具有不规则扩散的可压缩流体力学方程中的数学问题

This project will be devoted to study some equations from compressible fluid mechanics with fractional diffusion（also known as anomalous diffusion） term and corresponding inviscid problems. The fractional diffusion is more meaningful than the usual diffusion, it comes from Markov process with the rotation invariant in the stochastic process, and can be characterized by nonlocal pseudodifferential operator, which makes our problems become more difficult. We mainly discuss their well-posedness, including to blow-up criterion of local smooth solution, existence,uniqueness and long time behavior of global solution, regularity of global weak solutions and the inviscid limited problems etc., especially, we shall also look for the relation between the order of fractional Laplacian operator and well-posedness of the above problems, furthermore, we shall deeply study this class of operators, so that we can show some properties of solution to the corresponding inviscid problems and find different and similar points about the two classes of equations. We shall use the frequency localization technique, for instance of Littlewood-Paley decomposition (Bony paraproduct decomposition) and pseudodiffe- rential theory, and combine with energy methods, compactness method, operator semigroup theory and functional spaces theory etc. to consider our problems. We hope that our work of this project can be helpful and referencable to the diffusion equations with the fractional laplacian operator, inviscid equations, other fluid mechanics systems and operator theory.

本项目致力于研究含有分数阶扩散（又称不规则扩散）的可压缩流体力学方程以及它所对应的无粘性方程。分数阶扩散与通常的扩散相比更有意义，它来自随机过程中旋转不变的Markov过程,并可用非局部的拟微分算子来刻画它，这使得此类问题的研究变得更加困难。主要将讨论这两类问题的适定性，包括局部光滑解的Blow-up机制，全局弱解的正则性、全局解的存在唯一性与长时间性态以及无粘性极限等问题, 着重探讨分数阶Laplace算子的阶数对问题适定性的影响，并通过对该算子的讨论去探寻相应的无粘性方程解的性质并比较它们之间的差异和联系。主要利用频率局部化技术，如Littlewood-Paley分解(Bony仿积分解)和拟微分算子理论，并结合能量方法、紧致性方法、算子半群以及函数空间理论等来研究此类问题。期望本项目的研究能给相应的具有分数阶扩散方程、无粘性方程以及其他的流体动力学方程和算子理论的研究提供帮助和参考。

本项目为了研究具有不规则耗散项的可压缩流体动力学问题适定性，讨论了大量的相应的不可压缩流体动力学方程的适定性问题。主要研究了分数阶Boussinesq方程，各项异性的Boussinesq方程以及扩散项依赖温度的Boussinesq方程的整体适定性， 2维情形的MHD方程的整体适定性，2维分数阶Burgers方程的长时间行为，粘性项同时依赖温度和剪切力的可压缩Navier-Stokes方程的整体适定性等。目前对于不可压分数阶Boussinesq方程的耗散指标做到最大范围，这对进一步讨论无粘的Boussinesq方程有着很重要的意义，并得到了一些重要的不等式以及有意思的理论结果。用求解色散方程的方法和框架给出了仅速度方程带有damping项的MHD方程小解的整体适定性，并给出了衰减估计，这一方法和框架对类似的一类问题都是适用的。