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

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 $\gamma > 1$.