退化可压缩Navier-Stokes方程组的理论研究

Theoretical Research on Degenerate Compressible Navier-Stokes Equations

可压缩Navier-Stokes方程组(CNS)因其重要的物理背景和数学理论的挑战性,一直是偏微分方程研究的核心领域之一。在气体动力学中,CNS可通过Chapman-Enskog分解,从Boltzmann方程推导而来,且黏性和热传导系数均为绝对温度的函数,从而导致CNS的结构在真空附近会出现强的退化。在等熵情形,我们通过清晰地分析退化CNS系统的数学结构,且按照退化性的强弱,把动量方程分成奇异、正常和退化抛物方程组3种类型,从而分别找到在真空附近控制流体速度行为的方法,并通过引入合适的奇异—退化加权估计,系统地建立该方程组高维大初值正则解的局部适定性理论及与其相匹配的奇异性理论。本文聚焦于该退化系统解的存在性理论的发展,并探讨由此引出的一些相关公开问题。

The compressible Navier-Stokes equations(CNS)have always been one of the core fields in the research of partial differential equations due to their important physical background and theoretical challenges.In the theory of gas dynamics,the CNS can be derived from the Boltzmann equation by Chapman-Enskog expansion,and both the viscosity and the heat conductivity coefficients are functions of the absolute temperature,which leads to strong degeneracies of CNS’s structure near the vacuum.In the isentropic case,we clearly analyze the mathematical structure of the degenerate CNS system and classify the momentum equations into three types,namely,singular,normal,and degenerate parabolic equations according to the strength of the degeneracy,so as to find a way to control the behavior of fluid velocity near the vacuum in different cases.Moreover,by introducing suitable singular-degenerate weighted estimates,the local well-posedness theory of multi-dimensional regular solutions of the equations with large data and the matched singularity theory are systematically established.This paper focuses on the development of the existence theory of solutions to this degenerate system and explores some of the related open problems that arise from it.