多孔介质Navier-Stokes方程组的适定性与解的衰减性研究

Fractional derivatives are strong modeling tools in the field of physics. Physicians give the fractional Newton viscosity law by a rough induction from fractional Taylor formula. In addition, they construct fractional Darcy law through fractional Newton viscosity law and deeply studied the diffusion phenomenon in porous medium. The first goal of this project is to derive fractional Newton viscosity law rigorously from mathematical perspective. In addition, using conservation of mass, momentum and energy we plan to derive porous medium Navier-Stokes equations based on fractional derivative operator. Through this study, we plan to give a reasonable physical explanation about the fractional incompressible fluid equations which is an active research area. The second goal of this project is to construct the well-posedness and optimal time decay theory for compressible Navier-Stokes equations with fractional derivative operator. During this study we may propose a new form of effect viscous flux, extend the regularity theory about fractional parabolic equations and using split frequency idea, Littlewood-Paley decomposition, Besov space technique. In summary, through this project, we give mathematical foundations for physical study and applications about porous medium compressible Navier-Stokes equations.

在物理学领域，分数阶算子是一种新的有力建模工具，基于分数阶算子的Taylor公式以及粗略的推导，物理学家给出了分数阶Newton粘性法则，并基于分数阶Newton粘性法则建立了分数阶Darcy法则，从而深入地研究了多孔介质中的扩散现象。本项目一方面从数学的角度严格的推导分数阶Newton粘性法则，并运用质量、动量、能量守恒定律严格的推导基于分数阶算子的多孔介质Navier-Stokes方程组，从而为现有的关于分数次不可压缩Navier-Stokes方程组的研究提供相应的物理解释。另一方面，针对多孔介质等熵Navier-Stokes方程组提出相应的有效黏性流，并结合分数次抛物方程的正则性理论，Littlewood-Paley分解，Besov空间，分频估计等思想建立起多孔介质等熵Navier-Stokes方程组的适定性，解的最优衰减性理论。从而为这一更为广泛的模型的物理，工程研究提供数学理论。

本项目研究了可压缩Navier-Stokes方程组、磁流体方程组、短波-长波交互作用下的磁流体方程组在小初值假设下的整体解以及解的最优衰减速率。针对这些方程，证明了当初始密度、速度、温度等在一定的临界空间中靠近稳态解时，方程组的局部解可以延拓至整体解，借助于线性系统的Fourier分析、负正则指标Besov空间中的能量估计给出了非线性方程组解的最优衰减速率估计。本项目进一步在此基础上研究了带有分数阶算子的方程的适定性、正则性以及极大值原理等，并利用这些结果研究了逆源、逆系数等反问题的唯一性，贝叶斯反演适定性。本项目是问题驱动的应用数学研究，其研究成果有望为流体方程，分数阶方程的应用提供理论支撑与依据。