Global Existence and Uniqueness of Strong Solutions for the Magnetohydrodynamic Equations

Boundary Value Problems20072008:735846

DOI: 10.1155/2008/735846

Received: 21 June 2007

Accepted: 5 October 2007

Published: 27 November 2007

Abstract

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.

1. Introduction

Magnetohydrodynamics (MHD) concerns the motion of a conducting fluid in an electromagnetic field with a very wide range of applications. The dynamic motion of the fluids and the magnetic field strongly interact each other, and thus, both the hydrodynamic and electrodynamic effects have to be considered. The governing equations of the plane magnetohydrodynamic compressible flows have the following form (see, e.g., [15]):
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ1_HTML.gif
(1.1)

where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq1_HTML.gif denotes the density of the fluid, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq2_HTML.gif the longitudinal velocity, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq3_HTML.gif the transverse velocity, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq4_HTML.gif the transverse magnetic field, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq5_HTML.gif the temperature, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq6_HTML.gif the pressure, and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq7_HTML.gif the internal energy; http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq8_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq9_HTML.gif are the bulk and shear viscosity coefficients, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq10_HTML.gif is the magnetic viscosity, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq11_HTML.gif 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 [114] and the references cited therein. We mention that, when http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq12_HTML.gif , the system (1.1) reduces to the one-dimensional compressible Navier-Stokes equations for the flows between two parallel horizontal plates (see, e.g., [15]).

In this paper, we focus on a simpler case of (1.1), namely, we consider the magnetohydrodynamic equations for isentropic compressible fluids. Thus, instead of the equations in (1.1), we will study the following equations:
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ2_HTML.gif
(1.2)
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ3_HTML.gif
(1.3)
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ4_HTML.gif
(1.4)
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ5_HTML.gif
(1.5)
where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq13_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq14_HTML.gif being the adiabatic exponent and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq15_HTML.gif being the gas constant. We will study the initial boundary value problem of (1.2)–(1.5) in a bounded spatial domain http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq16_HTML.gif (without loss of generality) with the initial-boundary data:
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ6_HTML.gif
(1.6)
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ7_HTML.gif
(1.7)

where the initial data http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq17_HTML.gif satisfy certain compatibility conditions as usual and some additional assumptions below, and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq18_HTML.gif whenever http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq19_HTML.gif . 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 http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq20_HTML.gif may vanish in an open subset of the space-domain http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq21_HTML.gif , which evidently makes the existence and regularity questions more difficult than the usual case that the initial density http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq22_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq23_HTML.gif , 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).

Theorem 1.1.

Assume that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq24_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq25_HTML.gif satisfy the regularity conditions:
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ8_HTML.gif
(1.8)
Assume also that the following compatibility conditions hold for the initial data:
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ9_HTML.gif
(1.9)
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ10_HTML.gif
(1.10)
Then there exists a unique global strong solution http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq26_HTML.gif to the initial boundary value problem (1.2)–(1.7) such that for all http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq27_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ11_HTML.gif
(1.11)

Remark 1.2.

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 [1618] 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.

We end this section by introducing some notations which will be used throughout the paper. Let http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq28_HTML.gif denote the usual Sobolev space, and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq29_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq30_HTML.gif . For simplicity, we denote by http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq31_HTML.gif the various generic positive constants depending only on the data and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq32_HTML.gif , and use the following abbreviation:
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ12_HTML.gif
(1.12)

2. A Priori Estimates

This section is devoted to the derivation of a priori estimates of http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq33_HTML.gif . We begin with the observation that the total mass is conserved. Moreover, if we multiply (1.3), (1.4), and (1.5) by http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq34_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq35_HTML.gif , respectively, and sum up the resulting equations, we have by using (1.2) that
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ13_HTML.gif
(2.1)

Integrating (1.2) and (2.1) over http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq36_HTML.gif , we arrive at our first lemma.

Lemma 2.1.

For any http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq37_HTML.gif , one has
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ14_HTML.gif
(2.2)
where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq38_HTML.gif is the nonnegative function defined by
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ15_HTML.gif
(2.3)

The next lemma gives us an upper bound of the density http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq39_HTML.gif , which is crucial for the proof of Theorem 1.1.

Lemma 2.2.

For any http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq40_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq41_HTML.gif holds.

Proof.

Notice that (1.3) can be rewritten as
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ16_HTML.gif
(2.4)
Set
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ17_HTML.gif
(2.5)
from which and (2.4), we find that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq42_HTML.gif satisfies
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ18_HTML.gif
(2.6)
In view of Lemma 2.1 and (2.6), we have by using Cauchy-Schwarz's inequality that
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ19_HTML.gif
(2.7)
which imply
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ20_HTML.gif
(2.8)
Letting http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq43_HTML.gif denote the material derivative and choosing http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq44_HTML.gif , we obtain after a straightforward calculation that
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ21_HTML.gif
(2.9)

which, together with (2.8), yields Lemma 2.2 immediately.

