Global Existence and Optimal Decay Rates of Solutions for Compressible Hall-MHD Equations
aa r X i v : . [ m a t h . A P ] M a y Global Existence and Optimal Decay Rates of Solutions forCompressible Hall-MHD Equations
Jincheng Gao Zheng-an Yao
School of Mathematics and Computational Science, Sun Yat-sen University,510275, Guangzhou, PR China
Abstract
In this paper, we are concerned with global existence and optimal decay rates of solutions forthe three-dimensional compressible Hall-MHD equations. First, we prove the global existenceof strong solutions by the standard energy method under the condition that the initial data areclose to the constant equilibrium state in H -framework. Second, optimal decay rates of strongsolutions in L -norm are obtained if the initial data belong to L additionally. Finally, weapply Fourier splitting method by Schonbek [Arch.Rational Mech. Anal. 88 (1985)] to establishoptimal decay rates for higher order spatial derivatives of classical solutions in H -framework,which improves the work of Fan et al.[Nonlinear Anal. Real World Appl. 22 (2015)].Keywords: compressible Hall-MHD equations; global solutions; optimal decay rates; Fouriersplitting method.2010 Mathematics Subject Classification: 76W05, 35Q35, 35D05, 76X05. The application of Hall-magnetohydrodynamics system (in short, Hall-MHD) covers a very widerange of physical objects, for example, magnetic reconnection in space plasmas, star formulation,neutron stars, and geodynamo, refer to [1–6] and the references therein. Recently, Acheritogaray etal.[7] derived the Hall-MHD equations from the two-fluid Euler-Maxwell system for electrons andions through a set of scaling limits or from the kinetic equations by taking macroscopic quantities inthe equations under some closure assumptions. They also established the global existence of weaksolutions for periodic boundary condition. In this paper, we investigate the following compressibleHall-MHD equations in three dimensional whole space R (see [7]): ρ t + div( ρu ) = 0 , ( ρu ) t + div( ρu ⊗ u ) − µ ∆ u − ( µ + ν ) ∇ div u + ∇ P ( ρ ) = (curl B ) × B,B t − curl( u × B ) + curl (cid:20) (curl B ) × Bρ (cid:21) = ∆ B, div B = 0 , (1.1)where the functions ρ, u, and B represent density, velocity, and magnetic field respectively. Thepressure P ( ρ ) is a smooth function in a neighborhood of 1 with P ′ (1) = 1. The constants µ and ν Email: [email protected](J.C.Gao), [email protected](Z.A.Yao)..C.Gao, Z.A.Yao denote the viscosity coefficients of the flow and satisfy physical condition as follows µ > , µ + 3 ν ≥ . To complete the system (1.1), the initial data are given by( ρ, u, B )( x, t ) | t =0 = ( ρ ( x ) , u ( x ) , B ( x )) . (1.2)Furthermore, as the space variable tends to infinity, we assumelim | x |→∞ ( ρ − , u , B )( x ) = 0 . (1.3)Obviously, the compressible Hall-MHD equations transform into the well-known compressible MHDequations when the Hall effect term curl (cid:16) (curl B ) × Bρ (cid:17) is neglected.When the density is constant, Chae et al.[8] proved local existence of smooth solutions for largedata and global smooth solutions for small data in three dimensional whole space. They also showeda Liouville theorem for the stationary solutions. Chae and Lee [9] established an optimal blow-upcriterion for classical solutions and proved two global-in-time existence results of classical solutionsfor small initial data, the smallness conditions of which are given by the suitable Sobolev and Besovnorms respectively. Later, Fan et al.[10] also established some new regularity criteria, which alsoare built for density-dependent incompressible Hall-MHD equations with positive initial density byFan and Ozawa [11]. Maicon and Lucas [12] proved a stability theorem for global large solutionsunder a suitable integrable hypothesis and constructed a special large solution by assuming thecondition of curl-free magnetic fields. Fan et al. [13] established the global well-posedness of theaxisymmetric solutions. Chae and Schonbek [14] established temporal decay estimates for weaksolutions and obtained algebraic time decay for higher order Sobolev norms of small initial datasolutions as follows k∇ k u ( t ) k L + k∇ k B ( t ) k L ≤ C (1 + t ) − k , k ∈ N for all t ≥ T ∗ ( T ∗ is a positive constant). Furthermore, Weng [15] extended this result by providingupper and lower bounds on the decay of higher order derivatives. For the compressible Hall-MHDequations (1.1), Fan et al.[16] proved the local existence of strong solutions with positive initialdensity and global small solutions(classical solutions) with small initial perturbation. They alsoestablished optimal time decay rate for classical solutions as follows k ( ρ − , u, B )( t ) k L ≤ C (1 + t ) − . (1.4)Here, they required the initial perturbation is small in H -norm and bounded in L -norm.Recently, the study of decay rates for solutions to the MHD equations has aroused many re-searchers’ interest. First of all, under the H -framework, Li and Yu [17] and Chen and Tan [18] notonly established the global existence of classical solutions, but also obtained the time decay ratesfor the three-dimensional compressible MHD equations by assuming the initial data belong to L and L q ( q ∈ (cid:2) , (cid:1) ) respectively. More precisely, Chen and Tan [18] built the time decay rates k∇ k ( ρ − , u, B )( t ) k H − k ≤ C (1 + t ) − (cid:16) q − (cid:17) − k , (1.5)where k = 0 ,
1. The time decay rates (1.5) has also been established by Li and Yu [17] for the case q = 1. Motivated by the work of Guo and Wang [19], Tan and Wang [20] established the optimal lobal Existence and Optimal Decay Rates of Solutions for Compressible Hall-MHD Equations time decay rates for the higher order spatial derivatives of solutions if the initial perturbationbelongs to H N ∩ ˙ H − s (cid:0) N ≥ , s ∈ (cid:2) , (cid:1)(cid:1) . More precisely, they built the following time decay rates k∇ k ( ρ − , u, B )( t ) k H N − k ≤ C (1 + t ) − k + s , where k = 0 , , ..., N −
1. Motivated by the work [22], we (see [23]) establish the following timedecay rates for all t ≥ T ∗ ( T ∗ is a positive constant), k∇ k ( ρ − t ) k H − k + k∇ k u ( t ) k H − k ≤ C (1 + t ) − k , k∇ m B ( t ) k H − m ≤ C (1 + t ) − m , (1.6)where k = 0 , , , and m = 0 , , ,
3. It is easy to see that the time decay rates (1.6) is better thandecay rates (1.5) since (1.6) provides faster time decay rates for the higher order spatial derivativesof solutions.In this paper, we hope to establish the global existence and time decay rates of solutions forthe compressible Hall-MHD equations (1.1)-(1.3). First of all, we construct the global existence ofstrong solutions by the standard energy method under the condition that the initial data are closeto the constant equilibrium state (1 , ,
0) in H -norm. Second, if the initial data in L − norm arefinite additionally, the optimal time decay rates of strong solutions are established by the methodof Green function. Precisely, we obtain the following time decay rates for all t ≥ k ( ρ − t ) k H − k + k u ( t ) k H − k + k B ( t ) k H − k ≤ C (1 + t ) − k , where k = 0 ,
1. This framework of time convergence rates for compressible flows has been applied toother compressible models, refer to [24–27]. Although magnetic field equations (1.1) are nonlinearparabolic equations, we hope to establish optimal time decay rates for the second order spatialderivatives of magnetic field under the condition of small initial perturbation. In order to achievethis goal, we move the nonlinear terms to the right hand side of (1.1) and deal with the nonlinearterms as external force with the property on fast time decay rates. Then, the application of Fouriersplitting method by Schonbek [21] helps us to establish optimal time decay rate for the second orderspatial derivatives of magnetic field as follows k∇ B ( t ) k L ≤ C (1 + t ) − . Finally, one focus on establishing optimal time decay rates for higher order spatial derivatives ofclassical solutions to compressible Hall-MHD equations. More precisely, we prove that the globalclassical solution ( ρ, u, B ) of Cauchy problem (1.1)-(1.3) has the time decay rates (1.6). Obviously,these time decay rates improve the results (1.4) by Fan et al.[16] since we build faster time decayrates for higher order spatial derivatives of classical solutions.
Notation:
In this paper, we use H s ( R )( s ∈ R ) to denote the usual Sobolev spaces with norm k · k H s and L p ( R )(1 ≤ p ≤ ∞ ) to denote the usual L p spaces with norm k · k L p . The symbol ∇ l with an integer l ≥ l . For example, we define ∇ k v = { ∂ αx v i | | α | = k, i = 1 , , } , v = ( v , v , v ) . We also denote F ( f ) := ˆ f . The notation a . b means that a ≤ Cb for a universal constant C > t . The notation a ≈ b means a . b and b . a . For the sake of simplicity, wewrite R f dx := R R f dx. .C.Gao, Z.A.Yao First of all, we establish the global existence and optimal decay rates of strong solutions for thecompressible Hall-MHD equations (1.1)-(1.3).
Theorem 1.1.
Assume that the initial data ( ρ − , u , B ) ∈ H and there exists a small constant δ > such that k ( ρ − , u , B ) k H ≤ δ , then the problem (1.1) - (1.3) admits a unique global strong solution ( ρ, u, B ) satisfying for all t ≥ , k ( ρ − , u, B )( t ) k H + Z t ( k∇ ρ ( s ) k H + k∇ ( u, B )( s ) k H ) ds ≤ C k ( ρ − , u , B ) k H . Furthermore, if k ( ρ − , u , B ) k L is finite additionally, then the global strong solution ( ρ, u, B ) has following decay rates for all t ≥ , k∇ k ( ρ − t ) k H − k + k∇ k u ( t ) k H − k ≤ C (1 + t ) − k , k∇ m B ( t ) k H − m ≤ C (1 + t ) − m , (1.7) where k = 0 , , and m = 0 , , . Remark 1.1.
For any ≤ p ≤ , by virtue of Theorem 1.1 and the Sobolev interpolation inequality,we obtain time decay rates as follows k ( ρ − t ) k L p + k u ( t ) k L p ≤ C (1 + t ) − (cid:16) − p (cid:17) , k∇ k B ( t ) k L p ≤ C (1 + t ) − (cid:16) − p (cid:17) − k , where k = 0 , . Furthermore, in the same manner, we also have k ( ρ − t ) k L ∞ + k u ( t ) k L ∞ ≤ C (1 + t ) − , k B ( t ) k L ∞ ≤ C (1 + t ) − . Second, we build time decay rates for the time derivatives of global strong solutions.
