Local existence of smooth solutions to multiphase models in two space dimensions

TL;DR

This paper proves the local existence and uniqueness of smooth solutions for a class of multiphase fluid models in two dimensions using advanced mathematical techniques, with insights into challenges in three dimensions.

Contribution

It introduces an approximation method for multiphase models and establishes well-posedness and convergence results in two-dimensional space.

Findings

Proved local existence of smooth solutions in 2D

Developed an approximation based on Leray projection and paradifferential techniques

Discussed difficulties in extending results to 3D

Abstract

In this paper, we consider a class of models for multiphase fluids, in the framework of mixture theory. The considered system, in its more general form, contains both the gradient of a hydrostatic pressure, generated by an incompressibility constraint, and the gradient of a compressible pressure depending on the volume fractions of some of the different phases. To approach these systems, we define an approximation based on the \emph{Leray} projection, which involves the use of the \emph{Lax} symbolic symmetrizer for hyperbolic systems and paradifferential techniques. In two space dimensions, we prove its well-posedness and convergence to the unique classical solution to the original system. In the last part, we shortly discuss the difficulties in the three dimensional case.