To be continued, we need the following lemma because of the effect of magnetic http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq45_HTML.gif .

Lemma 2.3.

The magnetic field http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq46_HTML.gif satisfies the following estimates:
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ22_HTML.gif
(2.10)

Proof.

Multiplying (1.5) by http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq47_HTML.gif and integrating over http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq48_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ23_HTML.gif
(2.11)
where we have used Cauchy-Schwarz's inequality, Lemma 2.1, and the following inequalities:
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ24_HTML.gif
(2.12)

Since http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq49_HTML.gif 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.

To prove the second part, we multiply (1.5) by http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq50_HTML.gif and integrate the resulting equation over http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq51_HTML.gif to deduce that
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ25_HTML.gif
(2.13)

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.

Lemma 2.4.

The following estimates hold for the velocity http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq52_HTML.gif :
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ26_HTML.gif
(2.14)

Proof.

Multiplying (1.3) by http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq53_HTML.gif and then integrating over http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq54_HTML.gif , by Young's inequality we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ27_HTML.gif
(2.15)
It follows from (1.2) and (1.3) that
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ28_HTML.gif
(2.16)
Thus, inserting (2.16) into (2.15), and integrating over http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq55_HTML.gif , we see that
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ29_HTML.gif
(2.17)
where the terms on the right-hand side can be bounded by using Lemmas 2.1–2.3 as follows:
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ30_HTML.gif
(2.18)
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ31_HTML.gif
(2.19)
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ32_HTML.gif
(2.20)
Therefore, taking http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq56_HTML.gif appropriately small, we conclude from (2.17)–(2.20) that
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ33_HTML.gif
(2.21)
where, combined with the fact that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq57_HTML.gif due to Lemma 2.1, we obtain the first part of Lemma 2.4 by applying Gronwall's lemma and then Sobolev's inequality. Similarly, multiplying ((1.4) by http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq58_HTML.gif and integrating the resulting equation over http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq59_HTML.gif , we get that
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ34_HTML.gif
(2.22)
where we have also used Cauchy-Schwarz's inequality. Integration of (2.22) over http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq60_HTML.gif gives
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ35_HTML.gif
(2.23)

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.

Notice that (1.3), ((1.4) can be written as follows:
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ36_HTML.gif
(2.24)
where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq61_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq62_HTML.gif . Thus, by Lemmas 2.1–2.4, we see that
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ37_HTML.gif
(2.25)
which immediately implies
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ38_HTML.gif
(2.26)

Hence, we have the following lemma.

Lemma 2.5.

There exists a positive constant http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq63_HTML.gif , such that
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ39_HTML.gif
(2.27)

where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq64_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq65_HTML.gif .

To prove the uniqueness of strong solutions, we still need the following estimates.

Lemma 2.6.

The pressure http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq66_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq67_HTML.gif . Furthermore, if the compatibility conditions (1.9), (1.10) hold, then
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ40_HTML.gif
(2.28)

Proof.

It follows from the continuity equation (1.2) that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq68_HTML.gif satisfies
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ41_HTML.gif
(2.29)
which, differentiated with respect to http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq69_HTML.gif , leads to
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ42_HTML.gif
(2.30)
Multiplying the above equation by http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq70_HTML.gif and integrating over http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq71_HTML.gif , we deduce that
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ43_HTML.gif
(2.31)
where we have used the inequality
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ44_HTML.gif
(2.32)
which follows from the definition of http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq72_HTML.gif . Therefore, applying the previous Lemmas 2.1–2.5 and Gronwall's lemma, one has
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ45_HTML.gif
(2.33)

which proves the first part of the lemma.

We are now in a position to prove the second part. We first derive the estimate for the longitudinal velocity http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq73_HTML.gif . To this end, we firstly rewrite (1.3) as
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ46_HTML.gif
(2.34)
Differentiation of (2.34) with respect to http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq74_HTML.gif gives
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ47_HTML.gif
(2.35)
which, multiplied by http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq75_HTML.gif and integrated by parts over http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq76_HTML.gif , yields
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ48_HTML.gif
(2.36)

On the other hand, by virtue of (1.2) we have

http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ49_HTML.gif
(2.37)
from which and (2.36) we see that
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ50_HTML.gif
(2.38)
Using the previous lemmas and Young's inequality, we can estimate each term on the right-hand side of (2.38) as follows with a small positive constant http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq77_HTML.gif :
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ51_HTML.gif
(2.39)
Putting the above estimates into (2.38) and taking http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq78_HTML.gif sufficiently small, we arrive at
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ52_HTML.gif
(2.40)
so that, using the relation between http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq79_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq80_HTML.gif again, one infers from (2.40) that
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ53_HTML.gif
(2.41)
where the first term on the right-hand side of (2.41) is integrable on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq81_HTML.gif due to the previous lemmas. Thus, integrating (2.41) over http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq82_HTML.gif , we deduce from (1.3) that
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ54_HTML.gif
(2.42)
Letting http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq83_HTML.gif and using the compatibility condition (1.9), we easily obtain from (2.42) that
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ55_HTML.gif
(2.43)
which, together with http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq84_HTML.gif and Gronwall's lemma, immediately yields
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ56_HTML.gif
(2.44)

