ α u k L x + η k∇ ∂ α b k L x . Similarly, the term J can be bounded as J ≤ (cid:18) Cκ + Cκ (cid:19) k ∂ α θ k L x + Cκ k ∂ α θ k L x + Cν k ∂ α u k L x + ν k∇ ∂ α u k L x + κ k∇ ∂ α θ k L x . Adding the above estimates and denoting Q = k ∂ α u k L x + k ∂ α b k L x + k ∂ α θ k L x , we have dQdt ≤ C + CQ, where C depends on g , ν , η , κ , and the bounds on the H norms of u , b , and θ . Hence, using Gr¨onwall’s inequalityand combining all the above estimates, we finally obtain ( u, b, θ ) ∈ L ∞ t ((0 , T ′ ); H ∩ V ) . Furthermore, we have ν Z T ′ k∇ ∂ α u k L x dt + η Z T ′ k∇ ∂ α b k L x dt + κ Z T ′ k∇ ∂ α θ k L x dt remains finite for | α | = 3 , i.e., ( u, b, θ ) ∈ L t ((0 , T ′ ); H ∩ V ) . Therefore, by slightly modifying the proof of theuniqueness of the non-diffusive system below, we obtain the uniqueness of the solution and Theorem 2.8 is thusproven. (cid:3) Proof of uniqueness in Theorem 2.9.
In order to prove uniqueness, we use the fact that ( u, b, θ ) ∈ L ∞ ([0 , T ∗ ); H m ) .Suppose that ( u (1) , b (1) , θ (1) ) and ( u (2) , b (2) , θ (2) ) are two solutions to the non-diffusive MHD-Boussinesq system(1.2). By subtracting the two systems for the two solutions denoting e u = u (1) − u (2) , e p = p (1) − p (2) , e b = b (1) − b (2) ,12nd e θ = θ (1) − θ (2) , and by using H¨older’s inequality, Gagliardo-Nirenberg-Sobolev inequality, and Young’s in-equality, to obtain ∂ e u∂t − ν ∆ e u + ( e u · ∇ ) u (1) + ( u (2) · ∇ ) e u + ∇ e p = ( e b · ∇ ) b (1) + ( b (2) · ∇ ) e b + g e θe ,∂ e b∂t − η ∆ e b + ( e u · ∇ ) b (1) + ( u (2) · ∇ ) e b = ( e b · ∇ ) u (1) + ( b (2) · ∇ ) e u,∂ e θ∂t + ( e u · ∇ ) θ (1) + ( u (2) · ∇ ) e θ = 0 , with ∇ · e u = 0 = ∇ e b . Multiply the above equations by e u , e b , and e θ , respectively, integrate over T , and add, we get ddt (cid:16) k e u k L x + k e b k L x + k e θ k L x (cid:17) + ν k∇ e u k L x + η k∇ e b k L x = Z T ( e u · ∇ ) u (1) e u dx − Z T ( e b · ∇ ) b (1) e u dx + Z T g e θe e u dx + Z T ( e u · ∇ ) b (1) e b dx − Z T ( e b · ∇ ) u (1) e b dx + Z T ( e u · ∇ ) θ (1) e θ dx ≤ C k∇ u (1) k L x k e u k / L x k∇ e u k / L x + C k∇ b (1) k L x k e b k / L x k∇ e b k / L x k∇ e u k L x + g k e u k L x k e θ k L x + C k∇ b (1) k L x k e u k / L x k∇ e u k / L x k∇ e b k L x + C k∇ u (1) k L x k e b k / L x k∇ e b k / L x + C k e u k / L x k∇ e u k / L x k∇∇ θ (1) k L x k e θ k L x ≤ Cν k e u k L x + ν k∇ e u k L x + Cνη k e b k L x + ν k∇ e u k L x + η k∇ e b k L x + g k e θ k L x + g k e u k L x + Cνη k e b k L x + η k∇ e b k L x + ν k∇ e u k L x + Cη k e b k L x + η k∇ e b k L x + Cν k e u k L x + ν k∇ e u k L x + C k e θ k L x , where we used the bound in (3.9) and (3.11) on [0 , T ] for T < T ∗ . Let us denote X ( t ) = k e u k L x + k e b k L x + k e θ k L x , for ≤ t ≤ T < T ∗ . Then we have dX ( t ) dt ≤ CX ( t ) , Gr¨onwall’s inequality then gives continuity in the L ∞ (0 , T ; L ) norm. Integrating, we also obtain continuity inthe L (0 , T ; V ) norm. If the initial data is the same, then X (0) = 0 , so we obtain uniqueness of the solutions (cid:3) Proof of Theorem 2.6.