Theorem 1.2.
Under all the assumptions in Theorem 1.1, the global strong solution ( ρ, u, B ) ofCauchy problem (1.1) - (1.3) has the decay rates k ρ t ( t ) k H + k u t ( t ) k L ≤ C (1 + t ) − , k B t ( t ) k L ≤ C (1 + t ) − for all t ≥ . Furthermore, we establish optimal decay rates for the higher order spatial derivatives of classicalsolutions to the compressible Hall-MHD equations.
Theorem 1.3.
Assume that the initial data ( ρ − , u , B ) ∈ H ∩ L and there exists a smallconstant ε > such that k ( ρ − , u , B ) k H ≤ ε , (1.8) then the global classical solution ( ρ, u, B ) of the problem (1.1) - (1.3) has the time decay rates k∇ k ( ρ − t ) k H − k + k∇ k u ( t ) k H − k ≤ C (1 + t ) − k , k∇ m B ( t ) k H − m ≤ C (1 + t ) − m , (1.9) where k = 0 , , , and m = 0 , , , . lobal Existence and Optimal Decay Rates of Solutions for Compressible Hall-MHD Equations Remark 1.2.
Compared with the decay rates of linearized systems of (1.1) stated in Proposition2.5, (1.9) gives optimal decay rates of the solutions and its spatial derivatives (except for the thirdorder spatial derivatives of density and velocity) in L -norm to the nonlinear problem (1.1) - (1.3) .Here the decay rate of solutions to nonlinear system is optimal in the sense that it coincides withthe rate of solutions to the linearized systems. Remark 1.3.
By virtue of the Sobolev inequality and the results (1.9) in Theorem 1.3, then theglobal classical solution ( ρ, u, B ) has the time decay rates k ( ρ − t ) k L p + k u ( t ) k L p ≤ C (1 + t ) − (cid:16) − p (cid:17) , k∇ k B ( t ) k L p ≤ C (1 + t ) − (cid:16) − p (cid:17) − k , where k = 0 , , and p ∈ [2 , ∞ ] . Hence, the decay rate of classical solution ( ρ, u, B ) converging tothe equilibrium state (1 , , in L ∞ -norm is (1 + t ) − . Remark 1.4.
It is easy to see that (1.9) provides faster time decay rates for higher order spatialderivatives of global classical solutions than (1.4) . Hence, the results in Theorem 1.3 improve thework of Fan et al. [16].
Remark 1.5.
Although we only established the time decay rates under the H -framework in The-orem 1.3, the method here can be applied to the H N ( N ≥ -framework just following the idea asGao et al.[22]. Hence, if ( ρ − , u , B ) ∈ H N ∩ L ( N ≥ , then the global solution ( ρ, u, B ) hasthe time decay rates k∇ k ( ρ − t ) k H N − k + k∇ k u ( t ) k H N − k ≤ C (1 + t ) − k , k∇ m B ( t ) k H N − m ≤ C (1 + t ) − m , where k = 0 , , ..., N − , and m = 0 , , , ..., N. Finally, we build decay rates for the mixed space-time derivatives of global classical solutions.
Theorem 1.4.
Under all the assumptions in Theorem 1.3, the global classical solution ( ρ, u, B ) ofthe problem (1.1) - (1.3) satisfies the time decay rates k∇ k ρ t ( t ) k H − k + k∇ k u t ( t ) k L ≤ C (1 + t ) k , k∇ k B t ( t ) k L ≤ C (1 + t ) − k , where k = 0 , . This paper is organized as follows. In section 2, we establish some energy estimates that willplay an essential role for us to construct the global existence of strong solutions. Then, we close theestimates by the standard continuity argument and the global existence of strong solutions followsimmediately. Furthermore, we build the time decay rates by taking the method of Green functionand establish optimal time decay rates for the second order spatial derivatives of magnetic field.Finally, we also study decay rates for the time derivatives of density, velocity and magnetic field.In section 3, we establish the optimal decay rates for the higher order spatial derivatives of globalclassical solutions and mixed space-time derivatives of solutions. .C.Gao, Z.A.Yao In this section, we will establish global existence and optimal time decay rates of strong solutionsfor the compressible Hall-MHD equations. Indeed, computing directly, it is easy to deduce(curl B ) × B = ( B · ∇ ) B − ∇ ( | B | ) , and curl( u × B ) = u (div B ) − ( u · ∇ ) B + ( B · ∇ ) u − B (div u ) . Then, denoting ̺ = ρ −
1, we rewrite (1.1) in the perturbation form as ̺ t + div u = S ,u t − µ ∆ u − ( µ + ν ) ∇ div u + ∇ ̺ = S ,B t − ∆ B = S , div B = 0 , (2.1)where the function S i ( i = 1 , ,
3) is defined as S = − ̺ div u − u · ∇ ̺,S = − u ·∇ u − h ( ̺ )[ µ ∆ u + ( µ + ν ) ∇ div u ] − f ( ̺ ) ∇ ̺ + g ( ̺ ) (cid:20) B · ∇ B − ∇ ( | B | ) (cid:21) ,S = − u · ∇ B + B · ∇ u − B div u − curl (cid:20) g ( ̺ ) (cid:18) B · ∇ B − ∇ ( | B | ) (cid:19)(cid:21) . (2.2)Here the nonlinear function of ̺ is defined by h ( ̺ ) = ̺̺ + 1 , f ( ̺ ) = P ′ ( ̺ + 1) ̺ + 1 − , g ( ̺ ) = 1 ̺ + 1 . (2.3)The initial data are given as( ̺, u, B )( x, t ) | t =0 = ( ̺ , u , B )( x ) → (0 , ,
0) as | x | → ∞ . (2.4) First of all, suppose there exists a small positive constant δ satisfying following estimate k ( ̺, u, B )( t ) k H := k ̺ ( t ) k H + k u ( t ) k H + k B ( t ) k H ≤ δ, (2.5)which, together with Sobolev inequality, yields directly12 ≤ ̺ + 1 ≤ . Hence, we immediately have | f ( ̺ ) | , | h ( ̺ ) | ≤ C | ̺ | and | g ( k − ( ̺ ) | , | h ( k ) ( ̺ ) | , | f ( k ) ( ̺ ) | ≤ C for any k ≥ , (2.6)which will be used frequently to derive a priori estimates.We state the classical Sobolev interpolation of the Gagliardo-Nirenberg inequality, refer to [28]. lobal Existence and Optimal Decay Rates of Solutions for Compressible Hall-MHD Equations Lemma 2.1.
Let ≤ m, α ≤ l and the function f ∈ C ∞ ( R ) , then we have k∇ α f k L p . k∇ m f k − θL k∇ l f k θL , (2.7) where ≤ θ ≤ and α satisfy p − α (cid:18) − m (cid:19) (1 − θ ) + (cid:18) − l (cid:19) θ. First of all, we will derive following energy estimates.
Lemma 2.2.
Under the condition (2.5) , then for k = 0 , , we have ddt k∇ k ( ̺, u, B ) k L + C k∇ k +1 ( u, B ) k L . δ k∇ k +1 ̺ k L . (2.8) Proof.
Taking k -th spatial derivatives to (2.1) and (2.1) respectively, multiplying the resultingidentities by ∇ k ̺ and ∇ k u respectively and integrating over R (by parts), it is easy to obtain12 ddt Z ( |∇ k ̺ | + |∇ k u | ) dx + Z ( µ |∇ k +1 u | + ( µ + ν ) |∇ k div u | ) dx = Z ∇ k S · ∇ k ̺ dx + Z ∇ k S · ∇ k u dx. (2.9)Taking k -th spatial derivatives to (2.1) , multiplying the resulting identity by ∇ k B and integratingover R (by parts), we have12 ddt Z |∇ k B | dx + Z |∇ k +1 B | dx = Z ∇ k S · ∇ k B dx. (2.10)Adding (2.9) to (2.10), it follows immediately12 ddt Z ( |∇ k ̺ | + |∇ k u | + |∇ k B | ) dx + Z ( µ |∇ k +1 u | +( µ + ν ) |∇ k div u | + |∇ k +1 B | ) dx = Z ∇ k S · ∇ k ̺ dx + Z ∇ k S · ∇ k u dx + Z ∇ k S · ∇ k B dx. (2.11)For the case k = 0, then the differential identity (2.11) has the following form12 ddt Z ( | ̺ | + | u | + | B | ) dx + Z ( µ |∇ u | + ( µ + ν ) | div u | + |∇ B | ) dx = Z S · ̺ dx + Z S · u dx + Z S · B dx = I + I + I . (2.12)Applying (2.5), Holder, Sobolev and Young inequalities, it is easy to obtain I ≤ k ̺ k L k div u k L k ̺ k L + k ̺ k L k∇ ̺ k L k u k L . k ̺ k H k∇ u k L k∇ ̺ k L + k ̺ k H k∇ ̺ k L k∇ u k L . δ ( k∇ ̺ k L + k∇ u k L ) . (2.13) .C.Gao, Z.A.Yao Integrating by parts and applying (2.6), Holder, Sobolev and Young inequalities, it arrives at directly − Z h ( ̺ )( µ ∆ u + ( µ + ν ) ∇ div u ) udx ≈ Z ( h ′ ( ̺ ) ∇ ̺ · u + h ( ̺ ) ∇ u ) ∇ udx . k∇ ̺ k L k u k L k∇ u k L + k ̺ k L ∞ k∇ u k L . ( k ̺ k H + k∇ u k H )( k∇ ̺ k L + k∇ u k L ) . δ ( k∇ ̺ k L + k∇ u k L ) . (2.14)Hence, with the help of (2.6), (2.14), Holder, Sobolev and Young inequalities, we deduce I . ( k u k L k∇ u k L + k ̺ k L k∇ ̺ k L + k g ( ̺ ) k L ∞ k B k L k∇ B k L ) k u k L + δ ( k∇ ̺ k L + k∇ u k L ) . ( k u k H k∇ u k L + k ̺ k H k∇ ̺ k L + k B k H k∇ B k L ) k∇ u k L + δ ( k∇ ̺ k L + k∇ u k L ) . δ ( k∇ ̺ k L + k∇ u k L + k∇ B k L ) . (2.15)Integrating by part and applying (2.6), Holder and Sobolev inequalities, it arrives at − Z curl [ g ( ̺ )( B · ∇ B )] Bdx = − Z g ( ̺ )( B · ∇ B )curl Bdx . k g ( ̺ ) k L ∞ k B k L ∞ k∇ B k L k curl B k L . k B k H k∇ B k L . (2.16)Hence, with the help of (2.16), Holder, Sobolev and Young inequalities, we deduce I . ( k u k L k∇ B k L + k B k L k∇ u k L ) k B k L + k B k H k∇ B k L . ( k u k H + k B k H )( k∇ u k L + k∇ B k L ) + δ k∇ B k L . δ ( k∇ u k L + k∇ B k L ) . (2.17)Substituting (2.13), (2.15) and (2.17) into (2.12) and applying the smallness of δ , it arrives atdirectly ddt Z ( | ̺ | + | u | + | B | ) dx + Z ( µ |∇ u | + |∇ B | ) dx . δ k∇ ̺ k L . (2.18)For the case k = 1, then the differential identity (2.11) has the following form12 ddt Z ( |∇ ̺ | + |∇ u | + |∇ B | ) dx + Z ( µ |∇ u | + ( µ + ν ) |∇ div u | + |∇ B | ) dx = Z ∇ S · ∇ ̺ dx + Z ∇ S · ∇ u dx + Z ∇ S · ∇ B dx = II + II + II . (2.19)Applying Holder, Sobolev and Young inequalities, we obtain II ≤ ( k ̺ k L k div u k L + k u k L k∇ ̺ k L ) k∇ ̺ k L . ( k ̺ k H + k u k H )( k∇ ̺ k L + k∇ u k L ) . δ ( k∇ ̺ k L + k∇ u k L ) . (2.20) lobal Existence and Optimal Decay Rates of Solutions for Compressible Hall-MHD Equations Similarly, it is easy to deduce II ≤ ( k u k L k∇ u k L + k h ( ̺ ) k L ∞ k∇ u k L ) k∇ u k L + ( k f ( ̺ ) k L k∇ ̺ k L + k g ( ̺ ) k L ∞ k B k L k∇ B k L ) k∇ u k L . ( k ̺ k H + k u k H + k B k H )( k∇ ̺ k L + k∇ u k L + k∇ B k L ) . δ ( k∇ ̺ k L + k∇ u k L + k∇ B k L ) . (2.21)Integrating by part and applying (2.6), Holder and Sobolev inequalities, we obtain − Z ∇ curl[ g ( ̺ )( B · ∇ B )] ∇ B dx = − Z ∇ [ g ( ̺ )( B · ∇ B )] ∇ curl B dx . ( k∇ g ( ̺ ) k L k B k L k∇ B k L + k g ( ̺ ) k L ∞ k∇ B k L k∇ B k L ) k∇ curl B k L + k g ( ̺ ) k L ∞ k B k L ∞ k∇ B k L k∇ curl B k L . ( k∇ ̺ k L k∇ B k L + k∇ B k L + k B k H ) k∇ B k L . (2.22)Applying (2.22), Holder, Sobolev and Young inequalities, it arrives at directly II . ( k u k L k∇ B k L + k B k L k∇ u k L ) k∇ B k L + ( k∇ ̺ k L k∇ B k L + k∇ B k L + k B k H ) k∇ B k L . δ ( k∇ u k L + k∇ B k L ) . (2.23)Substituting (2.20), (2.21) and (2.23) into (2.19), then we obtain ddt Z ( |∇ ̺ | + |∇ u | + |∇ B | ) dx + Z ( µ |∇ u | + |∇ B | ) dx . δ k∇ ̺ k L , which, together with (2.18), completes the proof of the lemma.Next, we derive the second type of energy estimates involving the higher-order spatial derivativesof ̺ and u . Lemma 2.3.
Under the condition (2.5) , then we have ddt k∇ ( ̺, u, B ) k L + C k∇ ( u, B ) k L . δ k∇ ̺ k L . (2.24) Proof.
Taking k = 2 specially in (2.11), we deduce immediately12 ddt Z ( |∇ ̺ | + |∇ u | + |∇ B | ) dx + Z ( µ |∇ u | +( µ + ν ) |∇ div u | + |∇ B | ) dx = Z ∇ S · ∇ ̺ dx + Z ∇ S · ∇ u dx + Z ∇ S · ∇ B dx. (2.25) .C.Gao, Z.A.Yao Applying Holder, Sobolev and Young inequalities, it is easy to obtain − Z ∇ ( ̺ div u ) ∇ ̺dx = − Z ( ∇ ̺ div u + 2 ∇ ̺ ∇ div u + ̺ ∇ div u ) ∇ ̺dx . ( k∇ u k L ∞ k∇ ̺ k L + k∇ ̺ k L k∇ u k L + k ̺ k L ∞ k∇ u k L ) k∇ ̺ k L . ( k∇ u k L k∇ u k L k∇ ̺ k L + k∇ ̺ k H k∇ u k L + k ̺ k H k∇ u k L ) k∇ ̺ k L . ( k∇ u k L k∇ ̺ k L + k∇ ̺ k H + k ̺ k H )( k∇ ̺ k L + k∇ u k L ) . δ ( k∇ ̺ k L + k∇ u k L ) . (2.26)Integrating by part and applying Holder, Sobolev and Young inequalities, it arrives at − Z ∇ ( u · ∇ ̺ ) ∇ ̺ dx = Z (cid:20) − ( ∇ u ∇ ̺ + 2 ∇ u ∇ ̺ ) ∇ ̺ + 12 |∇ ̺ | div u (cid:21) dx . ( k∇ u k L k∇ ̺ k L + k∇ u k L ∞ k∇ ̺ k L ) k∇ ̺ k L . k∇ ̺ k H k∇ ̺ k L k∇ u k L + k∇ u k L k∇ u k L k∇ ̺ k L . ( k∇ u k L k∇ ̺ k L + k∇ ̺ k H )( k∇ ̺ k L + k∇ u k L ) . δ ( k∇ ̺ k L + k∇ u k L ) . (2.27)The combination of (2.26) and (2.27) gives rise to Z ∇ S · ∇ ̺ dx . δ ( k∇ ̺ k L + k∇ u k L ) . (2.28)Now, we give the estimate for the second term on the right hand side of (2.25). By virtue of Holderand Sobolev inequalities, we have Z ∇ ( u · ∇ u ) ∇ ∆ udx = Z ( ∇ u ∇ u + u ∇ u ) ∇ ∆ udx ≤ k∇ u k L k∇ u k L k∇ u k L + k u k L k∇ u k L k∇ u k L . k u k L k∇ u k L k∇ u k L k∇ u k L k∇ u k L + k u k H k∇ u k L . δ k∇ u k L . (2.29)In view of (2.6), Holder and Sobolev inequalities, we have Z ∇ ( h ( ̺ )( µ ∆ u + ( µ + ν ) ∇ div u )) ∇ ∆ udx . ( k∇ h ( ̺ ) k L k∇ u k L + k h ( ̺ ) k L ∞ k∇ u k L ) k∇ u k L . ( k∇ ̺ k H k∇ u k L + k ̺ k H k∇ u k L ) k∇ u k L . δ k∇ u k L , (2.30) lobal Existence and Optimal Decay Rates of Solutions for Compressible Hall-MHD Equations and Z ∇ ( f ( ̺ ) ∇ ̺ ) ∇ ∆ udx . ( k∇ ̺ k L + k f ( ̺ ) k L ∞ k∇ ̺ k L ) k∇ u k L . ( k∇ ̺ k L k∇ ̺ k L + k ̺ k H k∇ ̺ k L ) k∇ u k L . ( k∇ ̺ k L + k ̺ k H ) k∇ ̺ k L k∇ u k L . δ ( k∇ ̺ k L + k∇ u k L ) . (2.31)Similarly, it is easy to deduce Z ∇ (cid:20) g ( ̺ )( B · ∇ B − ∇ ( 12 | B | )) (cid:21) ∇ ∆ udx . ( k∇ g ( ̺ ) k L k B k L k∇ B k L + k g ( ̺ ) k L ∞ k∇ B k L k∇ B k L ) k∇ u k L + k g ( ̺ ) k L ∞ k B k L k∇ B k L k∇ u k L . ( k∇ ̺ k L k∇ B k L k∇ B k L + k B k L k∇ B k L k∇ B k L k∇ B k L ) k∇ u k L + k B k H k∇ B k L k∇ u k L . δ ( k∇ ̺ k L + k∇ u k L + k∇ B k L ) . (2.32)By virtue of the estimates (2.29)-(2.32), we obtain immediately Z ∇ S · ∇ u dx . δ ( k∇ ̺ k L + k∇ u k L + k∇ B k L ) . (2.33)Integrating by part and applying Holder and Sobolev inequalities, it arrives at Z ∇ ( u · ∇ B ) ∇ B dx . ( k∇ u k L k∇ B k L + k u k L k∇ B k L ) k∇ B k L . ( k u k L k∇ u k L k∇ B k L k∇ B k L + k u k H k∇ B k L ) k∇ B k L . δ ( k∇ u k L + k∇ B k L ) . (2.34)Similarly, it is easy to obtain Z ∇ ( B · ∇ u + B div u ) ∇ B dx . ( k∇ u k L k∇ B k L + k B k L k∇ u k L ) k∇ B k L . ( k u k L k∇ u k L k∇ B k L k∇ B k L + k B k H k∇ u k L ) k∇ B k L . δ ( k∇ u k L + k∇ B k L ) . (2.35) .C.Gao, Z.A.Yao Integrating by part and applying (2.6), Holder and Sobolev inequalities, we obtain − Z ∇ curl (cid:20) g ( ̺ ) (cid:18) B · ∇ B − ∇ ( 12 | B | ) (cid:19)(cid:21) ∇ B dx = − Z ∇ (cid:20) g ( ̺ ) (cid:18) B · ∇ B − ∇ ( 12 | B | ) (cid:19)(cid:21) ∇ curl B dx . ( k∇ g ( ̺ ) k L k B k L ∞ k∇ B k L ∞ + k∇ g ( ̺ ) k L k∇ B k L k∇ B k L ) k∇ curl B k L + ( k∇ g ( ̺ ) k L k B k L k∇ B k L + k g ( ̺ ) k L ∞ k∇ B k L k∇ B k L ) k∇ curl B k L + k g ( ̺ ) k L ∞ k B k L ∞ k∇ B k L k∇ curl B k L . ( k|∇ ̺ | + |∇ ̺ |k L k B k L ∞ k∇ B k L ∞ + k∇ ̺ k L k∇ B k L k∇ B k L ) k∇ B k L + ( k∇ ̺ k L k B k L k∇ B k L + k∇ B k L k∇ B k L + k B k L ∞ k∇ B k L ) k∇ B k L . ( k∇ ̺ k L k∇ ̺ k L + k∇ ̺ k L ) k B k H k∇ B k L k∇ B k L + k∇ B k H k∇ ̺ k L k∇ B k L + k∇ B k H k∇ B k L . δ ( k∇ ̺ k L + k∇ B k L ) . (2.36)In view of the estimates (2.34)-(2.36), we obtain directly Z ∇ S · ∇ B dx . δ ( k∇ ̺ k L + k∇ u k L + k∇ B k L ) . (2.37)Substituting (2.28), (2.33) and (2.37) into (2.25), then we have ddt Z ( |∇ ̺ | + |∇ u | + |∇ B | ) dx + Z ( µ |∇ u | + |∇ B | ) dx . δ k∇ ̺ k L , which completes the proof of the lemma.Finally, we will use the equations (2.1) to recover the dissipation estimate for ̺ . Lemma 2.4.
