Global solutions of compressible Navier-Stokes equations with small viscosity

Abstract

In this paper, we study the Cauchy problem for the compressible Navier-Stokes system in ${\mathbb{R}}^3$. Suppose that the viscosity coefficients satisfy $0<\max \{\mu ,\nu =\lambda +2\mu \}<1$, and set $\epsilon =\min \{\mu ,\nu =\lambda +2\mu \}$. We establish the global existence of classical solutions when the initial perturbations of the density and the curl-free part of the velocity are smaller than ${\epsilon }^{\frac{1}{2}+}$ (up to a logarithmic loss), while the divergence part of the initial velocity is smaller than $\epsilon$. This improves the classical global existence result of Matsumura-Nishida \cite{MaN80}, which requires all the initial data to be smaller than $\epsilon (<1)$. We expect that this result is representative of general Shizuta-Kawashima systems arising in physical applications. The improvement of the index from $1$ to $\frac{1}{2}+$ relies on exploiting the hidden Kawashima-type dissipation for the density and controlling the spacetime trace norm of the solution at the scale $\sqrt{\epsilon }$. These two ingredients are obtained through a weighted trace inequality and a Morawetz-type inequality for the perturbed sound speed and the divergence of the velocity.

33 pages