On Scaling Invariance and Type-I Singularities for the Compressible Navier-Stokes Equations

TL;DR

This paper introduces a new scaling invariance for the compressible Navier-Stokes equations, proves the non-occurrence of certain singularities under specific conditions, and constructs an explicit blowup solution for gamma greater than one.

Contribution

It identifies a novel scaling invariance, rules out type I singularities under certain divergence conditions, and constructs an explicit type II blowup solution for gamma > 1.

Findings

Type I singularities cannot occur under specified divergence bounds.

Regularity of solutions is established under density bounds involving (T - t)^(-kappa).

An explicit type II blowup solution is constructed for gamma > 1.

Abstract

We find a new scaling invariance of the barotropic compressible Navier-Stokes equations. Then it is shown that type I singularities of solutions with $$\limsup_{t \nearrow T}|{\rm div} u(t, x)|(T - t) \leq \kappa,$$ can never happen at time $T$ for all adiabatic number $\gamma \geq 1$. Here $\kappa > 0$ doesn't depend on the initial data. This is achieved by proving the regularity of solutions under $$\rho(t, x) \leq \frac{M}{(T - t)^\kappa},\quad M < \infty.$$ This new scaling invariance also motivates us to construct an explicit type II blowup solution for $\gamma > 1$.