We start by introducing the following notation. For the time interval ≤ t < t < ∞ , wedenote ( J ( t )) := sup τ ∈ ( t ,t ) (cid:8) k∇ h u ( τ ) k + k∇ h b ( τ ) k (cid:9) + Z t t k∇∇ h u ( τ ) k + k∇∇ h b ( τ ) k dτ, ∇ h = ( ∂ , ∂ ) , and ∆ h = ∂ + ∂ ). We also denote ( L ( t )) := sup τ ∈ ( t ,t ) (cid:8) k ∂ u ( τ ) k + k ∂ b ( τ ) k (cid:9) + Z t t k∇ ∂ u ( τ ) k + k∇ ∂ b ( τ ) k dτ. Aiming at a proof by contradiction, we denote the maximum time of existence and uniqueness of smooth solutionsby T max := sup { t ≥ | ( u, b, θ ) is smooth on (0 , t ) } . Since u , b , and θ are in H x , T max ∈ (0 , ∞ ] If T max = ∞ , the proof is done. Thus, we suppose T max < ∞ ,and show that the solution can be extended beyond T max , which is a contradiction. First, we choose ǫ > sufficiently small, say, ǫ < / (16 C max ) , where C max is the maximum of all the constants in the followingargument, depending on the space dimension, the constant g , the first eigenvalue λ of the operator − ∆ , as wellas the spatial-temporal L -norm of the solution up to T max . Then, we fix T ∈ (0 , T max ) such that T max − T < ǫ ,and Z T max T k∇ u ( τ ) k L x + k∇ b ( τ ) k L x + k θ k L x dτ < ǫ, (5.12)as well as Z T max T k u ( τ ) k rL sx + k u ( τ ) k rL sx + k b ( τ ) k rL sx + k b ( τ ) k rL sx dτ < ǫ. (5.13)We see that the proof is complete if we show that k∇ u ( T ) k + k∇ b ( T ) k + k∇ θ ( T ) k ≤ C < ∞ , for any T ∈ ( T , T max ) and C in independent of the choice of T . In fact, due to the continuity of integral, we can extendthe the regularity of u beyond T max and this becomes a contradiction to the definition of T max . Therefore, it issufficient to prove that J ( T ) + L ( T ) ≤ C < ∞ in view of the equation for θ in (1.2) for some constant C independent of T . We take the approach of [41], which first bounds L ( T ) by J ( T ) , then closes the estimates byobtaining an uniform upper bound on the latter. The regularity of θ thus follows from the higher order regularity of u and b . To start, we multiply the equations for u and b in (1.2) by − ∂ u and − ∂ b respectively, integrate over T × ( T , T ) , and sum to obtain (cid:16) k ∂ u ( T ) k L x + k ∂ b ( T ) k L x (cid:17) + Z T T Z T ν k∇ ∂ u k L x + η k∇ ∂ b k L x dx dτ = 12 (cid:16) k ∂ u ( T ) k L x + k ∂ b ( T ) k L x (cid:17) − X j,k =1 Z T T Z T ∂ u j ∂ j u k ∂ u k dx dτ + X j,k =1 Z T T Z T ∂ b j ∂ j b k ∂ u k dx dτ − X j,k =1 Z T T Z T ∂ u j ∂ j b k ∂ b k dx dτ + X j,k =1 Z T T Z T ∂ b j ∂ j u k ∂ b k dx dτ − g X k =1 Z T T Z T θe ∂ u k dx dτ, I , II , III , IV , and V , respectively. In order to estimate I we first rewrite it as I = − X j,k =1 Z T T Z T ∂ u j ∂ j u k ∂ u k dx dτ − X j =1 Z T T Z T ∂ u j ∂ j u ∂ u dx dτ − X k =1 Z T T Z T ∂ u ∂ u k ∂ u k dx dτ − Z T T Z T ∂ u ∂ u ∂ u dx dτ = X j,k =1 Z T T Z T u k (cid:0) ∂ u k ∂ j u j + ∂ u j ∂ j u k (cid:1) dx dτ − I a − I b − I c . By Lemma 2.1, the first two integrals on the right side