Localization of the formation of singularities in multidimensional compressible Euler equations

TL;DR

This paper investigates the formation of singularities in multidimensional compressible Euler equations, providing conditions for blowup and bounds on density based on initial data using integral functionals.

Contribution

It introduces a novel approach with integral functionals to analyze singularity formation and provides new bounds for solutions in multiple dimensions.

Findings

Blowup results for solutions with finite mass and energy.

Bounds on density in terms of initial data.

Analysis of solution dynamics using integral functionals.

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.