Vanishing capillarity limit of the Navier-Stokes-Korteweg system in one dimension with degenerate viscosity coefficient and discontinuous initial density

TL;DR

This paper proves the approximation of discontinuous solutions of the 1D Navier-Stokes system by solutions of the Navier-Stokes-Korteweg system as capillarity vanishes, even with degenerate viscosity and discontinuous initial density.

Contribution

It introduces new bounds for density and effective velocity, enabling the analysis of the vanishing capillarity limit with degenerate viscosity.

Findings

Approximation of discontinuous solutions as capillarity tends to zero.

Existence of unique global strong solutions under minimal initial conditions.

Uniform bounds for density and effective velocity are established.

Abstract

In the first main result of this paper we prove that one can approximate discontinious solutions of the 1d Navier Stokes system with solutions of the 1d Navier-Stokes-Korteweg system as the capilarity parameter tends to 0. Moreover, we allow the viscosity coefficients $\mu$ = $\mu$ ($\rho$) to degenerate near vaccum. In order to obtain this result, we propose two main technical novelties. First of all, we provide an upper bound for the density verifing NSK that does not degenerate when the capillarity coefficient tends to 0. Second of all, we are able to show that the positive part of the effective velocity is bounded uniformly w.r.t. the capillary coefficient. This turns out to be crucial in providing a lower bound for the density. The second main result states the existene of unique finite-energy global strong solutions for the 1d Navier-Stokes system assuming only that $\rho$0, 1/$\rho$0 $\in$ L $\infty$. This last result finds itself a natural application in the context of the mathematical modeling of multiphase flows.