Large-time behavior of the strong solution to nonhomogeneous incompressible MHD system with general initial data

System (1.1)-(1.4) has drawn many attentions of engineers and applied mathematicians due to its important physical background and mathematical feature. The Cauchy problem to (1.1)-(1.4) has been much studied in the literature, including the existence, uniqueness, and regularity of solutions of the system. Zhang [5] established local classical solutions of (1.1)-(1.4) and proved that as the viscosity (μ) and resistivity (ν) went to zero, the solution of (1.1)-(1.4) converged to the solution of ideal MHD system (i.e. \(\mu=\nu=0\)). Gerbeau and Le Bris [3] and Desjardins and Le Bris [6] considered the global existence of weak solutions of finite energy in the whole space or in the torus. Global existence of strong solutions with small initial data in some Besov spaces was considered by Abidi and Paicu [7]. For the initial-boundary problem of (1.1)-(1.4), Huang and Wang [8] recently proved the unique global strong solution with initial vacuum in dimension two without external force f. For more related results, we refer the reader to [1, 9–12] and the references therein.

For homogeneous incompressible MHD (\(\rho\equiv \mbox{const}\) in (1.1)-(1.4)), we first emphasize that the unique global classical solutions for every initial data \((\mathrm {u}_{0}, \mathrm {H}_{0})\in H^{m}\) with \(m\geq2\) have been established in [13, 14]. Furthermore, some sufficient conditions for smoothness were presented for weak solutions to the MHD equations in [15, 16]. For Cauchy problem, the authors in [17] studied the long time behavior of solutions to the MHD equations in two and three dimensions with some smallness conditions. In [18], the authors considered the asymptotic behavior of the strong solutions to MHD equations in a half space. We also notice that if the partial derivatives of the viscosity and resistivity are zero, the global regularity issue has been established in [19].

However, the large-time asymptotic behavior of strong solutions to (1.1)-(1.6) has not been well understood, especially for the case with general initial data and external force. The main purpose of this paper is to investigate the influence of the magnetic fields, viscosity, and boundary effects on the behavior of the solution of (1.1)-(1.6). We show that the velocity, the magnetic field, and their derivatives will converge to zero as time tends to infinity. The main result of this paper is stated as follows.

Our second main result gives an exponential decay rate for the solution on taking the external force \(f=0\), which is stated as follows.

We will prove Theorem 1.1 via the pure energy method, which is based on the uniform estimate for the local solution. This approach is motivated by the previous work on the nonhomogeneous incompressible Navier-Stokes equations due to Zhao [20]. Due to the strong coupling between the velocity, magnetic field and density equations, the nonhomogeneous problem (1.1)-(1.4) under consideration is much more involved. It should be pointed out that the analysis for (1.1)-(1.4) is very complicated and more efforts should be made as regards the estimates involving these coupling terms and space-time-dependent external force.

Next, we give a brief outline of the proof. To obtain the large-time behavior, we must establish the uniform estimate. As for the uniform estimate, it is difficult to estimate the higher-order derivative of u and H due to the lack of the spatial derivatives of the solution at the boundary. So it is divided into two steps: first of all applying the standard energy estimate on the temporal derivatives of the solution, and then spatial derivatives by applying regularity theory of Stokes equation repeatedly. With all the estimates in hand, we can establish the global existence by a standard method. Finally, the proof of the large-time behavior is mainly based on the following fact: if \(f(t)\in W^{1,1}(0,\infty)\), then \(f(t)\rightarrow0\) as \(t\rightarrow\infty\). By combining these uniform estimates in Lemmas 3.1-3.3 and the fact above we prove the desired large-time behavior of the solution in Section 4.

The rest of the paper is organized as follows. In Section 2, we state some well-known inequalities, and in Section 3 we deduce some uniform estimates and the time-dependent estimate of the higher-order derivative. Finally, we will complete the proof of Theorem 1.1 in Section 4.

In this section, we state some inequalities which will be frequently used in this paper. First of all, we recall some Sobolev embedding inequalities (see [21, 22]).

Next, we state the following Sobolev embedding (see [23, 24]), which will be used to deal with the estimate of H.

The local existence has been established by Chen et al. [9]. Hence, it only remains to establish some necessary a priori bounds for the strong solutions \((\rho, \mathrm {u},\mathrm {H})\) to the initial-boundary value problem (1.1)-(1.4).

Let \(T>0\) be a fixed time and \((\rho, \mathrm {u}, \mathrm {H})\) be the strong solution to (1.1)-(1.4) defined on \(\Omega\times(0,T]\). Throughout this paper, we will denote by C the various generic positive constants, which may depend on the initial data and ρ̄, \(\underline{\rho}\), f, and Ω but are independent of t. A special dependence will be pointed out explicitly in this paper if necessary.

With all the a priori estimates in Section 3 at hand, the existence and uniqueness of the strong solutions to (1.1)-(1.6) can be done by standard continuous arguments. Thus, this section is mainly devoted to the proof of the large-time behavior as (1.9)-(1.10) by the uniform estimates in Section 3. The proof is split into three steps as follows.