of I are bounded by C Z T T Z T | u || ∂ u ||∇ ∂ u | dx dτ ≤ C Z T T k u k L x k ∂ u k L x k∇ h ∂ u k L x dτ ≤ C Z T T k u k L x k ∂ u k L x k ∂ u k L x k∇ h ∂ u k L x dτ ≤ C k∇ h u k L ∞ t L x k ∂ u k L ∞ t L x k ∂ u k L t L x k∇ h ∂ u k L t L x k ∂ u k L t L x k∇ h ∂ u k L t L x ≤ CǫL ( T ) J ( T ) , where the L ∞ t norms are taken over the interval ( T , T ) and we used Lemma 2.2 in the second to the last inequality.Regarding I a , I b , and I c , we first integrate by parts, then estimate as I a + I b + I c = X j =1 Z T T Z T u ∂ u j ∂ j u dx dτ + X j =1 Z T T Z T u ∂ j u ∂ u j dx dτ + 2 X k =1 Z T T Z T u ∂ u k ∂ u k dx dτ + 2 Z T T Z T u ∂ u ∂ u dx dτ ≤ C Z T T | u ||∇ h u ||∇ ∂ u | dx dτ + C Z T T | u || ∂ u ||∇ ∂ u | dx dτ ≤ C Z T T k u k L sx k∇ h u k − s L x k∇ ∂ u k s L x dτ + C Z T T k u k L sx k ∂ u k − s L x k∇ ∂ u k s L x dτ ≤ C ( T − T ) − ( r + s ) k u k L rt L sx k∇ h u k − s L ∞ t L x k∇ ∂ u k s L t L x + C ( T − T ) − ( r + s ) k u k L rt L sx k ∂ u k − s L ∞ t L x k∇ ∂ u k s L t L x ≤ CǫJ − s ( T ) L s ( T ) + CǫL ( T ) , where we used the fact that k∇ u k / L t L x is small over the interval ( T , T ) and the constant C is independent of T .15ext, we estimate II . Proceeding similarly as the estimates for I , we first integrate by parts and rewrite II as II = X j =1 2 X k =1 Z T T Z T b k ∂ b j ∂ j u k dx dτ + X j =1 Z T T Z T b ∂ b j ∂ j u dx dτ ≤ C Z T T Z T | b || ∂ b ||∇ h ∂ u | dx dτ + C Z T T Z T | b || ∂ b ||∇ ∂ u | dx dτ. Therefore, by Lemma 2.1 and Lemma 2.2, we get II ≤ C Z T T k b k L x k ∂ b k L x k∇ h ∂ u k L x dτ + C Z T T Z T ( | u | + | b | )( | ∂ u | + | ∂ b | )( |∇ ∂ u | + |∇ ∂ b | ) dx dτ ≤ C Z T T k b k L x k ∂ b k L x k∇ ∂ b k L x k∇ h ∂ u k L x dτ + C Z T T ( k u k L sx + k b k L sx )( k ∂ u k L x + k ∂ b k L x ) − s ( k∇ ∂ u k L x + k∇ ∂ b k L x ) s dτ ≤ C k∇ h b k L ∞ t L x k ∂ b k L ∞ t L x k ∂ b k L t L x k∇ h ∂ b k L t L x k ∂ b k L t L x k∇ h ∂ u k L t L x + C ( T − T ) − ( r + s ) ( k u k L rt L sx + k b k L rt L sx )( k ∂ u k L ∞ t L x + k ∂ b k L ∞ t L x ) − s × ( k∇ ∂ u k L t L x + k∇ ∂ b k L t L x ) s ≤ CǫL ( T ) J ( T ) + CǫJ − s ( T ) L s ( T ) + CǫL ( T ) . The terms
III and IV are estimated analogously, i.e., we have III + IV ≤ CǫL ( T ) J ( T ) + CǫJ − s ( T ) L s ( T ) + CǫL ( T ) , where the constant C does not depend on T . We estimate the term V as V = − X k =1 Z T T Z T θe ∂ u k dτ ≤ C k θ k L x,t k ∂ u k L x,t ≤ C k θ k L x k ∂ u k L x,t ≤ CǫL ( T ) . Collecting the above estimate for I through V and using Young’s inequality, we obtain L ( T ) ≤ C + CǫL ( T ) J ( T ) + CǫL s ( T ) J − s ( T ) + CǫL ( T ) + CǫL ( T ) ≤ C + CǫL ( T ) + CǫJ ( T ) + CǫJ ( T ) + CǫL ( T ) . Thus, with our choice of ǫ > earlier, we get L ( T ) ≤ C + CJ ( T ) . (5.14)Next, in order to bound J ( T ) , we multiply the equation for u and b in 1.2 by − ∆ h u and − ∆ h b , respectively,16ntegrate over T × ( T , T ) , sum up, integrate by parts and