讲座题目：On continuous solution of compressible Euler equations

主办单位：三峡数学研究中心

报告专家：王天怡副教授

报告时间：2022年4月12日 16:00

报告地点：腾讯会议（ID: 113-992-934）

专家简介：王天怡，武汉理工大学理学院数学系，副教授。2014年在中国科学院数学与系统科学研究院获得理学博士学位。在博士阶段受国家留学基金委的资助，到英国牛津大学开展联合培养。毕业后赴香港中文大学和意大利萨索科学研究所进行博士后研究。主要研究方向：非线性偏微分方程、流体力学中的数学理论。论文发表在Advances in Mathematics，Archive for Rational Mechanics and Analysis等。主持国家自然科学基金项目2项，入选国家高层次青年人才。

报告摘要：In this talk, we will consider two progresses on the continuous solutions of compressible Euler equations. For the multiple space dimensional case, we prove the break-down of classical solutions with a large class of initial data by tracking the propagation of radially symmetric expanding wave including compression. The singularity formation is corresponding to the finite time shock formation. We also constructed two types of continuous solution with uniform bounds on velocity and density. For the vanishing pressure limit, which formulated as small parameter $\epsilon$ goes to $0$. Due to the characteristics are degenerated in the limiting process, the resonance may cause the mass concentration. It is shown that in the pressure vanishing process, for the isentropic Euler equations, the continuous solutions with compressive initial data converge to the mass concentration solution of pressureless Euler equations, and with rarefaction initial data converge to the continuous solutions globally. It is worth to point out: $u$ converges in $C^1$, while \rho$ converges in $C^0$, due the structure of pressureless Euler equations. To handle the blowup of density $\rho$ and spatial derivatives of velocity $u$, a new level set argument is introduced. Furthermore, we consider the convergence rate respect to $\epsilon$, both $u$ and the area of characteristic triangle are $\epsilon$ order, while the rates of $\rho$ and $u_x$ depend on the further regularity of the initial data of $u$. These are the joint work with Hong Cai, Geng Chen, and Wen-Jian Peng.