New weighted functional for non-existence of global solutions to the non-isentropic compressible Euler equations

Abstract

In this paper, we consider the Cauchy problem of the original three-dimensional non-isentropic compressible Euler equations. Among the two main results, a class of finite-time blowup conditions without assumptions on γ>1 and η(0)≔∫R3ρeS(0,x)∕γ(0,x)−ρ̄eS̄∕γdx is established. It is shown that together with the requirement on the sign of initial momentum, sufficiently strong F(0) will develop finite-time singularity and the C1 solutions cannot exist globally. Here, F(t)=∫R3[α(t)+f(r)]x⋅ρudx is a newly introduced functional weighted by a sum of a time-dependent parameter function α(t) and a radius-dependent parameter function f(t) satisfying some mild conditions. As one of the applications, it is analysed that stronger α implies that the necessary conditions for solutions of the original three-dimensional non-isentropic compressible Euler equations to exist on or before a given finite time is looser.