Vortex solutions for the compressible Navier-Stokes equations with general viscosity coefficients in 1D: regularizing effects or not on the density

TL;DR

This paper proves the existence of global weak solutions for 1D compressible Navier-Stokes equations with general viscosity, showing conditions under which initial shocks are regularized or persist, depending on the coupling strength.

Contribution

It establishes existence results for weak solutions with initial data including shocks and Dirac masses, and identifies conditions for instant regularization of the density.

Findings

Strong coupling leads to instant regularization of shocks

Weak coupling allows density discontinuities to persist

Existence of solutions with initial measure-valued momentum

Abstract

We consider Navier-Stokes equations for compressible viscous fluids in the one-dimensional case with general viscosity coefficients. We prove the existence of global weak solution when the initial momentum $\rho_0 u_0$ belongs to the set of the finite measure ${\cal M}(\mathbb{R})$ and when the initial density $\rho_0$ is in the set of bounded variation functions $BV(\mathbb{R})$. In particular it allows to deal with initial momentum which are Dirac masses and initial density which admit shocks. We can observe in particular that this type of initial data have infinite energy. Furthermore we show that if the coupling between the density and the velocity is sufficiently strong then the initial density which admits initially shocks is instantaneously regularized and becomes continuous. This coupling is expressed via the regularity of the so called effective velocity $v=u+\frac{\mu(\rho)}{\rho^2}\partial_x \rho$ with $\mu(\rho)$ the viscosity coefficient. Inversely if the coupling between the initial density and the initial velocity is too weak (typically $\rho_0 v_0\in{\cal M}(\mathbb{R})$) then we prove the existence of weak energy solution in finite time but the density remains a priori discontinuous on the time interval of existence.