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 $$\underset{t\nearrow T}{\mathrm{lim\hspace{0.17em}sup}}\mathrm{\mid }\mathrm{d}\mathrm{i}\mathrm{v}u(t,x)\mathrm{\mid }(T-t)\le \kappa ,$$ 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 $$\rho (t,x)\le \frac{M}{(T-t{)}^{\kappa }},M<\mathrm{\infty }\mathrm{.}$$ This new scaling invariance also motivates us to construct an explicit type II blowup solution for $\gamma >1$.