Optimal Convergence Rates for the Strong Solutions to the Compressible MHD Equations with Potential Force

as |x| 󳨀→ ∞. (2) Here ρ ≥ 0, u, b, θ ∈ R, and p(ρ, θ) are density, velocity, magnetic field, temperature, and pressure, respectively. The constants μ, λ are the shear and bulk viscosity coefficients of the flow, satisfying the physical restrictions μ > 0 and 2μ + 3λ ≥ 0. The resistivity coefficient ] > 0 is inversely proportional to the electrical conductivity constant and acts as the magnetic diffusivity of magnetic fields. By the positive constant κ we denote the heat conduction coefficient and cV > 0 is the specific heat at constant volume. In addition, F = F(x) is an external force and Ψ = Ψ[u] is the dissipation function:Ψ [u] = μ2 3 ∑

We only consider the potential force  = −∇Φ() in this work.Reference [1] has proofed the existence of the stationary solution to problem (1) and (2) under aforementioned assumptions.We give the solution ( * ,  * ,  * ,  * ) in a neighborhood of ( ∞ , 0, 0,  ∞ ) as The global unique solution is constructed to (1) near the steady state ( * , 0, 0,  ∞ ) when the initial perturbation belongs to the Sobolev space  2 (R 3 ).The main results are as follows.
Remark .In Theorem 1, using the Sobolev imbedding inequalities in Lemma 3, (7) together with ( 5) and ( 6) yields Due to its importance in mathematics and physics, there are a lot of literatures devoted to the mathematical theory of the MHD fluid system, including the global existence, unique, and time decay rates of solutions with or without external forces; see [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] and references therein.For the initial value problem for three-dimensional isentropic MHD system ((1) with  = .),Li and Yu [13] obtained the global existence and the  2 decay rate of classical solutions under the small oscillations on small initial data.Chen and Tan [4] obtained the decay estimate of solutions with the initial data in   ( ∈ [1, 6/5)).Noting the special construction of (1), the authors in [18] obtained decay estimate of solutions when the initial data belongs to negative Sobolev space.For the initial boundary value problem for three-dimensional isentropic MHD system (1), we refer to [11].Pu and Guo [17] extended the global existence and decay rate of solutions in [4,13] to three-dimensional incompressible nonisentropic MHD system (1).For the initial boundary value for threedimensional nonisentropic MHD system, the global existence of weak solutions has been established in [9,10].
If the magnetic field disappear ( = 0), system (1) deduces to Navier-Stokes equations.We only refer to time decay rates of solutions to the compressible Navier-Stokes equations.If there is no external force, Matsumura and Nishida [21] obtained the convergence rate for the compressible viscous and heat-conductive fluid in R 3 : For the same system, Ponce [22] gave the optimal   convergence rate for the space dimension  = 2 or 3. When there is an external potential force  = −∇Φ(), the first work to give explicit estimates for the decay rates for solutions to problem (1) and ( 2) was represented by Deckelnick [23].In [23], when the initial perturbation only belongs to the Sobolev space  3 (Ω), the following decay estimates were established in an unbounded domain or the exterior of a bounded domain with smooth boundary): Duan et al. [24] obtained a similar result for the nonisentropic case Recently, Wang [25] constructed the global unique solution near the stationary profile to the system for the small  2 initial data and the optimal   −  2 (1 ≤  ≤ 2) time decay rates of the solution: In this paper, we will generalize the result in [25] to the three-dimensional magnetomicropolar fluid system (1).We will give the global existence and the optimal   −  2 (1 ≤  ≤ 2) time decay rates of the solution.Notice that the first three equations in (1) have been studied in [25].The magnetic field (, ) satisfies the heat equation, which can be solved in Fourier space, b = exp(−|| 2 ) b0 , whose decay estimate has been proved in many papers, for instance, [26,27].However, compared with the work in [25], we have to deal with the difficulties in the present paper caused by the strong coupling between the velocity vector field and the magnetic field, for instance, the influence of the nonlinear term (∇ × ) ×  on the velocity vector field.
, and || = Σ 3 =1   .Denote ∇  as a set composed of all th partial derivatives with respect to the variable .  (R 3 ),  ∈ Z + , denotes the usual Sobolev space with its norm In particular, ‖ ⋅ ‖  means ‖ ⋅ ‖   (R 3 ) and ‖ ⋅ ‖  means ‖ ⋅ ‖  2 (R 3 ) .As usual, ⟨⋅, ⋅⟩ denotes the inner-product in  2 (R 3 ).The Fourier transform to the variable  ∈ R 3 is applied by f(, ) = ∫ R  (, ) −√−1⋅  and the inverse Fourier transform to the variable The outlet of paper is organized as follows.In Section 2, we reformulate the Cauchy problem (1)-( 2) into a more suitable form.In Section 3, we establish the global existence of solutions to the problem in  2 -framework.Section 4 is devoted to the proof of the   −  2 time decay rates of the solution and we complete the proof of Theorem 1.

