非等熵可压缩磁流体方程组解的研究

Compressible magnetohydrodynamic equations, which have deep physical background and practical significance, are the fundamental equations in the magnetic fluid mechanics. The well-posednes of strong solutions in multi-dimension is a hot issue in partial differential equations, and many famous mathematicians have made great contribution in this direction. When the initial data are close to a non-vacuum equilibrium, Japanese mathematician Kawashima obtained the global existence and uniqueness of classical solutions. However, his method does not work in the case of vacuum state, we need to some new theories and techniques. Our project will deal with global well-posedness and large time asymptotic behavior of strong solutions to the non-isentropic compressible magnetohydrodynamic equations. Details are as follows: I、global existence and large time asymptotic behavior of strong solutions to the two-dimensional Cauchy problem of non-isentropic compressible magnetohydrodynamic equations with vacuum as far-field density; II、global existence and large time asymptotic behavior of strong solutions to the three-dimensional Cauchy problem of non-isentropic compressible magnetohydrodynamic equations with vacuum as far-field density.

可压缩磁流体方程组是磁流体力学中的一个基本偏微分方程组，有着很强的物理背景和实际意义。高维可压缩磁流体方程组解的适定性是数学界一直关心的问题，也是近来国内外偏微分方程研究的一个热点问题之一。当初值在非真空平衡态附近扰动时，日本数学家Kawashima得到了非等熵可压缩磁流体方程组整体光滑解，但他的方法不再适用于含真空情形，这需要新的理论和技巧。本项目拟研究非等熵可压缩磁流体方程组强解的整体适定性以及解的长时间行为。具体如下：1、密度和温度在无穷远状态为零且初始密度允许带真空的二维非等熵可压缩磁流体方程组柯西问题强解的整体存在唯一性和解的长时间行为；2、密度和温度在无穷远状态为零且初始密度允许带真空的三维非等熵可压缩磁流体方程组柯西问题强解的整体存在唯一性和解的长时间行为。

项目负责人按照申请书中的研究计划，积极开展原创性研究。在项目执行期间，运用MHD方程组的结构、能量方法、奇异加权估计、拉格朗日坐标等工具取得了如下的研究成果：.(1)在非等熵可压缩MHD方程组解的适定性方面，项目负责人建立了二维含真空的零磁扩散Cauchy问题强解的存在性。此外，还与长春师范大学的刘洋合作，证明了三维远场含真空的Cauchy问题强解的整体适定性和熵有界解的存在性。.(2)在非均匀热传导MHD方程组强解的整体适定性方面，项目负责人证明了二维远场不含真空的大初值解的整体适定性,并得到了三维有界区域上解的整体存在性和长时间行为。.上述研究结果部分发表在Calculus of Variations and Partial Differential Equations、Journal of Differential Equations等国际知名学术期刊上。