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 $$ \Gamma(n)=\max\left\{\gamma(n), n-\frac{n\gamma}{\gamma(n)+1}\right\}\quad\textrm{and}\quad\gamma(n)=\frac{n(n-1)-n\gamma}{n-\gamma},$$ no concentration phenomenon occurs.