不可压磁流体力学方程组和一类广义Camassa-Holm方程的若干问题研究

In this project, we mainly study the Cauchy problem of the the incopressible magnetohydrodynamic equations and a genelazied Camassa-Holm equation. Because these two kinds of partial differential equations describe the motion of the plasma and shallow water waves respectively, they have important physical significance. For the incopressible magnetohydrodynamic equations, we mainly study the invisic limit for strong solutions in Besov spaces; continuous dependece property when there is no magnetic dissipation. For the genelazied Camassa-Holm eqaution, we mainly study non-uniformly continuous dependence property of strong solutions; the global existence and blowup phenomenon of strong solutions; existence and uniqueness of global weak solutions. The above problems have not yet be fully solved. Hence, in this project we will apply the harmonic analysis and weak convergence method to study these problems, and obtain some further theoretical results.

本项目拟研究不可压磁流体力学方程组和一类广义的Camassa-Holm方程柯西问题。这两类偏微分方程分别描述等离子体和浅水波的运动规律，有重要物理意义。对于不可压磁流体力学方程组，主要研究其在Besov空间的粘性极限；无磁场耗散时解对初值的连续依赖性。对于广义的Camassa-Holm方程，主要研究强解的非一致连续依赖性；强解的整体存在性和爆破现象；整体弱解的存在唯一性等问题。上述问题在这两个模型中仍未完全解决。因此本项目将运用调和分析和弱收敛方法来研究这些内容，取得进一步的理论成果。

本项目研究的不可压磁流体力学模型和Camassa-Holm方程是偏微分方程中所关注的问题，且具有重要的物理意义。本项目主要对磁流体力学模型和Camassa-Holm方程及其相关模型进行了广泛和深入的研究。具体针对磁流体力学方程，带Hall项的磁流体力学方程，Camassa-Holm方程，Novikov方程等多个新方程大初值解的整体存在性，强解的非一致连续依赖性，不适定性等问题进行了细致深入的研究。所得的研究成果在数学理论上对大初值解的整体存在性和解对初值的非一致连续依赖性的认识有较大帮助。相应的研究成果发表JDE、JMFM等SCI期刊论文中达12篇。