Construction of weak solutions to the equations of a compressible viscous model

TL;DR

This paper develops a novel approximation scheme for constructing weak solutions to a simplified compressible viscous fluid model, preserving the transport structure and handling complex pressures.

Contribution

It introduces a new approximation method that avoids classical regularization, enabling analysis of physically relevant models with non-monotone pressures.

Findings

Successfully constructs weak solutions for the model

Handles non-monotone pressure laws

Employs advanced compactness and renormalization techniques

Abstract

The paper aims on the construction of weak solutions to equations of a model of compressible viscous fluids, being a simplification of the classical compressible Navier-Stokes system. We present a novel scheme for approximating systems that preserves structural integrity by avoiding classical regularization with $ - \varepsilon \Delta \varrho $, thus maintaining the transport character of the continuity equation. Our approach, which necessitates specific conditions on the constitutive equation, accommodates physically relevant models such as isentropic and van der Waals gases, and globally handles non-monotone pressures. From an analytical perspective, our method synthesizes techniques from Feireisl-Lions and Bresch-Jabin to demonstrate the convergence of approximate densities using compensated compactness techniques. We also apply renormalization of the continuity equations and utilize weight techniques to manage unfavorable terms.