Weak solutions to the steady incompressible Euler equations with source terms

TL;DR

This paper demonstrates the non-uniqueness of weak solutions to the steady incompressible Euler equations with source terms by employing convex integration and Baire category methods, revealing multiple solutions with the same energy profile.

Contribution

It extends convex integration techniques to Euler equations with source terms, establishing non-uniqueness of stationary solutions with given energy profiles.

Findings

Existence of numerous weak solutions with prescribed energy profiles.

Application of convex integration to Euler equations with source terms.

Non-uniqueness result for steady incompressible Euler equations.

Abstract

In this paper, we prove the non-uniqueness of stationary solutions to steady incompressible Euler equations with source terms. Based on the convex integration scheme developed by De Lellis and Sz\'{e}kelyhidi, the Euler system is reformulated as a differential inclusion. The key point is to construct the corresponding plane-wave solutions via high frequency perturbations. Then we use iteration and Baire category argument to conclude that there exist a large amount of weak solutions with given energy profile.