On Classical Solutions of the Compressible Magnetohydrodynamic Equations with Vacuum
On Classical Solutions of the Compressible Magnetohydrodynamic Equations with Vacuum
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DOI：
10.1137/14095265x
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发表时间：
2014-01
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影响因子：
--
通讯作者：
Shengguo Zhu
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文献类型：
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作者：
Shengguo Zhu
本文考虑具有无穷大电导率的三维可压缩等熵MHD方程。首先建立了在初始数据任意大、包含真空且满足一定初始层相容条件时局部经典解的存在性；初始质量密度不需要被限定在远离零的范围内，并且可以在某个开集中消失。此外，我们证明了速度梯度变形张量的$L^\infty$范数控制了经典（或强）解的可能爆破（见\cite{olga}\cite{zx}），这意味着如果可压缩MHD方程的解最初是规则的，但在稍后的某个时间失去其规则性，那么奇点的形成一定是由于在临界时间接近时失去变形张量的界而引起的。我们的结果（见（1.12））与庞塞对$3$ -D不可压缩欧拉方程的判据\cite{pc}和黄立新对$3$ -D可压缩Navier-stokes方程的爆破判据\cite{hup}相同。
In this paper, we consider the 3-D compressible isentropic MHD equations with infinity electric conductivity. The existence of unique local classical solutions is firstly established when the initial data is arbitrarily large, contains vacuum and satisfies some initial layer compatibility condition. The initial mass density needs not be bounded away from zero and may vanish in some open set. Moreover, we prove that the $L^\infty$ norm of the deformation tensor of velocity gradients controls the possible blow-up (see \cite{olga}\cite{zx}) for classical (or strong) solutions, which means that if a solution of the compressible MHD equations is initially regular and loses its regularity at some later time, then the formation of singularity must be caused by the losing the bound of the deformtion tensor as the critical time approches. Our result (see (1.12) is the same as Ponce's criterion for $3$-D incompressible Euler equations \cite{pc} and Huang-Li-Xin's blow-up criterion for the $3$-D compressible Navier-stokes equations \cite{hup}.