On weak solutions to the compressible inviscid two-fluid model

TL;DR

This paper demonstrates the existence of infinitely many weak solutions for a three-dimensional compressible two-fluid model using convex integration, highlighting non-uniqueness and energy conservation under certain initial conditions.

Contribution

It introduces the application of convex integration to establish non-uniqueness of weak solutions for the compressible two-fluid model in three dimensions.

Findings

Existence of infinitely many global weak solutions for smooth initial data.

Existence of infinitely many weak solutions conserving energy for piecewise constant initial densities.

Local-in-time existence and uniqueness of classical solutions.

Abstract

In three space dimensions, we consider the compressible inviscid model describing the time evolution of two fluids sharing the same velocity and enjoying the algebraic pressure closure. By employing the technique of convex integration, we prove the existence of infinitely many global-in-time weak solutions for any smooth initial data. We also show that for any piecewise constant initial densities, there exists suitable initial velocity such that the problem admits infinitely many global-in-time weak solutions that conserve the total energy. The structure of the two-fluid model is crucial in our analysis. Adaptations of the main results to other two-fluid models, like the liquid-gas flow, are available. Local-in-time existence and uniqueness of classical solutions will also be shown.