Research Article | Open Access
Miao Ouyang, "Optimal Convergence Rates for the Strong Solutions to the Compressible MHD Equations with Potential Force", Mathematical Problems in Engineering, vol. 2019, Article ID 6710508, 12 pages, 2019. https://doi.org/10.1155/2019/6710508
Optimal Convergence Rates for the Strong Solutions to the Compressible MHD Equations with Potential Force
In this paper, the large-time behavior of solutions to the Cauchy problem for the 3D compressible MHD equations is considered with the effect of external force. We construct the global unique solution with the small initial data near the stationary profile. The optimal time decay rates of the solution to the system are built in multifrequency decompositions.
Consider the following Cauchy problem of the full compressible MHD equations affected by the external potential force in :where Here , , and are density, velocity, magnetic field, temperature, and pressure, respectively. The constants , are the shear and bulk viscosity coefficients of the flow, satisfying the physical restrictions and . The resistivity coefficient is inversely proportional to the electrical conductivity constant and acts as the magnetic diffusivity of magnetic fields. By the positive constant we denote the heat conduction coefficient and is the specific heat at constant volume. In addition, is an external force and is the dissipation function:
It will be assumed that is smooth in a neighborhood of with and throughout this paper. Both and are positive constants.
We only consider the potential force in this work. Reference  has proofed the existence of the stationary solution to problem (1) and (2) under aforementioned assumptions. We give the solution in a neighborhood of as and satisfy
The global unique solution is constructed to (1) near the steady state when the initial perturbation belongs to the Sobolev space . The main results are as follows.
Theorem 1. Suppose that , there exists constant such that ifthen the initial value problem (1) and (2) admits a unique solution globally in time which satisfiesMoreover, if the initial data is bounded in for any given , the solution satisfies the following decay-in-time estimates:for some positive constant .
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 [2–20] and references therein. For the initial value problem for three-dimensional isentropic MHD system ((1) with ), Li and Yu  obtained the global existence and the decay rate of classical solutions under the small oscillations on small initial data. Chen and Tan  obtained the decay estimate of solutions with the initial data in . Noting the special construction of (1), the authors in  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 . Pu and Guo  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 three-dimensional nonisentropic MHD system, the global existence of weak solutions has been established in [9, 10].
If the magnetic field disappear (), 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  obtained the convergence rate for the compressible viscous and heat-conductive fluid in :For the same system, Ponce  gave the optimal convergence ratefor the space dimension or . When there is an external potential force , the first work to give explicit estimates for the decay rates for solutions to problem (1) and (2) was represented by Deckelnick . In , when the initial perturbation only belongs to the Sobolev space , the following decay estimates were established in an unbounded domain (the half space or the exterior of a bounded domain with smooth boundary):Duan et al.  obtained a similar result for the nonisentropic caseRecently, Wang  constructed the global unique solution near the stationary profile to the system for the small initial data and the optimal time decay rates of the solution:
In this paper, we will generalize the result in  to the three-dimensional magnetomicropolar fluid system (1). We will give the global existence and the optimal time decay rates of the solution. Notice that the first three equations in (1) have been studied in . The magnetic field satisfies the heat equation, which can be solved in Fourier space, , whose decay estimate has been proved in many papers, for instance, [26, 27]. However, compared with the work in , 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.
Notations. Denote as a generic positive constant. For multi-indices and , with which means for all . ,,, and . Denote as a set composed of all th partial derivatives with respect to the variable . , , denotes the usual Sobolev space with its norm
In particular, means and means . As usual, denotes the inner-product in . The Fourier transform to the variable is applied by and the inverse Fourier transform to the variable by .
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 -framework. Section 4 is devoted to the proof of the time decay rates of the solution and we complete the proof of Theorem 1.
DefineThen, by using (4), the MHD equations (1) are transformed as the following:and the initial condition (2) turns into where , , , and are the source terms withFor obtaining a symmetric system, we denotewithThen (20) and (21) can be reformulated aswhere , , , , and
Then, we will focus on considering the global existence and time decay rates of the solution to the steady state , 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.
Lemma 3 (see ). Let . Then
Lemma 4 (see ). Let , then it holds thatfor an arbitrarily small .
3. Global Existence
We will use the energy method to establish the global existence of solutions to the problem (28) in -framework in this section. Let us define the solution space and the solution norm of the initial value problem (25) bywhere . 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.
Proposition 5 (local existence). Suppose that the initial data satisfy and (7). Then there exists a positive constant depending on , such that the initial value problem (25) has a unique solution which satisfies .
Proof. The proof can be stated by using the standard iteration arguments. Refer to , for instance.
Proposition 6 (a priori estimates). Suppose that the initial value problem (28) has a solution , where is a positive constant. Under the assumptions of Theorem 1, there exist a positive constant and a small positive constant which are independent of , such that if where and is given by (7), then it holds that, for any ,where is independent of .
Remark 7. 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 haveThis will be used in following statement. Before proving Proposition 6, we supply Lemmas 8 and 9.
Lemma 8. For , it holds that for any
Proof. For each multi-index with , operating to the first 4 equations of (28), then multiplying the resulting equations by , , , and , respectively, summing up, and then integrating the result over , we obtain where the nonlinear terms in (22)-(25) have the following equivalence properties:When , by using the Hölder inequality, Lemma 3, (12), the a priori assumption (33), and the Young inequality, we obtainwhere we have used the Hardy inequality as follows:In a similar way, we getPutting (39), (41) into (37) and taking sufficiently small, we haveWhen, the terms on the RHS of (37) can be estimated as follows. For , from (38), we getFor 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 where the terms including the sum of with will be vanished if . Similarly, we have By the Hölder inequality, the Young inequality, Lemma 3, and (12), the third term on the RHS of (43) can be estimated as follows: By a similar argument, the fourth term on the RHS of (43) has following estimations:Therefore, putting (44)-(47) into (43) yieldsFor , from (38), it is obvious thatFor the second term on the RHS of (49), based on integration by parts, the Hölder inequality, Lemma 3, and (33), we have