题目：Rough solutions of the 3-D compressible Euler equations

报告人：汪倩 (牛津大学) 

时间：5月20日周四晚上19：30-20:30

地点：Zoom会议  ID：7361907370  密码:122595

摘要：I will talk about my work arxiv:1911.05038. We prove the local-in-time existence the solution of the compressible Euler equations in $3$-D, for the Cauchy data of the velocity, density and vorticity $(v,\varrho, \omega) \in H^s\times H^s\times H^{s'}$, $2<s'<s$.  The result extends the  sharp  result of   Smith-Tataru and Wang,  established in the irrotational case, i.e  $ \omega=0$, which  is  known to be optimal for $s>2$. At the opposite extreme, in the incompressible case, i.e. with a constant density,  the result is known to  hold for $ \omega\in H^s$, $s>3/2$ and  fails for $s\le 3/2$, see the work of Bourgain-Li.  It is thus natural to conjecture that the optimal result should be  $(v,\varrho, \omega) \in H^s\times H^s\times H^{s'}$, $s>2, \, s'>\frac{3}{2}$. We view our work here as an important step in proving the  conjecture. The  main difficulty in establishing sharp well-posedness results for general compressible Euler flow is  due to the highly nontrivial interaction between  the  sound waves, governed by quasilinear wave equations, and vorticity which is transported by the flow. To overcome this difficulty, we separate the dispersive part of sound wave from the transported part, and gain regularity significantly by exploiting the nonlinear structure of the system and the geometric structures of the acoustic spacetime.