Global Existence and Uniqueness of Strong Solutions for the Magnetohydrodynamic Equations
© Jianwen Zhang. 2008
Received: 21 June 2007
Accepted: 5 October 2007
Published: 27 November 2007
This paper is concerned with an initial boundary value problem in one-dimensional magnetohydrodynamics. We prove the global existence, uniqueness, and stability of strong solutions for the planar magnetohydrodynamic equations for isentropic compressible fluids in the case that vacuum can be allowed initially.
where denotes the density of the fluid, the longitudinal velocity, the transverse velocity, the transverse magnetic field, the temperature, the pressure, and the internal energy; and are the bulk and shear viscosity coefficients, is the magnetic viscosity, is the heat conductivity. Notice that the longitudinal magnetic field is a constant which is taken to be identically one in (1.1).
The equations in (1.1) describe the macroscopic behavior of the magnetohydrodynamic flow. This is a three-dimensional magnetohydrodynamic flow which is uniform in the transverse directions. There is a lot of literature on the studies of MHD by many physicists and mathematicians because of its physical importance, complexity, rich phenomena, and mathematical challenges, see [1–14] and the references cited therein. We mention that, when , the system (1.1) reduces to the one-dimensional compressible Navier-Stokes equations for the flows between two parallel horizontal plates (see, e.g., ).
where the initial data satisfy certain compatibility conditions as usual and some additional assumptions below, and whenever . Here the boundary conditions in (1.7) mean that the boundary is nonslip and impermeable.
The purpose of the present paper is to study the global existence and uniqueness of strong solutions of problem (1.2)–(1.7). The important point here is that initial vacuum is allowed; that is, the initial density may vanish in an open subset of the space-domain , which evidently makes the existence and regularity questions more difficult than the usual case that the initial density has a positive lower bound. For the latter case, one can show the global existence of unique strong solution of this initial boundary value problem in a similar way as that in [3, 9, 14]. The strong solutions of the Navier-Stokes equations for isentropic compressible fluids in the case that initial vacuum is allowed have been studied in [16, 17]. In this paper, we will use some ideas developed in [16, 17] and extend their results to the problems (1.2)–(1.7). However, because of the additional nonlinear equations and the nonlinear terms induced by the magnetic field , our problem becomes a bit more complicated.
Our main result in this paper is given by the following theorem (the notations will be defined at the end of this section).
The compatibility conditions given by (1.9), (1.10) play an important role in the proof of uniqueness of strong solutions. Similar conditions were proposed in [16–18] when the authors studied the global existence and uniqueness of solutions of the Navier-Stokes equations for isentropic compressible fluids. In fact, one also can show the global existence of weak solutions without uniqueness if the compatibility conditions (1.9), (1.10) are not valid.
We will prove the global existence and uniqueness of strong solutions in Sections 3 and 4, respectively, while Section 2 is devoted to the derivation of some a priori estimates.
2. A Priori Estimates
Integrating (1.2) and (2.1) over , we arrive at our first lemma.
The next lemma gives us an upper bound of the density , which is crucial for the proof of Theorem 1.1.
For any , holds.
which, together with (2.8), yields Lemma 2.2 immediately.
To be continued, we need the following lemma because of the effect of magnetic .
Since because of Lemma 2.1, we thus obtain the first inequality indicated in this lemma from (2.11) by applying Gronwall's lemma and then Sobolev's inequality.
where we have used Cauchy-Schwarz's inequality, Sobolev's inequality (2.12), Lemma 2.1, and the first part of the lemma. This completes the proof of Lemma 2.3.
where Lemmas 2.1–2.3 and the first conclusion of this lemma have been used. Therefore, from the above inequality we obtain the second part, and so Lemma 2.4 is proved.
Hence, we have the following lemma.
where and .
To prove the uniqueness of strong solutions, we still need the following estimates.
which proves the first part of the lemma.
On the other hand, by virtue of (1.2) we have
In a same manner as that in the derivation of (2.44), we can show the analogous estimate for the transverse velocity by using the previous lemmas, (2.44), and the compatibility condition (1.10) as well. Thus, we complete the proof of Lemma 2.6.
From the a priori estimates established above, one sees that the compatibility conditions are used to obtain the second part of Lemma 2.6 only. However, this is crucial in the proof of the uniqueness of strong solutions.
3. Global Existence of Strong Solutions
By the uniform in bounds given in (3.5)–(3.8) we conclude that there exists a subsequence of which converges to a strong solution to the original problem and satisfies (3.5)–(3.8) as well. This completes the proof of Theorem 1.1 except the uniqueness assertion (because of the presence of vacuum), which will be proved in the next section.
4. Uniqueness and Stability of Strong Solutions
In this section, we will prove the following stability theorem, which consequently implies the uniqueness of strong solutions. Our proof is inspired by the uniqueness results due to Choe-Kim [16, 17] and Desjardins  for the isentropic compressible Navier-Stokes equations.
for some . Here and .
Summing up (4.4)–(4.12) and choosing appropriately small, we obtain Theorem 4.1 with by applying Gronwall's lemma.
This work is partly supported by NSFC (Grant no. 10501037 and no. 10601008).
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