Open Access

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]):
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq1_HTML.gif denotes the density of the fluid, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq2_HTML.gif the longitudinal velocity, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq3_HTML.gif the transverse velocity, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq4_HTML.gif the transverse magnetic field, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq5_HTML.gif the temperature, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq6_HTML.gif the pressure, and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq7_HTML.gif the internal energy; https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq8_HTML.gif and https://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, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq10_HTML.gif is the magnetic viscosity, https://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 https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ2_HTML.gif
(1.2)
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ3_HTML.gif
(1.3)
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ4_HTML.gif
(1.4)
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ5_HTML.gif
(1.5)
where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq13_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq14_HTML.gif being the adiabatic exponent and https://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 https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ6_HTML.gif
(1.6)
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ7_HTML.gif
(1.7)

where the initial data https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq18_HTML.gif whenever https://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 https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq24_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq25_HTML.gif satisfy the regularity conditions:
https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ9_HTML.gif
(1.9)
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq27_HTML.gif ,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq28_HTML.gif denote the usual Sobolev space, and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq29_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq30_HTML.gif . For simplicity, we denote by https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq32_HTML.gif , and use the following abbreviation:
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq34_HTML.gif , and https://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
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq37_HTML.gif , one has
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ14_HTML.gif
(2.2)
where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq38_HTML.gif is the nonnegative function defined by
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq40_HTML.gif , https://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
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ16_HTML.gif
(2.4)
Set
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq42_HTML.gif satisfies
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ19_HTML.gif
(2.7)
which imply
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ20_HTML.gif
(2.8)
Letting https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq43_HTML.gif denote the material derivative and choosing https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq45_HTML.gif .

Lemma 2.3.

The magnetic field https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq46_HTML.gif satisfies the following estimates:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq47_HTML.gif and integrating over https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq48_HTML.gif , we have
https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ24_HTML.gif
(2.12)

Since https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq50_HTML.gif and integrate the resulting equation over https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq51_HTML.gif to deduce that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq52_HTML.gif :
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq53_HTML.gif and then integrating over https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq54_HTML.gif , by Young's inequality we obtain
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq55_HTML.gif , we see that
https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ30_HTML.gif
(2.18)
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ31_HTML.gif
(2.19)
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ32_HTML.gif
(2.20)
Therefore, taking https://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
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq58_HTML.gif and integrating the resulting equation over https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq59_HTML.gif , we get that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq60_HTML.gif gives
https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ36_HTML.gif
(2.24)
where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq61_HTML.gif and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ37_HTML.gif
(2.25)
which immediately implies
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq63_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ39_HTML.gif
(2.27)

where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq64_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq66_HTML.gif satisfies https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq68_HTML.gif satisfies
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq69_HTML.gif , leads to
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq70_HTML.gif and integrating over https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq71_HTML.gif , we deduce that
https://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
https://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 https://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
https://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 https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq74_HTML.gif gives
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ47_HTML.gif
(2.35)
which, multiplied by https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq75_HTML.gif and integrated by parts over https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq76_HTML.gif , yields
https://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

https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq77_HTML.gif :
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq78_HTML.gif sufficiently small, we arrive at
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq79_HTML.gif and https://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
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq82_HTML.gif , we deduce from (1.3) that
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ54_HTML.gif
(2.42)
Letting https://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
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ55_HTML.gif
(2.43)
which, together with https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq84_HTML.gif and Gronwall's lemma, immediately yields
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq86_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq87_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ57_HTML.gif
(3.1)
Let https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq88_HTML.gif , satisfying https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq89_HTML.gif in https://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
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ58_HTML.gif
(3.2)
respectively, where
https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq93_HTML.gif , which satisfies
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq94_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ61_HTML.gif
(3.5)
where https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ62_HTML.gif
(3.6)
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ63_HTML.gif
(3.7)
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ64_HTML.gif
(3.8)

By the uniform in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq98_HTML.gif which converges to a strong solution https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq100_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq102_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq103_HTML.gif , respectively. If https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq104_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq106_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ65_HTML.gif
(4.1)

for some https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq107_HTML.gif . Here https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq108_HTML.gif and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ66_HTML.gif
(4.2)
Multiplying this by https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq110_HTML.gif and then integrating over https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq111_HTML.gif , we have
https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ68_HTML.gif
(4.4)

where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq114_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq115_HTML.gif , we easily deduce that
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ69_HTML.gif
(4.5)
which, multiplied by https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq116_HTML.gif and integrated by parts over https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq117_HTML.gif , gives
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ71_HTML.gif
(4.7)

where https://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
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ72_HTML.gif
(4.8)

where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq120_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq121_HTML.gif that
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ73_HTML.gif
(4.9)
which, multiplied by https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq122_HTML.gif and integrated by parts over https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq123_HTML.gif , gives
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ74_HTML.gif
(4.10)

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

Finally, from the continuity equations for https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq125_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_IEq126_HTML.gif , it is easy to see that
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F735846/MediaObjects/13661_2007_Article_812_Equ76_HTML.gif
(4.12)

where https://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 https://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 https://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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Copyright

© 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.