Preliminary
We are going to reformulate the Cauchy problem of the compressible MHD system (1) with the initial condition (2) and give some lemmas in this section.Define Then, by using (4), the MHD equations (1) are transformed as the following: and the initial condition (2) turns into ( ρ, ũ, b, θ) (, )     =0 = ( 0 −  * ,  0 ,  0 ,  0 ) () → (0, 0, 0, 0) , where S1 , S2 , S3 , and S4 are the source terms with For obtaining a symmetric system, we denote with Then ( 20) and ( 21) can be reformulated as where , and Then, we will focus on considering the global existence and time decay rates of the solution (, , , ) to the steady state ( * , 0, 0,  ∞ ), that is, the existence and decay rates of the perturbed solution (, , , ) to problem (28).First of all, some inequalities are listed as follows for late use.

Global Existence
We will use the energy method to establish the global existence of solutions to the problem (28) in  2 -framework in this section.Let us define the solution space and the solution norm of the initial value problem (25) by where  ∈ [0, +∞].By the standard continuity argument, the global existence of solutions to (25) will be obtained from the combination of the local existence result with some a priori estimates.
Proof.The proof can be stated by using the standard iteration arguments.Refer to [2], for instance.
Proposition 6 (a priori estimates).Suppose that the initial value problem ( ) has a solution (, , , ) ∈ (0, ), where  is a positive constant.Under the assumptions of eorem , there exist a positive constant  0 and a small positive constant  which are independent of , such that if where  > max{, √ 0 } and  is given by ( ), then it holds that, for any  ∈ [0, ], where  0 is independent of .
Remark .The global existence and uniqueness of the solutions stated in Theorem 1 follow from Propositions 5 and 6.By ( 12), (33), and the Sobolev inequality, we have This will be used in following statement.
where the nonlinear terms in ( 22)-( 25) have the following equivalence properties: When  = 0, by using the Hölder inequality, Lemma 3, (12), the a priori assumption (33), and the Young inequality, we obtain Putting (39), (41) into (37) and taking  sufficiently small, we have When1 ≤  ≤ 2, the terms on the RHS of (37) can be estimated as follows.For   1 (), from (38), we get For the first term on the RHS of (43), by using integration by parts, the Hölder inequality, the Young inequality, Lemma 3, and the a priori assumption (33), we have ( The other terms on the RHS of (49) can be estimated in some similar way.
Proof.From (28) 2 , we get By applying    (|| = ) to (58), multiplying it by ∇   , and then integrating over R 3 , we have Using integration by parts and (28) 1 , we obtain Adding ( 59) and (60) gives Then by using the Young inequality and integration by parts, it follows from the above equality that ) . (65) Thus, the proof of Lemma 9 is finished.
Let 0 ≤  ≤ 1.By summing up (36) from  =  to 2, since  is small, there exists a constant  1 > 0 such that Summing up (57) for from  =  to 1, since  is small, we have where  2 > 0 is a constant.Multiplying (67) by 2 1 / 2 < 1/2, adding it with (66), taking  = 0, and using the smallness of , we have Integrating (68) with respect to  and using the smallness of , we get (34).The proof of Proposition 6 is complete.

Decay Rates
In this section, we get decay rates of solutions to problem (25).
Let A be the following 8×8 matrix of the differential operators of the form If we set the linearized problem of ( 25) is written as By taking the Fourier transform of (71) with respect to the -variable and solving the ordinary differential equation with respect to t, we have where () =  −A ( ≥ 0) is the semigroup generated by the linear operator −A.We define () = F −1 ( − Â f(, )) for function (, ) with ) . ( In what following, we denote the solutions to the nonlinear problem (25) by  () = ( () ,  () ,  () ,  ())  .
(74) Then ( 28) can be rewritten as follows: where () ≜ ( where  is some fixed positive constant.Based on the Fourier transform and (77), we can define the low frequency and the high frequency decomposition (  (),   ()) for () as follows: To estimate   (), we need the   −   type of the time decay estimates on the low-frequency part of the semigroup ().Notice that the first, the second, and the fourth equations in (71) have been studied in [29].The magnetic field (, ) satisfies the heat equation, which can be solved in Fourier space, Ĥ = exp(−|| 2 ) Ĥ0 , whose decay estimate has been proved in many papers, for instance, [26,27].Then the semigroup () has the following properties on the decay in time.
. .Decay Estimates of the Low-Frequency Part.In this subsection, we have the decay estimates of   () with the   −   type estimate (82) as follows.