具有粗糙初值的Navier-Stokes方程的适定性

Navier-Stokes system is one of the fundamental systems in fluid mechanics. A question of longstanding interest to mathematicians and physicists is, given an initial data to the system, whether generates a unique global solution: the so-called “well-posedness”. We plan in this program to study the case that viscous coefficient depends on density, and initial density is non-smooth. This case is of great physical meaning, and has extensive use. The reference of this problem is enormous, but in most of them, density needs at least one order derivative, or the regularity of velocity is not critical. In this program, we further investigate the case where the density is just bounded and the velocity is with zero or negative regularity. The compressible case is also concerned. Velocity with negative regularity corresponds to the solution with high frequency oscillation, which has abundant physical phenomenon. And this project may help to understand these phenomenon.

Navier-Stokes方程是流体力学中最基本的方程组之一，其解的适定性问题一直是数学与力学界的研究热点。本项目计划研究该系统当粘性系数依赖于密度，且初始密度不光滑时的适定性问题，其具有广泛的物理与现实意义。粘性系数依赖于密度的研究成果目前为数不少，但已有相关结果中大部分的密度至少需要有一阶导数，速度正则性也不临界。本项目将在此基础上，进一步研究该情形密度仅为有界，且速度只有零阶或负导数的情况，并将讨论和研究可压情况。速度具有负导数对应于物理中的高频振荡解，本项目的研究能为理解众多物理现象提供参考。

流体力学方程组中初始密度正则性不足的问题在空气动力学、海洋运动等理论及工程等方面具有广泛的应用，一直以来时国际偏微分方程领域研究热点问题之一，然而其数学理论结果比较少。. 本项目在流体力学方程组低正则性问题取得了一系列成果，包括：证明了非齐次不可压缩Boussinesq方程大解的全局适定性；得到了各类可压缩流体方程的大解收敛到平衡态的最优速率；构造了三维可压缩Navier-Stokes方程在不可压缩流附近的一类特殊大解。. 本项目的研究有助于深刻理解众多物理现象，为未来开展探索、实验提供数学理论指导。