On the breakdown of 2D compressible Eulerian flows in bounded impermeable regions with corners

Abstract

We consider smooth solutions of the 2D compressible Euler equations with suitable external forces in impermeable domains with corners. If the corner angles are small enough, we obtain results which have the following corollary: under a mild condition on the equation of state and for suitable external forces, the solutions can be continued in time (with no loss of smoothness) as long as there is no accumulation of vorticity or velocity divergence or pressure gradient or entropy gradient. For some special (e.g. barotropic) flows, this formulation can be simplified. Our results in the small corner angles case correspond to similar results obtained by Chemin in all space (of any dimension), which contain in particular a version for compressible flows of the Beale-Kato-Majda and Ponce incompressible flows criteria. For larger corner angles, we give examples where our assumptions are satisfied but continuation in time is only possible with a loss of smoothness.