学术报告

Rough solutions of the $3$-D compressible Euler equations

发布人:发布时间: 2019-12-27

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

报告人:汪倩 教授 Mathematical Institute, University of Oxford)

时间:2020年1月3日(周五) 上午10:00-11:00

地点:数学系西大楼一楼108报告厅

摘要:I will talk about my recent work (arxiv:1911.05038). We prove the local-in-time well-posedness for 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. 

邀请人:李军 老师