全部

Well-posedness of Navier-Stokes equations with vacuum but without compatibility conditions

  • 演讲者:李进开(华南师范大学)

  • 时间:2021-12-14 14:30-15:30

  • 地点:腾讯会议 ID 568 124 215,密码 888888

Abstract

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 $q\in(3,6)$, $u_0\in H^1$, 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) carrying out 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 uniqueness is proved in the Lagrangian coordinates first and then transformed back to the Euler coordinates.

Short bio
李进开教授 2013 年博士毕业于香港中文大学数学科学研究所,导师为辛周平教授,2013 至 2016 于以色列威兹曼科学研究所(Weizmann Institute of Sciences)从事博士后研究工作,合作导师为 Edriss S. Titi 教授。2016 至 2018 在香港中文大学数学系任研究助理教授。2018 年至今在华南师范大学工作。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(2020 ICCM Best Paper Award)以及“第二届中国科协优秀科技论文”奖。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 30 余篇。