get (cid:16) k∇ h u ( T ) k L x + k∇ h b ( T ) k L x (cid:17) + Z T T Z T k∇∇ h u k L x + k∇∇ h b k L x = 12 (cid:16) k∇ h u ( T ) k L x + k∇ h b ( T ) k L x (cid:17) − X j,k =1 2 X i =1 Z T T Z T ∂ i u j ∂ j u k ∂ i u k dx dτ + X j,k =1 2 X i =1 Z T T Z T ∂ i b j ∂ j b k ∂ i u k dx dτ − X j,k =1 2 X i =1 Z T T Z T ∂ i u j ∂ j b k ∂ i b k dx dτ + X j,k =1 2 X i =1 Z T T Z T ∂ i b j ∂ j u k ∂ i b k dx dτ − g X k =1 2 X i =1 Z T T Z T θe ∂ ii u k dx dτ, where we used the divergence-free condition and Lemma 2.4. Denote by e I through e V the last five integrals on theright side of the above equation, respectively. Integrating by parts, we first rewrite e I as e I = − X i,j,k =1 Z T T Z T ∂ i u j ∂ j u k ∂ i u k dx dτ − X i,j =1 Z T T Z T ∂ i u j ∂ j u ∂ i u dx dτ − X i,k =1 Z T T Z T ∂ i u ∂ u k ∂ i u k dx dτ − X i =1 Z T T Z T ∂ i u ∂ u ∂ i u dx dτ = 12 X j,k =1 Z T T Z T u ∂ j u k ∂ j u k dx dτ − Z T T Z T u ∂ u ∂ u dx dτ − Z T T Z T u ∂ u ∂ u dx dτ + Z T T Z T u ∂ u ∂ u dx dτ + Z T T Z T u ∂ u ∂ u dx dτ + X i,j =1 Z T T Z T u ∂ i u j ∂ j u dx dτ + X i,j =1 Z T T Z T u ∂ j u ∂ i u j dx dτ + X i,k =1 Z T T Z T u ∂ u k ∂ ii u k dx dτ + X i,k =1 Z T T Z T u ∂ i u k ∂ u k dx dτ + 2 X i =1 Z T T Z T u ∂ i u ∂ i u dx dτ, where we applied Lemma 2.3 to the first term on the right side of the first equality above. Thus, by H¨older and17obolev inequalities, we bound e I as e I ≤ C Z T T Z T | u | ( |∇ h u | + | ∂ u | ) |∇∇ h u | dx dτ ≤ C Z T T k u k L sx k∇ h u k − s L x k∇∇ h u k s L x dτ + C Z T T k u k L sx k ∂ u k − s L x k∇ h ∂ u k s Lx k ∂ u k s Lx k∇∇ h u k L x dτ ≤ C ( T − T ) − ( r + s ) k u k L rt L sx k∇ h u k − s L ∞ t L x k∇∇ h u k s L t L x + C ( T − T ) − ( r + s ) k u k L rt L sx k ∂ u k s − s L t L x k ∂ u k s − s L ∞ t L x × k∇ ∂ u k s L t L x k∇∇ h u k s L t L x ≤ C + CǫJ ( T ) + CCǫJ
43 3 s − s +1+ s ≤ C + CǫJ ( T ) , where we used (5.14) and the fact that T − T < ǫ and /r + 3 /s = 3 / / (2 s ) for s > / . In order toestimate f II , we proceed a bit differently since Lemma 2.3 is not available for convective terms mixed with u and b . Instead, we integrate by parts and use the divergence-free condition ∂ b = − ∂ b − ∂ b and obtain f II = X j,k =1 2 X i =1 Z T T Z T ∂ i b j ∂ j b k ∂ i u k dx dτ = X i =1 Z T T Z T ∂ j b ∂ b ∂ i u dx dτ + X i =1 3 X k =2 Z T T Z T ∂ i b ∂ b k ∂ i u k dx dτ + X i =1 3 X k =1 3 X j =2 Z T T Z T ∂ i b j ∂ j b k ∂ i u k dx dτ = X i =1 Z T T Z T ∂ j b ( − b ∂ − b ∂ ) ∂ i u dx dτ − X i =1 3 X k =2 Z T T Z T u k ∂ i b ∂ i b k dx dτ − X i =1 3 X k =2 Z T T Z T u k ∂ b k ∂ ii b dx dτ − X i =1 3 X k =1 3 X j =2 Z T T Z T b j ∂ j b k ∂ ii u k dx dτ − X i =1 3 X k =1 3 X j =2 Z T T Z T b j ∂ i u k ∂ ij b k dx dτ. f II as f II ≤ C Z T T Z T ( | b | + | b | )( |∇ h u | + |∇ h b | + | ∂ u | + | ∂ b | )( |∇∇ h u | + |∇∇ h b | ) dx dτ ≤ C Z T T ( k b k L sx + k b k L sx )( k∇ h u k L x + k∇ h b k L x ) − s ( k∇∇ h u k L x + k∇∇ h b k L x ) s dτ + C Z T T ( k b k