The Shock Formation for 2D Isentropic Compressible Euler Equations with Damping

Abstract

In this paper, we investigate the blowup mechanism for the 2D isentropic compressible Euler equations with a damping term. Generally, damped Euler equations have a global classical solution for small smooth initial data. However, we identify a class of large initial data for the damped Euler equations, under the condition of azimuthal symmetry and by choosing a suitable self-similar transformation, resulting in the solution to the Cauchy problem blowing up in a finite time. Additionally, we introduce modulation variables to accurately describe the blowup time and location. Furthermore, the blowup profile exhibits a cusp singularity with Hölder $C^{1/3}$ regularity at the blowup point.