Universality in the Low Mach number limit via a convex integration framework

TL;DR

This paper demonstrates that solutions to the compressible Euler equations can be constructed via convex integration to converge to any given incompressible Euler solution as the Mach number approaches zero, revealing a universality in the low Mach limit.

Contribution

It introduces a convex integration framework to show that the incompressible Euler equations serve as a universal attractor for the low Mach number limit of the compressible Euler equations.

Findings

Constructed compressible solutions converge strongly to incompressible solutions as Mach number tends to zero.

Any incompressible flow can be realized as a limit of convex integration solutions to the compressible system.

Established a new universality principle in the singular limit of fluid dynamics.

Abstract

We study the low Mach number limit of the compressible Euler equations through the lens of convex integration. For any prescribed $L^2$ weak solution of the incompressible Euler equations, we construct a corresponding family of weak solutions to the compressible Euler equations via a refined convex integration scheme. We then prove that, as the Mach number tends to zero, this family of solutions converges strongly to the given incompressible solution. This result demonstrates that the incompressible system acts as a universal attractor in this setting: every incompressible flow can be realized as the limit of convex integration solutions to the compressible system. Our approach highlights a new form of universality for singular limits and provides a rigorous framework for understanding the incompressible limit from the perspective of weak solution theory.