Localization of the formation of singularities in multidimensional compressible Euler equations

Abstract

We consider the Cauchy problem with smooth data for compressible Euler equations in many dimensions and concentrate on two cases: solutions with finite mass and energy and solutions corresponding to a compact perturbation of a nontrivial stationary state. We prove the blowup results using the characteristics of the propagation of the solution in space and find upper and lower bounds for the density of a smooth solution in a given region of space in terms of the initial data. To solve the problems, we introduce a special family of integral functionals and study their temporal dynamics.

Self-Similar Solutions to the Compressible Euler Equations and their Instabilities

“…Some examples are the extension of these solutions [9,10], code verification [11,12], rigorous construction of smooth solutions [13,14], etc. On the other hand, recent results in singularity formation in the Euler equation, but out of Guderley problem, are [15][16][17][18][19][20][21]. See also [22][23][24] and references therein for stabilization of relativistic fluids.…”