Under the condition (2.5) , then for k = 0 , , we have ddt Z ∇ k u · ∇ k +1 ̺dx + C k∇ k +1 ̺ k L . k∇ k +1 u k L + k∇ k +2 u k L + k∇ k +2 B k L . (2.38) Proof.
Taking k -th spatial derivatives to the second equation of (2.1), multiplying by ∇ k +1 ̺ andintegrating over R , then we obtain Z ∇ k u t · ∇ k +1 ̺dx + Z |∇ k +1 ̺ | dx = Z ∇ k [ µ ∆ u + ( µ + ν ) ∇ div u ] ∇ k +1 ̺dx + Z ∇ k S · ∇ k +1 ̺dx. (2.39)In order to deal with R ∇ k u t ·∇ k +1 ̺dx , we turn the time derivatives of velocity to the density. Then, lobal Existence and Optimal Decay Rates of Solutions for Compressible Hall-MHD Equations applying the mass equation (2.1) , we can transform time derivatives to the spatial derivatives, i.e. Z ∇ k u t · ∇ k +1 ̺dx = ddt Z ∇ k u · ∇ k +1 ̺dx − Z ∇ k u · ∇ k +1 ̺ t dx = ddt Z ∇ k u · ∇ k +1 ̺dx + Z ∇ k u · ∇ k +1 (div u + div( ̺u )) dx = ddt Z ∇ k u · ∇ k +1 ̺dx − Z ∇ k div u · ∇ k (div u + div( ̺u )) dx = ddt Z ∇ k u · ∇ k +1 ̺dx − Z |∇ k div u | dx − Z ∇ k div u · ∇ k div( ̺u ) dx. (2.40)Substituting (2.40) into (2.39), it is easy to deduce ddt Z ∇ k u · ∇ k +1 ̺dx + Z |∇ k +1 ̺ | dx = Z |∇ k div u | dx + Z ∇ k div u · ∇ k div( ̺u ) dx + Z ∇ k S · ∇ k +1 ̺dx + Z ∇ k [ µ ∆ u + ( µ + ν ) ∇ div u ] ∇ k +1 ̺dx. (2.41)For the case k = 0 , then applying Holder, Sobolev and Young inequalities, we obtain Z div u · div( ̺u ) dx . k ̺ k L ∞ k∇ u k L + k u k L k div u k L k∇ ̺ k L . ( k ̺ k H + k u k H )( k∇ ̺ k L + k∇ u k L ) . δ ( k∇ ̺ k L + k∇ u k L ) . (2.42)By virtue of (2.6) and Holder inequality, it is easy to deduce Z S · ∇ ̺dx . ( k u k L k∇ u k L + k ̺ k L ∞ k∇ u k L ) k∇ ̺ k L + ( k ̺ k L ∞ k∇ ̺ k L + k g ( ̺ ) k L ∞ k B k L k∇ B k L ) k∇ ̺ k L . δ ( k∇ ̺ k L + k∇ u k L + k∇ B k L ) , (2.43)and Z [ µ ∆ u + ( µ + ν ) ∇ div u ] ∇ ̺dx . k∇ u k L + ε k∇ ̺ k L . (2.44)The combination of (2.42), (2.43) and (2.44) helps us complete the proof to (2.38) for the case of k = 0. As for the case k = 1 , applying Holder, Sobolev and Young inequalities, we deduce Z ∇ div u · ∇ div( ̺u ) dx . ( k∇ ̺ k L k div u k L + k ̺ k L ∞ k∇ div u k L ) k∇ u k L + ( k∇ ̺ k L k∇ u k L + k u k L ∞ k∇ ̺ k L ) k∇ u k L . δ ( k∇ ̺ k L + k∇ u k L ) . (2.45)With the help of Holder inequality and Lemma 2.3, it arrives at Z ∇ S · ∇ ̺dx . δ ( k∇ ̺ k L + k∇ u k L + k∇ B k L ) , (2.46) .C.Gao, Z.A.Yao and Z ∇ [ µ ∆ u + ( µ + ν ) ∇ div u ] ∇ ̺dx . k∇ u k L + ε k∇ ̺ k L . (2.47)The combination of (2.45), (2.46) and (2.47) gives rise to the proof of (2.38) for the case of k = 1. In this subsection, we shall combine the energy estimates that we have derived in the previoussection to prove the global existence of strong solutions. Summing up (2.8) from k = l ( l = 0 ,
1) to k = 1, then we obtain ddt k∇ l ( ̺, u, B ) k H − l + C k∇ l ( ∇ u, ∇ B ) k H − l . δ k∇ l +1 ̺ k H − l , which, together with (2.24), then it arrives at ddt k∇ l ( ̺, u, B ) k H − l + C k∇ l +1 ( u, B ) k H − l ≤ δC k∇ l +1 ̺ k H − l . (2.48)On the other hand, summing (2.38) from k = l ( l = 0 ,
1) to k = 1, we obtain immediately ddt X l ≤ k ≤ Z ∇ k u · ∇ k +1 ̺dx + C k∇ l +1 ̺ k H − l ≤ C (cid:16) k∇ l +1 u k H − l + k∇ l +2 B k H − l (cid:17) . (2.49)Multiplying (2.49) by 2 δC /C and adding the resulting inequality to (2.48), then it arrives at ddt E l ( t ) + C (cid:16) k∇ l +1 ̺ k H − l + k∇ l +1 ( u, B ) k H − l (cid:17) ≤ . (2.50)where E l ( t ) is defined as E l ( t ) = k∇ l ( ̺, u, B ) k H − l + 2 δC C X l ≤ k ≤ Z ∇ k u · ∇ k +1 ̺dx. By virtue of the smallness of δ , it is easy to obtain C − k∇ l ( ̺, u, B ) k H − l ≤ E l ( t ) ≤ C k∇ l ( ̺, u, B ) k H − l . (2.51)Choosing l = 0 in (2.50), integrating over [0 , t ] and applying the equivalent relation (2.51), we obtain k ( ̺, u, B )( t ) k H ≤ C k ( ̺ , u , B ) k H . Then, by the standard continuity argument (see Theorem 7.1 on page 100 in [29]), we close theestimate (2.5). Thus, we extend the local strong solutions to be global one and the uniqueness ofglobal strong solutions is guaranteed by the uniqueness of local solutions that has been prove by Fanet al. [16]. Therefore, choosing l = 0 in (2.50), integrating over [0 , t ] and applying the equivalentrelation (2.51), we obtain k ( ̺, u, B )( t ) k H + Z t ( k∇ ̺ ( τ ) k H + k ( ∇ u, ∇ B )( τ ) k H ) dτ ≤ C k ( ̺ , u , B ) k H , (2.52)which completes the proof of the global existence of strong solutions. lobal Existence and Optimal Decay Rates of Solutions for Compressible Hall-MHD Equations In this section, we will establish optimal decay rates for the compressible Hall-MHD equations(1.1)-(1.3). If the initial perturbation belongs to L additionally, we apply the method of Greenfunction to establish optimal time decay rates for the global strong solutions. Furthermore, theapplication of Fourier splitting method by Schonbek [21] helps us to build optimal time decay ratesfor the second order spatial derivatives of magnetic field.First of all, let us to consider the following linearized systems ̺ t + div u = 0 ,u t − µ ∆ u − ( µ + ν ) ∇ div u + ∇ ̺ = 0 ,B t − ∆ B = 0 , (2.53)with the initial data( ̺, u, B )( x, t ) | t =0 = ( ̺ , u , B )( x ) → (0 , ,
0) as | x | → ∞ . (2.54)Obviously, the solution ( ̺, u, B ) of the linearized problem (2.53)-(2.54) can be expressed as( ̺, u, B ) tr = G ( t ) ∗ ( ̺ , u , B ) tr , t ≥ . (2.55)Here G ( t ) := G ( x, t ) is the Green matrix for the systems (2.53) and the exact expression of theFourier transform ˆ G ( ξ, t ) of Green function G ( x, t ) asˆ G ( ξ, t ) = λ + e λ − t − λ − e λ + t λ + − λ − − iξ t ( e λ + t − e λ − t ) λ + − λ − − iξ ( e λ + t − e λ − t ) λ + − λ − λ + e λ + t − λ − e λ − t λ + − λ − ξξ t | ξ | + e λ t (cid:16) I × − ξξ t | ξ | (cid:17)
00 0 e λ t I × where λ = − µ | ξ | , λ = −| ξ | ,λ + = − (cid:18) µ + 12 ν (cid:19) | ξ | + i s | ξ | − (cid:18) µ + 12 ν (cid:19) | ξ | ,λ − = − (cid:18) µ + 12 ν (cid:19) | ξ | − i s | ξ | − (cid:18) µ + 12 ν (cid:19) | ξ | . Since the systems (2.57) is an independent coupling of the classical linearized Navier-Stokes equa-tions and heat equation, the representation of Green function ˆ G ( ξ, t ) is easy to verify. Furthermore,we have the following decay rates for the systems (2.53)-(2.54), refer to [30]. Proposition 2.5.