L sx + k b k L sx )( k ∂ u k L x + k ∂ b k L x ) − s ( k∇ h ∂ u k Lx + k∇ h ∂ b k Lx ) s × ( k ∂ u k Lx + k ∂ b k Lx ) s ( k∇∇ h u k L x + k∇∇ h b k L x ) dτ ≤ C ( T − T ) − ( r + s ) ( k b k L rt L sx + k b k L rt L sx ) × ( k∇ h u k L ∞ t L x + k∇ h b k L ∞ t L x ) − s ( k∇∇ h u k L t L x + k∇∇ h b k L t L x ) s + C ( T − T ) − ( r + s ) ( k b k L rt L sx + k b k L rt L sx ) × ( k ∂ u k L t L x + k ∂ b k L t L x ) s − s ( k ∂ u k L ∞ t L x + k ∂ b k L ∞ t L x ) s − s × ( k∇ ∂ u k L t L x + k∇ ∂ b k L t L x ) s ( k∇∇ h u k L t L x + k∇∇ h b k L t L x ) s ≤ C + CǫJ ( T ) + CǫJ
43 3 s − s +1+ s ≤ C + CǫJ ( T ) . Regarding g III , we proceed similarly as in the estimates for f II . Namely, we have g III = X j,k =1 2 X i =1 Z T T Z T ∂ i u j ∂ j b k ∂ i b k dx dτ = X i =1 Z T T Z T ∂ j u ∂ b ∂ i b dx dτ + X i =1 3 X k =2 Z T T Z T ∂ i u ∂ b k ∂ i b k dx dτ + X i =1 3 X k =1 3 X j =2 Z T T Z T ∂ i u j ∂ j b k ∂ i b k dx dτ = X i =1 Z T T Z T ∂ j u ( − b ∂ − b ∂ ) ∂ i b dx dτ − X i =1 3 X k =2 Z T T Z T b k ∂ i u ∂ i b k dx dτ − X i =1 3 X k =2 Z T T Z T b k ∂ b k ∂ ii u dx dτ − X i =1 3 X k =1 3 X j =2 Z T T Z T u j ∂ j b k ∂ ii b k dx dτ − X i =1 3 X k =1 3 X j =2 Z T T Z T u j ∂ i b k ∂ ij b k dx dτ ≤ C Z T T Z T ( | u | + | u | + | b | + | b | )( |∇ h u | + |∇ h b | + | ∂ u | + | ∂ b | )( |∇∇ h u | + |∇∇ h b | ) dx dτ. Whence, by H¨older’s inequality and Gagliardo-Nirenberg-Sobolev inequality the far right side of the above in-equality is also bounded by C + CǫJ ( T ) + CǫJ
43 3 s − s +1+ s hence by C + CǫJ ( T ) in view of (5.14). The term f IV is bounded similarly as g III by C + CǫJ ( T ) , thus, we19mit the details. Next we estimate e V . Observing Theorem 2.8, we have e V = g X k =1 2 X i =1 Z T T Z T θe ∂ ii u k dx dτ ≤ C k θ k L x,t k∇∇ h u k L x,t ≤ CǫJ ( T ) , due to (5.12). Combining the above estimates for e I through e V , we get (cid:16) k∇ h u ( T ) k L x + k∇ h b ( T ) k L x (cid:17) + Z T T Z T k∇∇ h u k L x + k∇∇ h b k L x dx dτ ≤ (cid:16) k∇ h u ( T ) k L x + k∇ h b ( T ) k L x (cid:17) + C + CǫJ ( T ) + CǫJ ( T ) , where is the constant C is independent of T . Therefore, we get J ( T ) = sup τ ∈ ( t ,t ) (cid:8) k∇ h u ( τ ) k + k∇ h b ( τ ) k (cid:9) + Z t t k∇∇ h u ( τ ) k + k∇∇ h b ( τ ) k dx dτ ≤ (cid:16) k∇ h u ( T ) k L x + k∇ h b ( T ) k L x (cid:17) + CǫJ ( T ) + CǫJ ( T ) + C, where we applied the ǫ -Young inequality. Hence, by choosing ǫ < / C we obtain
14 sup τ ∈ ( t ,t ) (cid:8) k∇ h u ( τ ) k + k∇ h b ( τ ) k (cid:9) + Z t t k∇∇ h u ( τ ) k + k∇∇ h b ( τ ) k dx dτ (5.15) ≤ (cid:16) k∇ h u ( T ) k L x + k∇ h b ( T ) k L x (cid:17) + C, (5.16)Finally, we have k∇ h u ( T ) k L x + k∇ h b ( T ) k L x ≤ (cid:16) k∇ h u ( T ) k L x + k∇ h b ( T ) k L x (cid:17) + C, for any T ∈ ( T , T max ) . Therefore we have sup T ∈ ( T ,T max ) k∇ h u ( T ) k L x ≤ C < ∞ , and by (5.14) and (5.16), we obtain sup T ∈ ( T ,T max ) (cid:0) J ( T ) + L ( T ) (cid:1) ≤ C < ∞ , which