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

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 gamma > 1 and eta(0) := integral(R3) (rho e(s(0,x))/(gamma)(0, x) (rho) over bare (s) over bar/gamma) dx 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 C-1 solutions cannot exist globally. Here, F(t) = integral(R3)[alpha(t) + f(r)]x. rho udx is a newly introduced functional weighted by a sum of a time-dependent parameter function alpha(t) and a radius-dependent parameter function f(t) satisfying some mild conditions. As one of the applications, it is analysed that stronger a 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. (C) 2019 Elsevier Masson SAS. All rights reserved.