Young Measures Generated by Ideal Incompressible Fluid Flows

TL;DR

This paper demonstrates that all measure-valued solutions of the incompressible Euler equations can be approximated by sequences of exact weak solutions, expanding the understanding of solution sets in fluid dynamics.

Contribution

It proves that any measure-valued solution can be generated by weak solutions, revealing a vast set of solutions for the incompressible Euler equations.

Findings

Any measure-valued solution can be approximated by weak solutions.

The set of weak solutions is very large, possibly too large.

Provides a link between measure-valued solutions and actual weak solutions.

Abstract

In their seminal paper "Oscillations and concentrations in weak solutions of the incompressible fluid equations", R. DiPerna and A. Majda introduced the notion of measure-valued solution for the incompressible Euler equations in order to capture complex phenomena present in limits of approximate solutions, such as persistence of oscillation and development of concentrations. Furthermore, they gave several explicit examples exhibiting such phenomena. In this paper we show that any measure-valued solution can be generated by a sequence of exact weak solutions. In particular this gives rise to a very large, arguably too large, set of weak solutions of the incompressible Euler equations.