implies u, b ∈ L ∞ t ([0 , T ); H ∩ V ) ∩ L t ([0 , T ); H ∩ V ) . Thus, by our arguments in previous sections, u and b are smooth up to time T . In particular, u and b are boundedin H ∩ V . Whence, we multiply the equation for θ in (1.2) by − ∆ θ , integrate by parts over T and obtain ddt k∇ θ k L x = X i,j =1 Z T u j ∂ j θ∂ ii θ dx ≤ C Z T |∇ u ||∇ θ | dx ≤ C k∇ u k L ∞ x k∇ θ k L x ≤ C k u k H x k∇ θ k L x , where we used ∇ · u = 0 and the Sobolev embedding H ֒ → L ∞ . Integrating in time from T to T and by thefact that u is bounded in H independent of T , we have θ ∈ L ∞ t ([0 , T ); H ∩ V ) due to Gr¨onwall’s inequality.The proof of Theorem 2.6 is thus complete. (cid:3) Results regarding the fully inviscid case
We provide a proof following a similar argument to the one given for the existence and uniqueness for the three-dimensional Euler equations in [23] and [30].
Proof of Theorem 2.7.
The first part of the proof follows similarly to that of Theorem 2.9 and we use the samenotation here, except that we choose the orthogonal projection P N from H to its subspaces H σ generated by thefunctions { e πik · x | | k | = max k i ≤ N } , for integer N > and k ∈ Z . For u N , b N ∈ H σ , and θ N and p N in the corresponding projected space for scalarfuntions, respectively, we consider solutions of the following ODE system, du N dt + P N B ( u N , u N ) + ∇ p N = P N B ( b N , b N ) + gθ N e ,db N dt + P N B ( u N , b N ) = P N B ( b N , u N ) ,dθ N dt + P N B ( u N , θ N ) = 0 , where we slightly abuse the notation by using B and B to denote the same type of nonlinear terms as wereintroduced in Section 2. We show that the limit of the sequence of solutions exists and solves of original system(1.3). First, we observe that the above ODE system has solution for any time T > since all terms but the timederivatives are at least locally Lipschitz continuous. In particular, by similar arguments as in Section 3, the solutionremains bounded in L ∞ t ((0 , e T ); H ) ∩ L ∞ t ((0 , e T ); H m ∩ V ) for some e T depending on the H -norm of the initialdata. Next, we show that ( u N , b N , θ N ) is a Cauchy sequence in L . For N ′ > N , by subtracting the correspondingequations for ( u N , b N , θ N ) and ( u N ′ , b N ′ , θ N ′ ) , we obtain ddt ( u N − u N ′ ) = − P N B ( u N , u N ) + P N ′ B ( u N ′ , u N ′ ) + P N B ( b N , b N ) − P N ′ B ( b N ′ , b N ′ ) − ∇ ( p N − p N ′ ) + g ( θ N − θ N ′ ) e ,ddt ( b N − b N ′ ) = − P N B ( u N , b N ) + P N ′ B ( u N ′ , b N ′ ) + P N B ( b N , u N ) − P N ′ B ( b N ′ , u N ′ ) ,ddt ( θ N − θ N ′ ) = − P N B ( u N , θ N ) + P N ′ B ( u N ′ , θ N ′ ) . ( u N − u N ′ ) , ( b N − b N ′ ) , and ( θ N − θ N ′ ) . Adding allthree equations, and using (2.5a) and (2.5b) from Lemma 2.1, we obtain ddt (cid:16) k u N − u N ′ k L x + k b N − b N ′ k L x + k θ N − θ N ′ k L x (cid:17) = g (( u N − u N ′ ) e )( θ N − θ N ′ ) − ( P N B ( u N , u N ) , u N ′ ) − ( P N ′ B ( u N ′ , u N ′ ) , u N ) − ( P N B ( b N , b N ) , u N ′ ) − ( P N ′ B ( b N ′ , b N ′ ) , u N ) + ( P N B ( u N , b N ) , b N ′ ) + ( P N ′ B ( u N ′ , b N ′ ) , b N ) − ( P N B ( b N , u N ) , b N ′ ) − ( P N ′ B ( b N ′ , u N ′ ) , b N ) + ( P N B ( u N , θ N ) , θ N ′ ) − ( P N ′ B ( u N ′ , θ N ′ ) , θ N )= g (( u N − u N ′ ) e )( θ N − θ N ′ ) + ((1 − P N ) B ( u N , u N ) , u N ′ ) + ( B ( u N − u N ′ , u N ′ − u N ) , u N )+ ((1 − P N ) B ( b N , b N ) , u N ′ ) + ( B ( b N − b N ′ , u N ′ − u N ) , u N )+ ((1 − P N ) B ( b N , u N ) , b N ′ ) + ( B ( b N − b N ′ , b N ′ − b N ) , u N ) − ((1 − P N ) B ( u N , b N ) , u N ′ ) + ( B ( u N − u N ′ , b N ′ − b N ) , b N ) − ((1 − P N ) B ( u N , θ N ) , θ N ′ ) + ( B ( u N − u N ′ , θ N ′ − θ N ) , θ N )= S + X i =1 S i , where we integrated by parts and used the divergence free condition ∇ · u N = ∇ · u N ′ = ∇ · b N = ∇ · b N ′ = 0 .Then we estimate S and the two types of terms S i , i = 1 , . . . , separately. After integration by parts, we firsthave S + X i even S i ≤ g k u N − u N ′ k L x k θ N − θ N ′ k L x + k∇ u N k L ∞ x k u N − u N ′ k L x + 2 k∇ b N k L ∞ x k u N − u N ′ k L x k b N − b N ′ k L x + k∇ u N k L ∞ x k b N − b N ′ k L x + k∇ θ N k L ∞ x k u N − u N ′ k L x k θ N − θ N ′ k L x ≤ C (cid:16) k u N − u N ′ k L x + k b N − b N ′ k L x + k θ N − θ N ′ k L x (cid:17) , where we used H¨older’s inequality and the Sobolev embedding H ֒ → L ∞ . Here the constant C depends onlyon the H norm of u , b , and θ . Regarding the remaining terms, we denote by ˆ f , the Fourier transform of f ∈ L ( T ) ˆ f ( k ) = 1(2 π ) / Z T e − ik · x f ( x ) dx, X i odd S i ≤ k ( u N · ∇ ) u N k L x k (1 − P N ) u N ′ k L x + k ( b N · ∇ ) b N k L x k (1 − P N ) u N ′ k L x + k ( b N · ∇ ) u N k L x k (1 − P N ) b N ′ k L x + k ( u N · ∇ ) b N k L x k (1 − P N ) b N ′ k L x + k ( u N · ∇ ) θ N k L x k (1 − P N ) θ N ′ k L x ≤ C k∇ u N k L ∞ x k u N k L x X | k | >N | ˆ u N ′ ( k ) | (1 + | k | ) N ) / + C k∇ b N k L ∞ x k b N k L x X | k | >N | ˆ u N ′ ( k ) | (1 + | k | ) N ) / + C k∇ u N k L ∞ x k b N k L x X | k | >N | ˆ b N ′ ( k ) | (1 + | k | ) N ) / + C k∇ b N k L ∞ x k u N k L x X | k | >N | ˆ b N ′ ( k ) | (1 + | k | ) N ) / + C k∇ θ N k L ∞ x k u N k L x X | k | >N | ˆ θ N ′ ( k ) | (1 + | k | ) N ) / ≤ CN , where C depends on the initial datum, and we used the fact that k f k H x = X k ∈ Z | ˆ f ( k ) | (1 + | k | ) . Summing up the above estimates we have ddt (cid:16) k u N − u N ′ k L x + k b N − b N ′ k L x + k θ N − θ N ′ k L x (cid:17) ≤ C (cid:16) k u N − u N ′ k L x + k b N − b N ′ k L x + k θ N − θ N ′ k L x (cid:17) + CN , which by Gr¨onwall’s inequality implies k u N − u N ′ k L x + k b N − b N ′ k L x + k θ N − θ N ′ k L x ≤ CN . Sending N → ∞ , we obtain the desired Cauchy sequence. Namely, ( u N , b N , θ N ) → ( u, b, θ ) with u, b ∈ H and θ ∈ L x . Due to the above convergence and the fact that u N , b N ∈ H