Assume that ( ̺, u, B ) is the solution of the linearized systems (2.53) - (2.54) withthe initial data ( ̺ , u , B ) ∈ L ∩ H , then k∇ k ̺ k L ≤ C (cid:16) k ( ̺ , u ) k L + k∇ k ( ̺ , u ) k L (cid:17) (1 + t ) − − k , k∇ k u k L ≤ C (cid:16) k ( ̺ , u ) k L + k∇ k ( ̺ , u ) k L (cid:17) (1 + t ) − − k , k∇ k B k L ≤ C (cid:16) k B k L + k∇ k B k L (cid:17) (1 + t ) − − k for ≤ k ≤ . .C.Gao, Z.A.Yao In the sequel, we want to verify some estimates that play an important role for us to derivedecay rates for the compressible Hall-MHD equations (2.1)-(2.4). k ( S , S , S ) k L . δ ( k∇ ̺ k L + k∇ u k H + k∇ B k H ) , k ( S , S , S ) k L . δ ( k∇ ̺ k L + k∇ u k H + k∇ B k H ) , k∇ ( S , S , S ) k L . δ ( k∇ ̺ k L + k∇ u k L + k∇ B k L ) + k∇ ( ̺, B ) k H k∇ ( u, B ) k H . (2.56)Now, we establish the decay rates for the compressible Hall-MHD equations (2.1)-(2.4). Lemma 2.6.
Under the assumptions of Theorem 1.1, the global strong solution ( ̺, u, B ) of problem (2.1) - (2.4) has the time decay rates k∇ k ̺ ( t ) k H − k + k∇ k u ( t ) k H − k + k∇ k B ( t ) k H − k ≤ C (1 + t ) − − k (2.57) for k = 0 , .Proof. Taking l = 1 specially in (2.50), it arrives at directly ddt E ( t ) + C (cid:0) k∇ ̺ k L + k∇ u k H + k∇ B k H (cid:1) ≤ , (2.58)where E ( t ) is defined as E ( t ) = k∇ ̺ k H + k∇ u k H + k∇ B k H + 2 C δC Z ∇ u · ∇ ̺dx. With the help of Young inequality and smallness of δ , it is easy to deduce C − k∇ ( ̺, u, B ) k H ≤ E ( t ) ≤ C k∇ ( ̺, u, B ) k H . (2.59)Adding both hand sides of (2.58) by k∇ ( ̺, u, B ) k L and applying the equivalent relation (2.59),then we have ddt E ( t ) + C E ( t ) ≤ k∇ ( ̺, u, B ) k L . In view of the Gronwall inequality, it follows immediately E ( t ) ≤ E (0) e − Ct + Z t e − C ( t − τ ) k∇ ( ̺, u, B )( τ ) k L dτ. (2.60)In order to derive the time decay rate for E ( t ), we need to control the term k∇ ( ̺, u, B ) k L . Infact, by Duhamel principle, we can represent the solutions for the problem (2.1)-(2.4) as( ̺, u, B ) tr ( t ) = G ( t ) ∗ ( ̺ , u , B ) tr + Z t G ( t − s ) ∗ ( S , S , S ) tr ( s ) ds. (2.61)Denoting E ( t ) = sup ≤ τ ≤ t (1 + τ ) ( k∇ ̺ ( τ ) k H + k∇ u ( τ ) k H + k∇ B ( τ ) k H ) , lobal Existence and Optimal Decay Rates of Solutions for Compressible Hall-MHD Equations which, together with (2.61), (2.56) and Proposition 2.5, gives directly k∇ ( ̺, u, B )( t ) k L ≤ C (1 + t ) − + C Z t ( k ( S , S , S )( τ ) k L + k∇ ( S , S , S )( τ ) k L ) (1 + t − τ ) − dτ ≤ C (1 + t ) − + C Z t δ ( k∇ ̺ ( τ ) k H + k∇ u ( τ ) k H + k∇ B ( τ ) k H ) (1 + t − τ ) − dτ + C Z t k∇ ( ̺, B )( τ ) k H k∇ ( u, B )( τ ) k H (1 + t − τ ) − dτ ≤ C (1 + t ) − + Cδ p E ( t ) Z t (1 + t − τ ) − (1 + τ ) − dτ + C p E ( t ) (cid:20)Z (1 + t − τ ) − (1 + τ ) − dτ (cid:21) (cid:20)Z t k∇ ( u, B )( τ ) k H dτ (cid:21) ≤ C (1 + t ) − + Cδ p E ( t )(1 + t ) − ≤ (1 + t ) − (1 + δ p E ( t )) , where we have used the fact Z t (1 + t − τ ) − r (1 + τ ) − r dτ = Z t + Z t t (1 + t − τ ) − r (1 + τ ) − r dτ ≤ (cid:18) t (cid:19) − r Z t (1 + τ ) − r dτ + (cid:18) t (cid:19) − r Z t t (1 + t − τ ) − r dτ ≤ (1 + t ) − r , for r = and r = respectively. Thus, we have the estimate k∇ ( ̺, u, B )( t ) k L ≤ C (1 + t ) − (1 + δE ( t )) . (2.62)Inserting (2.62) into (2.60), it follows immediately E ( t ) ≤ E (0) e − Ct + C Z t e − C ( t − τ ) (1 + τ ) − (1 + δE ( τ )) dτ ≤ E (0) e − Ct + C (1 + δE ( t )) Z t e − C ( t − τ ) (1 + τ ) − dτ ≤ E (0) e − Ct + C (1 + δE ( t ))(1 + t ) − ≤ C (1 + δE ( t ))(1 + t ) − , (2.63)where we have used the fact Z t e − C ( t − τ ) (1 + τ ) − dτ = Z t + Z t t e − C ( t − τ ) (1 + τ ) − dτ ≤ e − c t Z t (1 + τ ) − dτ + (cid:18) t (cid:19) − Z t t e − C ( t − τ ) dτ ≤ C (1 + t ) − . .C.Gao, Z.A.Yao Hence, by virtue of the definition of E ( t ) and (2.63), it follows immediately E ( t ) ≤ C (1 + δE ( t )) , which, in view of the smallness of δ , gives E ( t ) ≤ C. Therefore, we have the following time decay rates k∇ ̺ ( t ) k H + k∇ u ( t ) k H + k∇ B ( t ) k H ≤ C (1 + t ) − . (2.64)On the other hand, by (2.61), (2.56), (2.64) and Proposition 2.5, it is easy to deduce k ( ̺, u, B ) k L ≤ C (1 + t ) − + C Z t (cid:0) k ( S , S , S ) k L + k ( S , S , S ) k L (cid:1) (1 + t − τ ) − dτ ≤ C (1 + t ) − + C Z t δ (cid:0) k∇ ̺ ( τ ) k L + k∇ u ( τ ) k H + k∇ B ( τ ) k H (cid:1) (1 + t − τ ) − dτ ≤ C (1 + t ) − + C Z t (1 + t − τ ) − (1 + τ ) − dτ ≤ C (1 + t ) − , where we have used the fact Z t (1 + t − τ ) − (1 + τ ) − dτ ≤ C (1 + t ) − . Hence, we have the following decay rate k ̺ ( t ) k L + k u ( t ) k L + k B ( t ) k L ≤ C (1 + t ) − . (2.65)Therefore, the combination of (2.64) and (2.65) completes the proof of the lemma.Finally, we establish optimal decay rates for the second order derivatives of magnetic field. Lemma 2.7.
Under the assumptions of Theorem 1.1, then the magnetic field has the followingdecay rate k∇ B ( t ) k L ≤ C (1 + t ) − . (2.66) Proof.
Taking k = 2 in (2.10), it follows immediately12 ddt Z |∇ B | dx + Z |∇ B | dx = Z ∇ S · ∇ B dx. (2.67)By Holder, Sobolev and Young inequalities, we obtain Z ∇ ( B · ∇ u + B div u ) ∇ B dx . ( k∇ u k L k∇ B k L + k B k L ∞ k∇ u k L ) k∇ B k L . ( k∇ u k H k∇ B k L + k∇ B k H k∇ u k L ) k∇ B k L . k∇ ( u, B ) k H k∇ ( u, B ) k L + δ k∇ B k L . (2.68) lobal Existence and Optimal Decay Rates of Solutions for Compressible Hall-MHD Equations It follows from (2.34) and (2.36) that Z ∇ ( u · ∇ B ) ∇ B dx . k∇ u k H k∇ B k L + δ k∇ B k L , (2.69)and − Z ∇ curl (cid:20) g ( ̺ ) (cid:18) B · ∇ B − ∇ ( 12 | B | ) (cid:19)(cid:21) ∇ B dx . k∇ ̺ k L k∇ B k H + δ k∇ B k L . (2.70)Substituting (2.68)-(2.70) into (2.67) and applying the time decay rates (2.57), then we obtain ddt Z |∇ B | dx + Z |∇ B | dx . k∇ ( u, B ) k H k∇ ( u, B ) k L + k∇ ̺ k L k∇ B k H . (1 + t ) − (1 + t ) − + (1 + t ) − (1 + t ) − . (1 + t ) − . (2.71)For some constant R defined below, denoting the time sphere(see [21]) S = ( ξ ∈ R (cid:12)(cid:12) | ξ | ≤ (cid:18) R t (cid:19) ) , it follows immediately Z R |∇ B | dx ≥ Z R /S | ξ | | ˆ B | dξ ≥ R t Z R /S | ξ | | ˆ B | dξ ≥ R t Z R | ξ | | ˆ B | dξ − (cid:18) R t (cid:19) Z S | ξ | | ˆ B | dξ, or equivalently Z R |∇ B | dx ≥ R t Z R |∇ B | dx − (cid:18) R t (cid:19) Z R |∇ B | dx. (2.72)The combination of (2.71), (2.72) and (2.57) yields directly ddt Z |∇ B | dx + 41 + t Z |∇ B | dx ≤ t ) Z |∇ B | dx + C (1 + t ) − . (1 + t ) − (1 + t ) − + (1 + t ) − ≤ C (1 + t ) − , (2.73)where we have chosen R = 4 in (2.72). Multiplying (2.73) by (1 + t ) , we obtain ddt (cid:2) (1 + t ) k∇ B k L (cid:3) ≤ C (1 + t ) − . (2.74) .C.Gao, Z.A.Yao Integrating (2.74) over [0 , t ], then we have the following decay rate k∇ B ( t ) k L ≤ C (1 + t ) − . Therefore, we complete the proof of the lemma.
Proof of Theorem 1.1:
With the help of (2.52), Lemma 2.6 and Lemma 2.7, we complete theproof of Theorem 1.1.
In this subsection, we establish the decay rates for the time derivatives of strong solutions.
Lemma 2.8.
Under the assumptions of Theorem 1.1, the global strong solution ( ̺, u, B ) of problem (2.1) - (2.4) satisfies k ̺ t ( t ) k H + k u t ( t ) k L ≤ C (1 + t ) − , k B t ( t ) k L ≤ C (1 + t ) − . (2.75) Proof.
