Regularity of large solutions for the compressible magnetohydrodynamic equations
© Liu et al; licensee Springer. 2011
Received: 10 June 2011
Accepted: 10 October 2011
Published: 10 October 2011
In this paper, we consider the initial-boundary value problem of one-dimensional compressible magnetohydrodynamics flows. The existence and continuous dependence of global solutions in H1 have been established in Chen and Wang (Z Angew Math Phys 54, 608-632, 2003). We will obtain the regularity of global solutions under certain assumptions on the initial data by deriving some new a priori estimates.
Here, v, u, w , b , θ, and p are the specific volume, the longitudinal velocity, the transverse velocity, the transverse magnetic filed, the absolute temperature, and the pressure, respectively; λ, μ, ν, and κ are the bulk viscosity coefficient, the shear viscosity coefficient, the magnetic diffusivity, and the heat conductivity, respectively.
whose internal energy e and pressure p are coupled by the standard thermodynamical relation (1.8). In this case, Kawohl  obtained the existence of global solutions with the exponents r ∈ [0, 1], q ≥ 2r + 2. Jiang  also established the global existence with basically same constitutive relations as those in  but with the exponents r ∈ [0, 1], q ≥ r + 1. When the exponents q, r satisfy the more general constitutive relations than those in [8, 9], Qin  established the regularity and asymptotic behavior of global solutions with arbitrary initial data for a one-dimensional viscous heat-conductive real gas.
For the radiative and reactive gas, Ducomet  established the global existence and exponential decay in H1 of smooth solutions, and Umehara and Tani  proved the global existence of smooth solutions for a self-gravitating radiative and reactive gas.
for some , Zhang and Xie  investigated the existence of global smooth solutions.
Fan et al.  investigated the uniqueness of the weak solutions of MHD with Lebesgue initial data. Fan et al.  also considered a one-dimensional plane compressible MHD flows and proved that as the shear viscosity goes to zero, global weak solutions converge to a solution of the original equations with zero shear viscosity. The uniqueness and continuous dependence of weak solutions for the Cauchy problem have been proved by Hoff and Tsyganov .
As mentioned above, the global existence in of global solutions has never been studied for Equations (1.1)-(1.5) of the nonlinear one-dimensional compressible magnetohydrodynamics flows with initial-boundary conditions (1.6)-(1.7). The main aim of this paper is to prove the regularity of solutions in the subspace of (H i [0, 1])7(i = 2, 4) for systems (1.1)-(1.7). In order to obtain higher regularity of global solutions, there are many complicated estimates on higher derivations of solutions to be involved, this is our main difficulty. To overcome this difficulty, we should use some proper embedding theorems, the interpolation techniques as well as many delicate estimates. This is the novelty of the paper.
The notation in this paper will be stated as follows:
L p , 1 ≤ p ≤ +∞, W m, p , m ∈ N, H1 = W1,2, denote the usual (Sobolev) spaces on Ω. In addition, ||·|| B denotes the norm in the space B, we also put . Constants C i (i = 1, 2, 3, 4) depend on the norm of the initial data (v0, u0, w 0 , b0, θ0) and T > 0.
Now we are in a position to state our main results.
2 Proof of Theorem 1.1
In this section, we study the global existence of problem (1.1)-(1.7) in by establishing a series of priori estimates. Without loss of generality, we take c v = R = 1. We begin with the following lemma.
Proof. See, e.g., .
Thus, (2.3) follows from (2.4)-(2.5).
Thus, (2.6) follows from (2.8)-(2.10).
which, together with Lemmas 2.1-2.3, yields (2.11).
Proof of Theorem 1.1. By Lemmas 2.1-2.4, we complete the proof of Theorem 1.1.
3 Proof of Theorem 1.2
In this section, we study the global existence of problem (1.1)-(1.7) in by establishing a series of priori estimates. We begin with the following lemmas.
Thus, (3.1) follows from (3.3), (3.7), (3.11) and (3.16), and (3.2) from (3.5), (3.9), (3.13), (3.18) and (3.20)-(3.23).
Thus, taking supremum in t on the left-hand side of (3.29), picking ε ∈ (0, 1) small enough, and using (3.23), we can derive estimate (3.26).
Thus, inserting (3.40) into (3.39) implies estimate (3.30).
Analogously, we can obtain estimates (3.31)-(3.32). □
which, by Gronwall's inequality, gives the estimate (3.41).