x ∩ V and θ ∈ H x , we see that u and b arealso bounded in H x ∩ V while θ is bounded in H x . Thus, the existence part of the theorem is proved by easilyverifying that ( u, b, θ ) satisfies system (1.3) with some pressure p as discussed below. In fact, for a test function23 ( x ) ∈ V and < t < e T , ( u N , b N , θ N ) satisfies ( u N ( · , t ) , φ ) = ( u N (( · , , φ ) + Z t ( P N (( u N · ∇ ) φ, u N ) dτ − Z t ( P N (( b N · ∇ ) φ ) , b N ) dτ + g Z t ( θ N e , φ ) dτ, ( b N (( · , t ) , φ ) = ( b N (( · , , φ ) + Z t ( P N (( u N · ∇ ) φ ) , b N ) dτ − Z t ( P N (( b N · ∇ ) φ ) , u N ) dτ, ( θ N (( · , t ) , φ ) = ( θ N (( · , , φ ) + Z t ( B ( u N , φ ) , θ N ) . Sending N → ∞ and extracting a subsequence if necessary, we have that the integrals of nonlinear terms convergeweakly to the corresponding integrals of nonlinear terms in (1.3). Also, we see that the nonlinear terms are weaklycontinuous in time. Whence by differentiating the first equation in time, we conclude that the limit indeed satisfiesthe equations for u in (1.3) in the weak sense, i.e., ddt ( u (( · , t ) , φ ) = − (( u · ∇ ) u, φ ) + (( b · ∇ ) b, φ ) + ( gθe , φ ) , which in turn implies that there exists some p ∈ C ([0 , e T ]; H ) , such that dudt + ( u · ∇ ) u + ∇ p = ( b · ∇ ) b + gθe . Regarding uniqueness, suppose there are two solutions ( u (1) , b (1) , θ (1) ) and ( u (2) , b (2) , θ (2) ) with the sameinitial data ( u , b , θ ) for (1.3). Subtracting the corresponding equations for the two solutions and denoting e u , e b ,and e θ for u (1) − u (2) , b (1) − b (2) , and θ (1) − θ (2) , respectively, we obtain ∂ e u∂t + ( e u · ∇ ) u (1) + ( u (2) · ∇ ) e u + ∇ e p = ( e b · ∇ ) b (1) + ( b (2) · ∇ ) e b + g e θe ,∂ e b∂t + ( e u · ∇ ) b (1) + ( u (2) · ∇ ) e b = ( e b · ∇ ) u (1) + ( b (2) · ∇ ) e u,∂ e θ∂t + ( e u · ∇ ) θ (1) + ( u (2) · ∇ ) e θ = 0 , with ∇ · e u = 0 = ∇ e b and e u (0) = e b (0) = e θ (0) = 0 . Multiply the above equations by e u , e b , and e θ , respectively,integrate over T , and add, we get ddt (cid:16) k e u k L x + k e b k L x + k e θ k L x (cid:17) = Z T ( e u · ∇ ) u (1) e u dx − Z T ( e b · ∇ ) b (1) e u dx + Z T g e θe e u dx + Z T ( e u · ∇ ) b (1) e b dx − Z T ( e b · ∇ ) u (1) e b dx + Z T ( e u · ∇ ) θ (1) e θ dx ≤ C k u (1) k L ∞ x k e u k L x + C k b (1) k L ∞ x k e u k L x k e b k L x + C k u (1) k L ∞ x k e u k L x k e b k L x + C k θ (1) k L ∞ x k e u k L x k e θ k L x , where we applied H¨older’s inequality and the Sobolev-Nirenberg inequality. Now due to the embedding H ֒ → L ∞ ( T ) , and Young’s inequality, we have ddt (cid:16) k e u k L x + k e b k L x + k e θ k L x (cid:17) ≤ C (cid:16) k e u k L x + k e b k L x + k e θ k L x (cid:17) , C depends on g and H norm of ( u (1) , b (1) , θ (1) ) . Thus, by Gr¨onwall’s inequality, ( e u ( t ) , e b ( t ) , e θ ( t )) remains for ≤ t ≤ ¯ T . Uniqueness is proved. 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