Blow-up of the 3-D compressible Navier-Stokes equations for monatomic gases

TL;DR

This paper demonstrates finite-time blow-up of solutions to the 3-D compressible Navier-Stokes equations for monatomic gases with specific adiabatic exponent, using self-similar solutions derived from Euler equations.

Contribution

It establishes the first proof of blow-up for the 3-D compressible Navier-Stokes equations in the monatomic gas case with b3=5/3, leveraging self-similar solutions.

Findings

Proved blow-up of solutions for b3=5/3

Constructed self-similar imploding solutions for Euler equations

Built asymptotically self-similar blow-up solutions for Navier-Stokes

Abstract

In this paper, we prove the blow-up of the $3$-D isentropic compressible Navier-Stokes equations for the adiabatic exponent $\gamma=5/3$, which corresponds to the law of monatomic gases. This is the degenerate case in the sense of [Merle, Rapha\"el, Rodnianski and Szeftel, Ann. of Math. (2), 196 (2022), 567-778; Ann. of Math. (2), 196 (2022), 779-889]. Motivated by these breakthrough works, we first establish the existence of a sequence of smooth, self-similar imploding solutions to the compressible Euler equations for $\gamma=5/3$. Subsequently, we utilize these self-similar profiles to construct smooth, asymptotically self-similar blow-up solutions to the compressible Navier-Stokes equations for monatomic gases.