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

Add to your list(s) Download to your calendar using vCal

• Qian Wang (Oxford)
• Tuesday 25 February 2020, 14:00-15:00
• WATN-R17 18, Watson Building.

If you have a question about this talk, please contact Hong Duong.

I will talk about my 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 $(v,\varrho, \omega) \in H^{s\times H}s\times H^{$, $2 2$. 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.

This talk is part of the Analysis seminar series.

This talk is included in these lists:

• Analysis seminar
• School of Mathematics events
• WATN-R17 18, Watson Building

Note that ex-directory lists are not shown.