报告题目：Local well-posedness of heat conductive compressible Navier-Stokes equations in the presence of vacuum without compatibility conditions

报 告 人：李进开教授（华南师范大学）

报告地点：数学研究中心528

报告时间：2024年4月6日  16:00

报告摘要：

In this talk, we consider the initial-boundary value problem to the heat conductive compressible Navier-Stokes equations. Local existence and uniqueness of strong solutions will be presented for any such initial data that the initial density $\rho_0$, velocity $u_0$, and temperature $\theta_0$ satisfy $rho_0\in W^{1, q}$, with $q\in(3,6)$, $u_0\in H^l$, and $\sqrt{\rho_0}\theta_0\in L^2$. The initial density is assumed to be only nonnegative and thus the initial vacuum is allowed. In addition to the necessary regularity assumptions, we do not require any initial compatibility conditions such as those proposed by Cho and Kim, which although are widely used in many previous works but put some inconvenient constraints on the initial data. Due to the weaker regularities of the initial data and the absence of the initial compatibility conditions, leading to weaker regularities of the solutions compared with those in the previous works, the uniqueness of solutions obtained in this talk does not follow from the arguments used in the existing literatures. Our proof of the uniqueness of solutions is based on the following new idea of two-stages argument: (i) showing that the difference of two solutions (or part of their components) with the same initial data is controlled by some power function of the time variable; (ii) carryingo ut some singular-in-time weighted energy differential inequalities fulfilling the structure of the Gr\"onwall inequality. The existence is established in the Euler coordinates, while the uniquenessis proved in the Lagrangian coordinates first and then transformed back to the Euler coordinates.

报告人简介：

李进开，华南师范大学数学科学学院院长，教授，博士生导师。2022年入选“国家高层次人才特殊支持计划”科技创新领军人才，2018年入选“国家海外高层次人才引进计划”青年项目，曾获得“2020世界华人数学家联盟最佳论文奖”金奖、“第二届中国科协优秀科技论文”奖，被授予第二十五届广东省青年五四奖章。目前已在包括CPAM, Adv. Math, JFA, ARMA, CPDE, SIMA等国际学术期刊上发表SCI论文40多篇。