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

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 $$\underset{t\nearrow T}{\mathrm{limsup}}|\mathrm{div}u(t,x)|(T-t)\le \kappa ,$$ can never happen at time $T$ for all adiabatic number $\gamma \ge 1$. Here $\kappa >0$ doesn't depend on the initial data. This is achieved by proving the regularity of solutions under $$\rho (t,x)\le \frac{M}{(T-t{)}^{\kappa }},M<\infty .$$ This new scaling invariance also motivates us to construct an explicit type II blowup solution for $\gamma >1$.