By virtue of the equation (2.1) and decay rates (1.7), we have k ̺ t k L = k div u + ̺ div u + u · ∇ ̺ k L ≤ k div u k L + k ̺ k L ∞ k div u k L + k∇ ̺ k L k u k L ≤ C (1 + t ) − . (2.76)Similarly, it follows immediately k∇ ̺ t k L = k∇ div u + ∇ ̺ div u + ̺ ∇ div u + ∇ u · ∇ ̺ + u · ∇ ̺ k L . k∇ div u k L + k∇ ̺ k L k∇ u k L + k ( ̺, u ) k L ∞ k∇ ( ̺, u ) k L . k∇ u k L + k ( ̺, u ) k H k∇ ( ̺, u ) k L ≤ C (1 + t ) − . (2.77)By virtue of the equation (2.1) , decay rates (1.7) and estimate (2.56), we have k u t k L = k µ ∆ u + ( µ + ν ) ∇ div u − ∇ ̺ + S k L . k∇ u k L + k∇ ̺ k L + δ ( k∇ ̺ k L + k∇ ( u, B ) k H ) . (1 + t ) − + (1 + t ) − ≤ C (1 + t ) − . (2.78)By virtue of (2.1) , (1.7), Holder and Sobolev inequalities, we obtain k B t k L = (cid:13)(cid:13)(cid:13)(cid:13) ∆ B − u · ∇ B + B · ∇ u − B div u − curl (cid:20) g ( ̺ ) (cid:18) B · ∇ B − ∇ ( | B | ) (cid:19)(cid:21)(cid:13)(cid:13)(cid:13)(cid:13) L . k ∆ B k L + k u k L k∇ B k L + k B k L k∇ u k L + k∇ g ( ̺ ) k L k B k L k∇ B k L + k g ( ̺ ) k L ∞ k∇ B k L k∇ B k L + k g ( ̺ ) k L ∞ k B k L ∞ k∇ B k L . k∇ B k L + k ( u, B ) k H k∇ ( u, B ) k L + k∇ ̺ k L k∇ B k H + k∇ B k H k∇ B k L . (1 + t ) − + (1 + t ) − (1 + t ) − + (1 + t ) − (1 + t ) − ≤ C (1 + t ) − . (2.79)In view of the decay rates (2.76)-(2.79), we complete the proof of the lemma. lobal Existence and Optimal Decay Rates of Solutions for Compressible Hall-MHD Equations Proof of Theorem 1.2:
With the help of Lemma 2.8, we complete the proof of Theorem 1.2.
In this section, we first establish optimal time decay rates for the higher order spatial derivativesof global classical solutions under the condition of small initial perturbation in H -norm and finiteinitial perturbation in L -norm. Furthermore, we also study the decay rates for the mixed space-time derivatives of global classical solutions.First of all, Fan et al.(see (3 .
2) on Page 430 in [16]) have established following estimate k ( ̺, u, B )( t ) k H ≤ C k ( ̺ , u , B ) k H ≤ Cε . (3.1)Thus, the inequality (2.6) also holds on under the condition of (1.8). Just following the idea as Lemma 2.6, it is easy to establish optimal decay rates for the globalclassical solutions. For the sake of brevity, we only state the results in the following lemma.
Lemma 3.1.
Under the assumptions of Theorem 1.3, the global classical solution ( ̺, u, B ) of prob-lem (2.1) - (2.4) satisfies for all t ≥ , k∇ k ̺ ( t ) k H − k + k∇ k u ( t ) k H − k + k∇ k B ( t ) k H − k ≤ C (1 + t ) − − k , (3.2) where k = 0 , . Next, we establish optimal time decay rates for the second order spatial derivatives of magneticfield and enhance the time decay rates for the third order spatial derivatives of magnetic field.
Lemma 3.2.
Under the assumptions of Theorem 1.3, then the magnetic field has following decayrate for all t ≥ , k∇ B ( t ) k H ≤ C (1 + t ) − . (3.3) Proof.
Taking k = 3 in (2.10), it follows immediately12 ddt Z |∇ B | dx + Z |∇ B | dx = Z ∇ (cid:20) − u · ∇ B + B · ∇ u − B div u − curl (cid:18) g ( ̺ ) (cid:18) B · ∇ B − ∇ ( | B | ) (cid:19)(cid:19)(cid:21) ∇ Bdx = III + III + III + III . (3.4)By virtue of (3.1), Holder and Sobolev inequalities, it arrives at III = Z ( ∇ u ∇ B + 2 ∇ u ∇ B + u ∇ B ) ∇ Bdx . ( k∇ u k L k∇ B k L + k∇ u k L k∇ B k L + k u k L k∇ B k L ) k∇ B k L . k∇ u k L k∇ B k L + k∇ u k L k∇ B k L + ( ε + ε ) k∇ B k L . k∇ u k H k∇ B k H + ( ε + ε ) k∇ B k L . (3.5) .C.Gao, Z.A.Yao In view of the Sobolev and Young inequalities, we obtain
III = − Z ( ∇ B ∇ u + 2 ∇ B ∇ u + B ∇ u ) ∇ Bdx . ( k∇ B k L k∇ u k L + k∇ B k L k∇ u k L + k B k L ∞ k∇ u k L ) k∇ B k L . ( k∇ B k L k∇ u k H + k∇ B k L k∇ u k H + k∇ B k H k∇ u k L ) k∇ B k L . k∇ B k H k∇ u k H + k∇ B k H k∇ u k L + ε k∇ B k L . (3.6)In the same manner, we get III . k∇ B k H k∇ u k H + k∇ B k H k∇ u k L + ε k∇ B k L . (3.7)Applying (2.6), (3.1), Holder, Sobolev and Young inequalities, it is easy to deduce III . ( k∇ g ( ̺ ) k L k B k L ∞ k∇ B k L ∞ + k∇ g ( ̺ ) k L k∇ B k L k∇ B k L ∞ ) k∇ B k L + ( k∇ g ( ̺ ) k L k B k L ∞ k∇ B k L + k∇ g ( ̺ ) k L k∇ B k L k∇ B k L ) k∇ B k L + ( k∇ g ( ̺ ) k L k B k L k∇ B k L + k g ( ̺ ) k L ∞ k∇ B k L k∇ B k L ) k∇ B k L + k g ( ̺ ) k L ∞ k B k L ∞ k∇ B k L . k∇ B k H k∇ B k H + k∇ B k L k∇ B k H + k∇ B k H k∇ B k L + ( ε + ε ) k∇ B k L . k∇ B k H k∇ B k H + ( ε + ε ) k∇ B k L . (3.8)Substituting (3.5)-(3.8) into (3.4) and applying the smallness of ε and ε , it is easy to deduce ddt Z |∇ B | dx + Z |∇ B | dx . k∇ ( u, B ) k H k∇ ( u, B ) k H , (3.9)which, together with the time decay rates (3.2), yields directly ddt Z |∇ B | dx + Z |∇ B | dx . (1 + t ) − . (3.10)Similar to (2.72), it is easy to deduce Z |∇ B | dx ≥
51 + t Z |∇ B | dx − (cid:18)
51 + t (cid:19) Z |∇ B | dx. (3.11)The combination of (3.2), (3.10) and (3.11) gives directly ddt Z |∇ B | dx + 51 + t Z |∇ B | dx . t ) Z |∇ B | dx + (1 + t ) − . (1 + t ) − (1 + t ) − + (1 + t ) − . (1 + t ) − , which, together with (2.73), yields directly ddt Z ( |∇ B | + |∇ B | ) dx + 41 + t Z ( |∇ B | + |∇ B | ) dx . (1 + t ) − . (3.12) lobal Existence and Optimal Decay Rates of Solutions for Compressible Hall-MHD Equations Multiplying (3.12) by (1 + t ) , it arrives at ddt (cid:2) (1 + t ) k∇ B k H (cid:3) ≤ C (1 + t ) − , which, integrating over [0 , t ], gives k∇ B ( t ) k H ≤ C (1 + t ) − . Therefore, we complete the proof of the lemma.In order to establish optimal decay rate for the third order spatial derivatives of magnetic field,we need to improve the decay rate for the second and third order spatial derivatives of velocity.Indeed, following the idea as the compressible MHD equations(see [23]), it is easy to verify the timedecay rates (3.2) also holds on for k = 2. For the convenience of readers, we also introduce themethod to improve the decay rates for the second order spatial derivatives of density and velocityhere. Lemma 3.3.
Under the assumptions of Theorem 1.3, the global classical solution ( ̺, u, B ) of Cauchyproblem (2.1) - (2.4) has ddt k∇ ( ̺, u ) k H + µ k∇ u k H ≤ C (cid:2) (1 + t ) − + ε k∇ ̺ k L (cid:3) . (3.13) Proof.
