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 $\mathbb{R}<sup>3$. Suppose that the viscosity coefficients satisfy $0&lt;\max{μ, ν=λ+2μ}&lt;1$, and set $\varepsilon=\min{μ, ν=λ+2μ}$. 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 $\varepsilon<sup>{\frac12+}$ (up to a logarithmic loss), while the divergence part of the initial velocity is smaller than $\varepsilon$. This improves the classical global existence result of Matsumura-Nishida \cite{MaN80}, which requires all the initial data to be smaller than $\varepsilon (&lt;1)$. We expect that this result is representative of general Shizuta-Kawashima systems arising in physical applications. The improvement of the index from $1$ to $\frac12+$ relies on exploiting the hidden Kawashima-type dissipation for the density and controlling the spacetime trace norm of the solution at the scale $\sqrt{\varepsilon}$. 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.