稳态可压的磁流体力学方程的弱解存在性

Magnetohydrodynamic(MHD) is the study of electrically conductive fluid motion in the electromagnetic field.MHD equations is governed by the compressible Navier Stokes equations and the Maxwell equations of the magnetic field.For this type of coupling problems many of which are non-linear.The compactness question of weak solutions to the magnetohydrodynamic equations for the viscous,compressible,heat conducting fluids has been one of the most concern problems.In this issue,we mainly concerned with the study of the existence of weak solutions for the boundary value problems in a bounded three-dimensional domain.It is assumed that the pressure is a function of density and temperature,to give the slip and no-slip boundary condition for the velocity,and for the temperature given Newton boundary conditions (hot soliton model).Exsistence of weak solution is determined by dealing with the following problems:firstly,because the research question is steady flow problem,we need get the priori estimate of the density,velocity,temperature and magnetic field in the model by constructing energy estimate.Secondly,the system is non-linear,we get the higher regularity of approximation solution by using iterative methods.finally,taking the limits for the approximation system,then we acquire the existence of weak solutions for non-isentropic steady compressible MHD equations.In this process,due to restrictions of related estimated we need to attention energy equation.

磁流体是研究导电流体在电磁场中的运动, 因而磁流体力学方程是可压的Navier-Stokes方程和磁场中的Maxwell方程的耦合. 对于这类耦合问题很多都是非线性的.在本课题中, 将会考虑一直备受关注的三维有界区域内粘性可压热传导磁流体力学方程的弱解的紧性问题. 这里假定压力是密度和温度的函数, 对于速度赋予滑移和无滑移边界条件, 而对于温度赋予牛顿边值条件(热孤立子模型). 对上述边值问题将得到弱解的存在性. 为此我们需要解决以下三方面的问题: 首先, 由于所研究的问题是稳态流问题, 所以需要通过构造能量估计来得到此模型中关于密度, 速度, 温度和磁场强度的先验估计. 其次, 研究体系是非线性的, 需要用迭代法理论将其转化为线性问题, 进而得到逼近解的更高正则性. 最后, 我们需要对逼近体系取极限进而得到弱解的存在性. 在这个过程中由于相关量的估计限制需要特别关注能量方程.

近年来，由于科学技术的飞速发展，流体力学和其他学科相互渗透，形成了一系列交叉学科，磁流体力学就是现在研究最广泛的一类，它是结合经典流体力学和电动力学的方法研究导电流体和磁场相互作用的。它的基本方程是流体力学中的纳维-斯托克斯方程和电动力学中的麦克斯韦方程组。这类微分方程具有很强的非线性性，所以研究这类方程的解的存在性具有实际意义。.本项目主要研究三维有界区域内粘性可压热传导磁流体力学方程的弱解的紧性问题。目前，在本项目的支持下，我们获得两个结果：.1.利用稳态流的相关理论，研究对于速度赋予狄利克雷边界条件，温度赋予牛顿边值条件是磁流体力学的弱解存在性。.2.应用上述理论研究当速度赋予无滑移边界条件时磁流体力学方程的弱解存在性。.这两个结果被放在一篇论文中研究并且发表在高水平SCI杂志:.Qiu Meng. Existence of weak solutions to steady compressible magnetohydrodynamic equations. J.Math.Anal.Appl.447(2017)1027-1050.