Taking k = 2 specially in (2.9), then we obtain12 ddt Z ( |∇ ̺ | + |∇ u | ) dx + Z ( µ |∇ u | + ( µ + ν ) |∇ div u | ) dx = Z ∇ S · ∇ ̺ dx + Z ∇ S · ∇ u dx. (3.14)Integrating by part and applying (3.2), Holder, Sobolev and Young inequalities, we obtain Z ∇ S · ∇ ̺ dx = Z ∇ ( ̺ div u + u · ∇ ̺ ) · ∇ ̺dx . ( k∇ ̺ k L k∇ u k L + k∇ u k L k ̺ k L + k∇ ̺ k L k u k L ) k∇ ̺ k L . (1 + t ) − + ε k∇ ̺ k L . (3.15)From the estimates (2.29) and (2.30), it is obtain the estimates Z ∇ [ − u ·∇ u − h ( ̺ )[ µ ∆ u + ( µ + ν ) ∇ div u ]] ∇ udx . ε k∇ u k L . (3.16)Integrating by part and applying (2.6), (3.2), Holder and Sobolev inequalities, we obtain Z ∇ [ − f ( ̺ ) ∇ ̺ ] ∇ u dx . ( k f ( ̺ ) k L ∞ k∇ ̺ k L + k∇ f ( ̺ ) k L k∇ ̺ k L ) k∇ u k L . ( k ̺ k L ∞ + k∇ ̺ k L ) k∇ ̺ k L k∇ u k L . k∇ ̺ k H k∇ ̺ k L + ε k∇ u k L . (1 + t ) − + ε k∇ u k L . (3.17) .C.Gao, Z.A.Yao In the same manner, it is easy to deduce Z ∇ (cid:20) g ( ̺ ) (cid:18) B · ∇ B − ∇ ( | B | ) (cid:19)(cid:21) ∇ u dx ≈ Z ( ∇ g ( ̺ ) B ∇ B + g ( ̺ ) ∇ B ∇ B + g ( ̺ ) B ∇ B ) ∇ udx . ( k∇ ̺ k L k B k L k∇ B k L + k g ( ̺ ) k L ∞ k∇ B k L k∇ B k L ) k∇ u k L + k g ( ̺ ) k L ∞ k B k L k∇ B k L k∇ u k L . k∇ ̺ k L k∇ B k L k∇ B k L + k∇ B k L k∇ B k L + k∇ B k L k∇ B k L + ε k∇ u k L . (1 + t ) − + ε k∇ u k L . (3.18)In view of the estimates (3.16) − (3.18), it is easy deduce Z ∇ S · ∇ u dx . (1 + t ) − + ε k∇ u k L . (3.19)Substituting (3.15) and (3.19) into (3.14) and applying the smallness of ε and ε , we obtain ddt Z ( |∇ ̺ | + |∇ u | ) dx + µ Z |∇ u | dx . (1 + t ) − + ε k∇ ̺ k L . (3.20)Taking k = 3 in (2.9) specially, then we have12 ddt Z ( |∇ ̺ | + |∇ u | ) dx + Z ( µ |∇ u | + ( µ + ν ) |∇ div u | ) dx = Z ∇ ( − ̺ div u − u · ∇ ̺ ) ∇ ̺dx + Z ∇ [ − u ·∇ u − h ( ̺ )( µ ∆ u + ( µ + ν ) ∇ div u )] ∇ u dx + Z ∇ (cid:20) − f ( ̺ ) ∇ ̺ + g ( ̺ )( B · ∇ B − ∇ ( | B | )) (cid:21) ∇ u dx = IV + IV + IV + IV + IV + IV + IV . (3.21)Applying the Holer and Sobolev inequalities, it is easy to deduce IV . ( k∇ ̺ k L k∇ u k L ∞ + k∇ ̺ k L k∇ u k L ) k∇ ̺ k L + ( k∇ ̺ k L k∇ u k L + k ̺ k L ∞ k∇ u k L ) k∇ ̺ k L . ε ( k∇ ̺ k L + k∇ u k L ) . (3.22)Similarly, it is easy to deduce IV . ε ( k∇ ̺ k L + k∇ u k L ) . (3.23)Integrating by part and applying decay rates (3.2), Holder and Sobolev inequalities, it arrives at IV = Z ∇ ( u · ∇ u ) ∇ u dx . ( k∇ u k L k∇ u k L + k u k L k∇ u k L ) k∇ u k L . k∇ u k L k∇ u k L + ( ε + ε ) k∇ u k L . (1 + t ) − + ( ε + ε ) k∇ u k L . (3.24) lobal Existence and Optimal Decay Rates of Solutions for Compressible Hall-MHD Equations In view of (2.6), (3.2), Holer and Sobolev inequalities, we obtain IV ≈ Z ∇ ( h ( ̺ ) ∇ u ) ∇ u dx = Z ( ∇ h ( ̺ ) ∇ u + 2 ∇ h ( ̺ ) ∇ u + h ( ̺ ) ∇ u ) ∇ udx . ( k∇ ̺ k L k∇ u k L + k∇ ̺ k L k∇ u k L ) k∇ u k L + ( k∇ ̺ k L k∇ u k L + k h ( ̺ ) k L ∞ k∇ u k L ) k∇ u k L . k∇ ̺ k H k∇ u k L + ( ε + ε ) k∇ u k L . (1 + t ) − + ( ε + ε ) k∇ u k L . (3.25)Similarly, it is easy to deduce immediately IV = Z ( f ( ̺ ) ∇ ̺ + 2 ∇ f ( ̺ ) ∇ ̺ + ∇ f ( ̺ ) ∇ ̺ ) ∇ udx . ( k f ( ̺ ) k L ∞ k∇ ̺ k L + k∇ ̺ k L k∇ ̺ k L + k∇ ̺ k L k∇ ̺ k L ) k∇ u k L . k∇ ̺ k H k∇ ̺ k L + k∇ ̺ k L + k∇ ̺ k L k∇ ̺ k L + ε k∇ u k L . (1 + t ) − + ε k∇ u k L . (3.26)Integrating by part and applying (2.6), (3.2), Holder and Sobolev inequalities, we get IV = − Z ( ∇ g ( ̺ ) B ∇ B + 2 ∇ g ( ̺ ) ∇ ( B ∇ B ) + g ( ̺ ) ∇ ( B ∇ B )) ∇ udx . ( k∇ g ( ̺ ) k L k B k L k∇ B k L + k∇ g ( ̺ ) k L k∇ B k L ) k∇ u k L + ( k∇ g ( ̺ ) k L k B k L k∇ B k L + k g ( ̺ ) k L ∞ k∇ B k L k∇ B k L ) k∇ u k L + k g ( ̺ ) k L ∞ k B k L ∞ k∇ B k L k∇ u k L . k∇ B k L k∇ B k L + k∇ B k L k∇ B k H + ε k∇ u k L . k∇ B k H k∇ B k H + ε k∇ u k L . (1 + t ) − + ε k∇ u k L . (3.27)In the same manner, it arrives at directly IV . (1 + t ) − + ε k∇ u k L . (3.28)Substituting (3.22)-(3.28) into (3.21), we obtain ddt Z ( |∇ ̺ | + |∇ u | ) dx + µ Z |∇ u | dx . (1 + t ) − + ε k∇ ̺ k L , which, together with (3.20), completes the proof of the lemma.Next, we establish the inequality to recover the dissipation estimate for ̺ . Lemma 3.4.
Under the assumptions in Theorem 1.3, the global classical solution ( ̺, u, B ) of Cauchyproblem (2.1) - (2.4) satisfies ddt Z ∇ u · ∇ ̺dx + C Z |∇ ̺ | dx ≤ C (cid:2) (1 + t ) − + k∇ u k H (cid:3) . (3.29) .C.Gao, Z.A.Yao Proof.
Taking ∇ operator on both hand sides of (2.1) , multiplying by ∇ ̺ and integrating over R , then we have Z ( ∇ u t · ∇ ̺ + |∇ ̺ | ) dx = Z [ µ ∆ ∇ u + ( µ + ν ) ∇ div u + ∇ S ] ∇ ̺ dx. (3.30)In order to deal with the term R ∇ u t · ∇ ̺dx , we turn the time derivatives of velocity to densityand apply the transport equation (2.1) . More precisely, we get Z ∇ u t · ∇ ̺dx = ddt Z ∇ u · ∇ ̺dx − Z ∇ u · ∇ ̺ t dx = ddt Z ∇ u · ∇ ̺dx + Z ∇ div u · ∇ ̺ t dx = ddt Z ∇ u · ∇ ̺dx − Z ∇ div u · ∇ (div u + ̺ div u + u ∇ ̺ ) dx. (3.31)Substituting (3.31) into (3.30), it arrives at ddt Z ∇ u · ∇ ̺dx + Z |∇ ̺ | dx = Z |∇ div u | dx + Z ∇ div u · ∇ ( ̺ div u + u ∇ ̺ ) dx + Z ∇ S · ∇ ̺dx + Z [ µ ∆ ∇ u + ( µ + ν ) ∇ div u ] ∇ ̺dx. (3.32)With the help of time decay rates (3.2), Holder and Sobolev inequalities, we obtain Z ∇ div u · ∇ ( ̺ div u + u ∇ ̺ ) dx = − Z ∇ div u · ∇ ( ̺ div u + u ∇ ̺ ) dx . ( k∇ ̺ k L k∇ u k L + k ̺ k L k∇ u k L + k u k L k∇ ̺ k L ) k∇ u k L . k∇ ̺ k H k∇ u k H + k∇ u k L k∇ ̺ k H + ε k∇ u k L . (1 + t ) − + k∇ u k L . (3.33)On the other hand, just following the idea as (3.24)-(3.27), we have Z ∇ S · ∇ ̺dx . (1 + t ) − + k∇ u k L + ε k∇ ̺ k L (3.34)and Z [ µ ∆ ∇ u + ( µ + ν ) ∇ div u ] ∇ ̺dx . k∇ u k L + ε k∇ ̺ k L . (3.35)Plugging (3.33)-(3.35) into (3.32), we complete the proof of lemma.Now, we establish optimal decay rates for the second order spatial derivatives of density andvelocity. Lemma 3.5.
Under the assumptions in Theorem 1.3, then the density and velocity have followingdecay rate k∇ ̺ ( t ) k H + k∇ u ( t ) k H ≤ C (1 + t ) − (3.36) for all t ≥ T ∗ ( T ∗ is a constant defined below ) . lobal Existence and Optimal Decay Rates of Solutions for Compressible Hall-MHD Equations Proof.
Multiplying (3.29) by C ε C and adding to (3.13), then we have ddt E ( t ) + C Z ( |∇ ̺ | + |∇ u | + |∇ u | ) dx ≤ C (1 + t ) − , (3.37)where E ( t ) is defined as E ( t ) = k∇ ̺ k H + k∇ u k H + 2 C ε C Z ∇ u · ∇ ̺dx. (3.38)By virtue of the smallness of ε , we have C − k∇ ( ̺, u ) k H ≤ E ( t ) ≤ C k∇ ( ̺, u ) k H . (3.39)It follows directly from (3.37) that ddt E ( t ) + C Z ( |∇ ̺ | + |∇ ̺ | + |∇ u | + |∇ u | ) dx ≤ C (1 + t ) − . (3.40)In the same manner as (2.72), we have Z |∇ ̺ | dx ≥ R t Z |∇ ̺ | dx − (cid:18) R t (cid:19) Z |∇ ̺ | dx, (3.41)and k∇ u k H ≥ R t k∇ u k H − (cid:18) R t (cid:19) k∇ u k H . (3.42)Plugging (3.41) and (3.42) into (3.40), it follows ddt E ( t ) + C (cid:20) R t Z ( |∇ ̺ | + |∇ u | + |∇ u | ) dx + Z |∇ ̺ | dx (cid:21) . (cid:18) R t (cid:19) Z ( |∇ ̺ | + |∇ u | + |∇ u | ) dx + (1 + t ) − . (1 + t ) − (1 + t ) − + (1 + t ) − . (1 + t ) − . (3.43)For some large time t ≥ R −
1, we have R t ≤ , which implies R t Z |∇ ̺ | dx ≤ Z |∇ ̺ | dx. (3.44)Combining (3.43) with (3.44), it is easy to deduce ddt E ( t ) + C R t ) k∇ ( ̺, u ) k H . (1 + t ) − , which, together with the equivalent relation (3.39), yields ddt E ( t ) + C R C (1 + t ) E ( t ) . (1 + t ) − . (3.45) .C.Gao, Z.A.Yao If choosing R = C C in (3.45), it is easy to deduce ddt E ( t ) + 41 + t E ( t ) . (1 + t ) − , (3.46)for all t ≥ T ∗ := C C −
1. Multiplying (3.46) by (1 + t ) , then we have ddt (cid:2) (1 + t ) E ( t ) (cid:3) . (1 + t ) − . (3.47)Integrating (3.47) over [0 , t ], then we get E ( t ) . (1 + t ) − , which, together with the equivalent relation (3.39), gives k∇ ̺ ( t ) k H + k∇ u ( t ) k H ≤ C (1 + t ) − . Therefore, we complete the proof of the lemma.Finally, we establish optimal decay rate for the third order spatial derivatives of magnetic field.