In a same manner as that in the derivation of (2.44), we can show the analogous estimate for the transverse velocity http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq85_HTML.gif by using the previous lemmas, (2.44), and the compatibility condition (1.10) as well. Thus, we complete the proof of Lemma 2.6.

Remark 2.7.

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

In this section, we prove the global existence of strong solutions to the problem (1.2)–(1.7) by applying the a priori estimates given in the previous section. As usual, we first mollify the initial data to get the existence of smooth approximate solutions. For this purpose, we choose the smooth approximate functions http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq86_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq87_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ57_HTML.gif
(3.1)
Let http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq88_HTML.gif , satisfying http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq89_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq90_HTML.gif , be the unique solution to the boundary value problems
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ58_HTML.gif
(3.2)
respectively, where
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ59_HTML.gif
(3.3)
Thus, with the regularized initial data http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq91_HTML.gif satisfying the compatibility conditions as above, we can follow the similar arguments as in [3, 9, 14] (because of http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq92_HTML.gif ) to show that the problems (1.2)–(1.7) admit a global strong solution http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq93_HTML.gif , which satisfies
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ60_HTML.gif
(3.4)
Applying the a priori estimates obtained in the previous section, we conclude that the approximate solution http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq94_HTML.gif satisfies
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ61_HTML.gif
(3.5)
where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq95_HTML.gif depends on the norms of initial data given in Theorem 1.1, but not on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq96_HTML.gif . With the help of (1.2)–(1.5) and (3.5), it is easy to see that
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ62_HTML.gif
(3.6)
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ63_HTML.gif
(3.7)
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ64_HTML.gif
(3.8)

By the uniform in http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq97_HTML.gif bounds given in (3.5)–(3.8) we conclude that there exists a subsequence of http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq98_HTML.gif which converges to a strong solution http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq99_HTML.gif 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 [19] for the isentropic compressible Navier-Stokes equations.

Theorem 4.1.

Let http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq100_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq101_HTML.gif be global solutions to problems (1.2)–(1.7) with initial data http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq102_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq103_HTML.gif , respectively. If http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq104_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq105_HTML.gif satisfy the regularity given in Theorem 1.1, then for any http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq106_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ65_HTML.gif
(4.1)

for some http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq107_HTML.gif . Here http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq108_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq109_HTML.gif .

Proof.

From the continuity equation it follows that
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ66_HTML.gif
(4.2)
Multiplying this by http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq110_HTML.gif and then integrating over http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq111_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ67_HTML.gif
(4.3)
where we have used the inequality http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq112_HTML.gif . Thus, by Cauchy-Schwarz's inequality, one has
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ68_HTML.gif
(4.4)

where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq113_HTML.gif .

By virtue of the equations satisfied by http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq114_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq115_HTML.gif , we easily deduce that
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ69_HTML.gif
(4.5)
which, multiplied by http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq116_HTML.gif and integrated by parts over http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq117_HTML.gif , gives
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ70_HTML.gif
(4.6)
so that we have by using Sobolev's inequality and Cauchy-Schwarz's inequality that
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ71_HTML.gif
(4.7)

where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq118_HTML.gif .

Proceeding the similar argument as that in the derivation of (4.7), we also have
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ72_HTML.gif
(4.8)

where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq119_HTML.gif .

Furthermore, it follows from the equations for the magnetic fields http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq120_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq121_HTML.gif that
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ73_HTML.gif
(4.9)
which, multiplied by http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq122_HTML.gif and integrated by parts over http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq123_HTML.gif , gives
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ74_HTML.gif
(4.10)

where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq124_HTML.gif .

Finally, from the continuity equations for http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq125_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq126_HTML.gif , it is easy to see that
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ75_HTML.gif
(4.11)
and hence we obtain in a manner similar to the derivation of (4.4) that
http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ76_HTML.gif
(4.12)

where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq127_HTML.gif .

Summing up (4.4)–(4.12) and choosing http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq128_HTML.gif appropriately small, we obtain Theorem 4.1 with http://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq129_HTML.gif by applying Gronwall's lemma.

Declarations

Acknowledgment

This work is partly supported by NSFC (Grant no. 10501037 and no. 10601008).

Authors’ Affiliations

(1)
School of Mathematical Sciences, Xiamen University

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© Jianwen Zhang. 2008

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.