Weak solutions to problems involving inviscid fluids

TL;DR

This paper demonstrates the existence of infinitely many weak solutions to a class of fluid mechanics equations derived from the pressure-less Euler system, using convex integration methods.

Contribution

It introduces a new application of convex integration to establish multiple weak solutions for a generalized fluid dynamics problem.

Findings

Existence of infinitely many weak solutions for the system.

Applicability of convex integration to variable coefficient Euler-like equations.

Prescribed initial data and kinetic energy can be achieved with these solutions.

Abstract

We consider an abstract functional-differential equation derived from the pressure-less Euler system with variable coefficients that includes several systems of partial differential equations arising in the fluid mechanics. Using the method of convex integration we show the existence of infinitely many weak solutions for prescribed initial data and kinetic energy.