Weak martingale solutions to the stochastic 1D Quantum-Navier-Stokes equations

TL;DR

This paper establishes the existence of global weak martingale solutions for a stochastic 1D quantum-Navier-Stokes model with density-dependent viscosity and capillarity, accommodating vacuum regions and randomness.

Contribution

It introduces a novel approach to prove existence of solutions for a stochastic quantum-Navier-Stokes system with vacuum and density-dependent viscosity.

Findings

Existence of global weak dissipative martingale solutions proven.

Solutions accommodate vacuum regions and stochastic forcing.

Method involves approximation, truncation, and stochastic compactness.

Abstract

In this paper we prove the existence of global weak dissipative martingale solutions for a one-dimensional compressible fluid model with capillarity and density dependent viscosity, driven by random initial data and a stochastic forcing term. These solutions are weak in both PDEs and Probability sense and may have vacuum regions. The proof relies on the construction of an approximating system which provides extra dissipation properties and the convergence is based on an appropriate truncation of the velocity field in the momentum equation and a stochastic compactness argument