Development of singularities for the compressible Euler equations with external force in several dimensions

TL;DR

This paper investigates the development of singularities in solutions to the compressible Euler equations with external forces, establishing conditions under which solutions blow up in finite time and exploring effects of damping and rotation.

Contribution

It introduces a comprehensive sufficient condition for singularity formation in Euler equations with external forces, extending previous results and analyzing damping and rotational influences.

Findings

Solutions develop singularities in finite time under broad conditions.

Explicit construction of smooth solutions that avoid singularities.

Damping and rotation significantly influence singularity formation.

Abstract

We consider solutions to the Euler equations in the whole space from a certain class, which can be characterized, in particular, by finiteness of mass, total energy and momentum. We prove that for a large class of right-hand sides, including the viscous term, such solutions, no matter how smooth initially, develop a singularity within a finite time. We find a sufficient condition for the singularity formation, "the best sufficient condition", in the sense that one can explicitly construct a global in time smooth solution for which this condition is not satisfied "arbitrary little". Also compactly supported perturbation of nontrivial constant state is considered. We generalize the known theorem by Sideris on initial data resulting in singularities. Finally, we investigate the influence of frictional damping and rotation on the singularity formation.