Hausdorff dimension of concentration for isentropic compressible Navier-Stokes equations

TL;DR

This paper investigates the Hausdorff dimension of the set where kinetic energy concentrates in solutions to the isentropic compressible Navier-Stokes equations, establishing bounds on the size of such sets in terms of Hausdorff dimension.

Contribution

It provides a novel Hausdorff dimension estimate for the concentration set of kinetic energy in isentropic compressible Navier-Stokes solutions, extending understanding of energy concentration phenomena.

Findings

Concentration occurs outside a set with Hausdorff dimension ≤ Γ(n)+1.

The Hausdorff dimension bound depends on the adiabatic constant γ and space dimension n.

No concentration phenomenon occurs outside the specified Hausdorff dimension set.

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.