高维激波以及非线性守恒律方程的低正则解

There are important theoretical and realistic meanings for the studies on multidimensional shock waves, low regularity solutions of nonlinear conservation law equations and related singularity structures. The typical nonlinear conservation laws include the compressible/incompressible Euler/Navier-Stokes equations. For the compressible/incompressible Euler systems, when the classical solutions blow up, people generally think that there will appear shock waves for the compressible fluids and appear turbulences for the incompressible fluids, respectively. It is always a difficult topic for the study on blowup mechanisms of classical solutions to compressible Euler equation, there are few results on the formations and developments of multidimensional shock waves, many specialists are also extensively concerned with the problems of related piecewise smooth solutions. Meanwhile, the well-posedness problems of piecewise smooth solutions (including the vertex sheets, free boundaries and so on) also appear in the incompressible Euler equation and compressible/incompressible Navier-Stokes equation. The studies on these problems are also closely related to degenerate elliptic equations, degenerate hyperbolic equations, numerical analysis of partial differential equations, moreover, there are deep relations between the compressible equation and incompressible equation. Through these studies, we intend to make essential progresses in the research areas of multidimensional shock waves and low regularity solutions of nonlinear partial differential equations.

研究高维激波、非线性守恒律方程低正则解和解的奇性结构有着重要的理论价值和实际意义。非线性守恒律方程的典型代表是可压缩/不可压缩流体的无粘Euler方程及有粘Navier-Stokes方程。对于无粘的Euler方程，当经典解出现奇性时，一般的观点是：可压缩流将产生激波，不可压缩流将产生湍流。关于可压缩Euler方程经典解爆破机制的研究是高维双曲方程理论中的困难课题，与之相应的高维激波形成和构造仍未取得实质结果，围绕此问题及相关分片光滑解（包括真空界面）的研究一直是人们高度关注的，同时分片光滑解（包括窝度片、自由界面等）的适定性也出现在不可压缩Euler方程及有粘/无粘Navier-Stokes方程中。这些问题的研究与退化椭圆、退化双曲、偏微分方程数值解等紧密相关，而且可压/不可压流方程之间也存在深刻联系。通过该类问题的研究，希望推进有关高维激波及非线性方程低正则解适定性的理论创新。

非线性守恒律方程的典型代表是可压缩/不可压缩流体的无粘Euler方程及有粘Navier-Stokes方程。对于无粘的Euler方程，当经典解出现奇性时，可压缩流将产生激波，与之相应的高维激波形成和构造仍未取得实质结果，围绕此问题及相关分片光滑解（包括真空界面）的研究一直是人们高度关注的，同时分片光滑解的适定性也出现在不可压缩Euler方程及有粘/无粘Navier-Stokes方程中。通过该类问题的研究，我们推进了有关高维激波及非线性方程低正则解适定性的理论研究，并取得了系列的创新成果。