Rough solutions of the 3-D compressible Euler equations


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 $(v,\varrho, \mathscr{w}) \in H^s\times H^s\times H^{s’}$, $2 < s'< s$.  The result extends the  sharp  results of   Smith-Tataru  and of Wang,  established in the irrotational case, i.e. $\mathscr{w}=0$, 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  $\mathscr{w}\in H^s$, $s>3/2$ and  fails for $s\le 3/2$.  We therefore conjecture that the optimal result  should be  $(v,\varrho, \mathscr{w}) \in H^s\times H^s\times H^{s’}$, $s>2$, $s’>\frac{3}{2}$. 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 acoustical spacetime.


Qian Wang

Mathematical Institute, University of Oxford, Oxford, United Kingdom