On formation of singularity for non-isentropic Navier-Stokes equations without heat-conductivity

TL;DR

This paper investigates the blowup mechanism of smooth solutions to non-isentropic Navier-Stokes equations without heat-conductivity, showing that density or temperature concentration leads to finite-time singularity formation.

Contribution

It introduces a simple continuation principle and confirms a strong version of Nash's conjecture regarding finite-time blowup.

Findings

Finite-time concentration of density or temperature causes singularity formation.

A continuation principle for the system is established.

Affirmative evidence for Nash's conjecture on blowup mechanisms.

Abstract

It is known that smooth solutions to the non-isentropic Navier-Stokes equations without heat-conductivity may lose their regularities in finite time in the presence of vacuum. However, in spite of the recent progress on such blowup phenomenon, it remain to give a possible blowup mechanism. In this paper, we present a simple continuation principle for such system, which asserts that the concentration of the density or the temperature occurs in finite time for a large class of smooth initial data, which is responsible for the breakdown of classical solutions. It also give an affirmative answer to a strong version of conjecture proposed by J.Nash in 1950s