Development of Singularities for the Compressible Euler Equations with External Force in Several Dimensions

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 [1] on initial data resulting in singularities. Finally, we investigate the influence of frictional damping and rotation on the singularity formation.