Lemma 3.6.
Under the assumption of Theorem 1.3, then the magnetic field has following decayrate for all t ≥ T ∗ , k∇ B ( t ) k L ≤ C (1 + t ) − . (3.48) Proof.
By virtue of (3.9) and applying time decay rates (3.3) and (3.36), we obtain ddt Z |∇ B | dx + Z |∇ B | dx . k∇ ( u, B ) k H k∇ ( u, B ) k H . (1 + t ) − (1 + t ) − . (1 + t ) − , which, together with (3.11), gives directly ddt Z |∇ B | dx + 51 + t Z |∇ B | dx . (1 + t ) − k∇ B k L + (1 + t ) − . (1 + t ) − (1 + t ) − + (1 + t ) − . (1 + t ) − . (3.49)Multiplying (3.49) by (1 + t ) and integrating the resulting inequality over [0 , t ], we obtain k∇ B ( t ) k L . (1 + t ) − . Therefore, we complete the proof of the lemma.
Proof of Theorem 1.3:
With the help of Lemma 3.1, Lemma 3.2, Lemma 3.5 and Lemma3.6, we complete the proof of Theorem 1.3. lobal Existence and Optimal Decay Rates of Solutions for Compressible Hall-MHD Equations In this section, we establish the time decay rates for the mixed space-time derivatives of globalclassical solutions.
Lemma 3.7.
Under the assumptions in Theorem 1.3, the global classical solution ( ̺, u, B ) of Cauchyproblem (2.1) - (2.4) has the time decay rates k∇ k ̺ t ( t ) k H − k + k∇ k u t ( t ) k L ≤ C (1 + t ) − k , k∇ k B t ( t ) k L ≤ C (1 + t ) − k , where k = 0 , .Proof. First of all, applying the estimate (2.77) and time decay rates (1.9), we obtain k∇ ̺ t k L . (1 + t ) − . (3.50)Applying the equation (2.1) , time decay rates (1.9), Holder and Sobolev inequalities, it arrives at k∇ ̺ t k L = k − ∇ div u − ∇ ( ̺ div u + u · ∇ ̺ ) k L . k∇ u k L + k∇ ( ̺, u ) k L k∇ ( ̺, u ) k L + k ( ̺, u ) k L ∞ k∇ ( ̺, u ) k L . (1 + t ) − . (3.51)Combining (3.50)-(3.51) with (2.76), it is easy to obtain k∇ k ̺ t ( t ) k H − k ≤ (1 + t ) − k , (3.52)where k = 0 ,
1. Secondly, in view of the equation (2.1) , (2.56) and Holder inequality, we get k∇ u t k L = k µ ∆ ∇ u + ( µ + ν ) ∇ div u − ∇ ̺ + ∇ S k L . k∇ u k L + k∇ ̺ k L + k∇ ( ̺, B ) k H k∇ ( u, B ) k H + δ ( k∇ ̺ k L + k∇ u k L + k∇ B k L ) . (1 + t ) − , which, together with (2.78), gives directly k∇ k u t ( t ) k L ≤ C (1 + t ) − k , (3.53)where k = 0 ,
1. Finally, it follows from (2.1) , (2.68)-(2.70), Holder and Sobolev inequalities that k∇ B t k L = k∇ ∆ B + ∇ S k L . k∇ B k L + k∇ ( u, B ) k H k∇ ( u, B ) k L + k∇ ̺ k L k∇ B k H . (1 + t ) − + (1 + t ) − (1 + t ) − + (1 + t ) − (1 + t ) − . (1 + t ) − , which, together with (2.79), gives directly k∇ k B t ( t ) k L ≤ C (1 + t ) − k , (3.54)where k = 0 ,
1. Combining (3.52), (3.53) with (3.54), then we complete the proof of lemma.
Proof of Theorem 1.4:
With the help of Lemma 3.7, we complete the proof of Theorem 1.4. .C.Gao, Z.A.Yao Acknowledgements
This research was supported in part by NNSFC(Grant No.11271381) and China 973 Program(GrantNo. 2011CB808002).
References [1] T. G. Forbes, Magnetic reconnection in solar flares, Geophys. Astrophys. Fluid Dyn. 62 (1991)15-36.[2] H. Homann, R. Grauer, Bifurcation analysis of magnetic reconnection in Hall-MHD systems,Phys. D 208 (2005) 59-72.[3] M. Wardle, Star formation and the Hall effect, Astrophys. Space Sci. 292 (2004) 317-323.[4] S. A. Balbus, C. Terquem, Linear analysis of the Hall effect in protostellar disks, Astrophys.J. 552 (2001) 235-247.[5] D. A. Shalybkov, V. A. Urpin, The Hall effect and the decay of magnetic fields, Astron.Astrophys. (1997) 685-690.[6] P. D. Mininni, D. O. G`omez, S. M. Mahajan, Dynamo action in magnetohydrodynamics andHall magnetohydrodynamics, Astrophys. J. 587 (2003) 472-481.[7] M. Acheritogaray, P. Degond, A. Frouvelle, J. G. Liu, Kinetic formulation and global existencefor the Hall-Magneto-hydrodynamics system, Kinet. Relat. Models 4 (2011) 901-918.[8] D. Chae, P. Degond, J. G. Liu, Well-posedness for Hall-magnetohydrodynamics, Ann. Inst. H.Poincar´e Anal. Non Lin´eaire 31 (2014) 555-565.[9] D. Chae, J. Lee, On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics, J. Differential Equations 256 (2014) 3835-3858.[10] J. S. Fan, F. C. Li, G. Nakamura, Regularity criteria for the incompressible Hall-magnetohydrodynamic equations, Nonlinear Anal. 109 (2014) 173-179.[11] J. S. Fan, T. Ozawa, Regularity criteria for the density-dependent Hall-magnetohydrodynamics,Appl. Math. Lett. 36 (2014) 14-18.[12] Maicon J. Benvenutti, Lucas C. F. Ferreira, Existence and stability of global large strongsolutions for the Hall-MHD system, arXiv:1412.8516.[13] J. S. Fan, S. X. Huang, G. Nakamura, Well-posedness for the axisymmetric incompressibleviscous Hall-magnetohydrodynamic equations, Appl. Math. Lett. 26 (2013) 963-967.[14] D. Chae, M. E. Schonbek, On the temporal decay for the Hall-magnetohydrodynamic equations,J. Differential Equations 255 (2013) 3971-3982.[15] S. K. Weng, On analyticity and temporal decay rates of solutions to the viscous resistiveHall-MHD system, arXiv:1412.8239. lobal Existence and Optimal Decay Rates of Solutions for Compressible Hall-MHD Equations [16] J. S. Fan, A. Alsaedi, T. Hayat, G. Nakamura, Y. Zhou, On strong solutions to the compressibleHall-magnetohydrodynamic system, Nonlinear Anal. Real World Appl. 22 (2015), 423-434.[17] F. C. Li, H. J. Yu, Optimal decay rate of classical solutions to the compressible magnetohy-drodynamic equations, Proc. Roy. Soc. Edinburgh Sect. A 141 (2011) 109-126.[18] Q. Chen, Z. Tan, Global existence and convergence rates of smooth solutions for the compress-ible magnetohydrodynamic equations, Nonlinear Anal. 72 (2010) 4438-4451.[19] Y. Guo, Y. J. Wang, Decay of dissipative equations and negative Sobolev spaces, Comm. PartialDifferential Equations 37 (2012) 2165-2208.[20] Z. Tan, H. Q.Wang, Optimal decay rates of the compressible magnetohydrodynamic equations,Nonlinear Anal. Real World Appl. 14 (2013) 188-201.[21] M. E. Schonbek, L decay for weak solutions of the Navier-Stokes equations, Arch. RationalMech. Anal. 88 (1985) 209-222.[22] J. C. Gao, Q. Tao, Z. A. Yao, Long-time Behavior of Solution for the Compressible NematicLiquid Crystal Flows in R , arXiv:1503.02865.[23] J. C. Gao, Y. H. Chen, Z. A. Yao. Long-time Behavior of Solution to the Compressible Mag-netohydrodynamic Equations, Preprint.[24] Y. J. Wang, Z. Tan, Global existence and optimal decay rate for the strong solutions in H tothe compressible Navier-Stokes equations, Appl. Math. Lett. 24 (2011) 1778-1784.[25] X. P. Hu, G. C. Wu, Global existence and optimal decay rates for three-dimensional compress-ible viscoelastic flows, SIAM J. Math. Anal. 45 (2013) 2815-2833.[26] W. J. Wang, W. K. Wang, Decay rates of the compressible Navier-Stokes-Korteweg equationswith potential forces, Discrete Contin. Dyn. Syst. 35 (2015) 513-536.[27] W. J. Wang, Large time behavior of solutions to the compressible Navier-Stokes equations withpotential force, J. Math. Anal. Appl. 423 (2015) 1448-1468.[28] L. Nirenberg, On elliptic partial differential euations, Ann.Scuola Norm. Sup. Pisa 13 (1959)115-162.[29] A. Matsumura, T. Nishida, The initial value problems for the equations of motion of viscousand heat-conductive gases, J.Math.Kyoto Univ. 20 (1980) 67-104.[30] R. J. Duan, H. X. Liu, S. J. Ukai, T. Yang, Optimal L p − L q convergence rates for the compress-ible Navier-Stokes equations with potential force, J.Differential Equations 238 (2007) 220-233.convergence rates for the compress-ible Navier-Stokes equations with potential force, J.Differential Equations 238 (2007) 220-233.