非等熵可压缩 Navier-Stokes 方程的零耗散极限问题

Euler equations, Navier-Stokes equations and etc. are a class of all-important nonlinear partial differential equations in fluid dynamics, because of the wide applications in Aeronautics and Astronautics, Astrophysics, Geological mechanics, Weather forcasting, Oil and gas detection, Image processing and etc, the research of the theories and applications of these equations has been one of the most popular topics in the fields of partial differential equations. This project will study the zero dissipation limit problem of the non-isentropic compressible Navier-Stokes equations with the composite wave initial data. Previous related results focused on the "smooth initial perturbation", but few results on the problem of "non-smooth initial perturbation", so this project is a great challenge. We will use the semi-difference method to get the local existence, regularity and jump estimation of the solution; apply energy estimate, pointwise estimate as well as our research methods used in studying the isentropic case to obtain the accurate description of the new wave phenomenon caused by the "non-smooth initial perturbation" and the uniform estimates of the soution (independently of time) ; apply mathematical tools coming from the Riemann problem to the nonlinear hyperbolic evolution equations to establish the stability theory of wave interactions.

Euler 方程，Navier-Stokes 方程等是流体力学中一类非常重要的非线性偏微分方程，由于其在航空航天、天体物理、地质力学、天气预报、油气探测和图像处理等诸多方面中的广泛应用，这些方程的理论及其应用的研究一直是偏微分方程领域同行研究的热点。本项目将研究一维非等熵可压缩 Navier-Stokes 方程具有复合波初值的零耗散极限问题。以前相关的结果主要集中在“光滑初始扰动”问题上，关于“非光滑初始扰动”问题的结果却很少，所以本项目的研究具有很大的挑战。我们将利用半差分方法得到解的局部存在性、 正则性及跳跃估计；利用能量估计、逐点估计以及我们研究等熵情形时所用的方法有机结合起来得到由非光滑初始扰动引起的新的波现象的精确描述和解的一致（独立于时间）估计；利用发展非线性双曲方程 Riemann 问题的数学工具来建立波干扰的稳定性理论。

Navier-Stokes方程的粘性消失极限问题一直是偏微分方程领域同行研究的热点。本项目的研究成果主要包含两个方面的内容: 1. 等熵 Navier-Stokes 方程具有组合激波初值的粘性消失极限问题; 2. 非等熵 Navier-Stokes 方程具有组合激波初值的粘性消失极限问题。项目得到了等熵和非等熵 Navier-Stokes 方程组合激波初值问题解的整体存在性、正则性、跳跃不连续关于粘性的衰减性质及波之间的干扰估计，最后证明了当粘性消失时，Navier-Stokes 方程的解一致收敛到相应的 Euler 方程的 Riemann 解。