Hausdorff dimension of concentration for isentropic compressible Navier-Stokes equations

Abstract

The concentration phenomenon of the kinetic energy, $\rho |\mathbf{u}{|}^2$, associated to isentropic compressible Navier-Stokes equations, is addressed in ${\mathbb{R}}^n$ with $n=2,3$ and the adiabatic constant $\gamma \in [1,\frac{n}{2}]$. Except a space-time set with Hausdorff dimension less than or equal to $\Gamma (n)+1$ with $$\Gamma (n)=\max \left\{\gamma (n),n-\frac{n\gamma }{\gamma (n)+1}\right\}\text{and}\gamma (n)=\frac{n(n-1)-n\gamma }{n-\gamma },$$ no concentration phenomenon occurs.