Weak solutions to the steady compressible Euler equations with source terms

TL;DR

This paper demonstrates the existence of infinitely many weak solutions with prescribed energy profiles for the steady compressible Euler equations with source terms, using convex integration techniques.

Contribution

It introduces a novel application of convex integration to construct multiple weak solutions with specific energy profiles for these equations.

Findings

Existence of infinitely many solutions with given energy profiles

Construction of solutions via convex integration and localized plane waves

Solutions are compactly supported with prescribed density and pressure

Abstract

In this paper, we showed that for some given suitable density and pressure, there exist infinitely many compactly supported solutions with prescribed energy profile. The proof is mainly based on the convex integration scheme. We construct suitable subsolutions and localized plane-wave solutions to the reformulated system, and weak solutions are obtained by iterating these subsolutions.