Well-posedness of a model of nonhomogeneous compressible-incompressible fluids

TL;DR

This paper establishes local existence and uniqueness of solutions for a density-dependent compressible-incompressible fluid model, using paradifferential techniques, and analyzes convergence of numerical approximation methods.

Contribution

It introduces a new analytical approach for a simplified fluid model, proving well-posedness and convergence of numerical schemes.

Findings

Proved local existence and uniqueness in Sobolev spaces.

Demonstrated convergence of projection and artificial compressibility methods.

Provided a new framework for analyzing density-dependent fluid models.

Abstract

We propose a model of a density-dependent compressible-incompressible fluid, which is intended as a simplified version of models based on mixture theory as, for instance, those arising in the study of biofilms, tumor growth and vasculogenesis. Though our model is, in some sense, close to the density-dependent incompressible Euler equations, it presents some differences that require a different approach from an analytical point of view. In this paper, we establish a result of local existence and uniqueness of solutions in Sobolev spaces to our model, using paradifferential techniques. Besides, we show the convergence of both a continuous version of the Chorin-Temam projection method, viewed as a singular perturbation type approximation, and the 'artificial compressibility method'.