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

Robin Ming Chen, Alexis Vasseur, Dehua Wang, Cheng Yu

TL;DR

The paper proves a universal low-Mach limit for the isentropic compressible Euler equations by leveraging a refined convex integration framework. From any $L^2$-weak solution ${\bf u}$ of the incompressible Euler equations, the authors construct a smooth incompressible subsolution and lift it to a compressible subsolution with density near unity, then perform a controlled convex integration that enforces a new $L^2$-type constraint to achieve uniform energy bounds. This yields infinitely many compressible weak solutions whose density converges to 1 and momentum converges to ${\bf u}$ in the low-Mach limit, for all $\gamma>1$, demonstrating that the incompressible system acts as a universal attractor within this weak-solution setting. The result provides a rigorous universality principle for singular limits and offers a robust framework for understanding the incompressible limit through weak solution theory, independent of small-scale oscillations modeled by convex integration.

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.

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

TL;DR

The paper proves a universal low-Mach limit for the isentropic compressible Euler equations by leveraging a refined convex integration framework. From any -weak solution of the incompressible Euler equations, the authors construct a smooth incompressible subsolution and lift it to a compressible subsolution with density near unity, then perform a controlled convex integration that enforces a new -type constraint to achieve uniform energy bounds. This yields infinitely many compressible weak solutions whose density converges to 1 and momentum converges to in the low-Mach limit, for all , demonstrating that the incompressible system acts as a universal attractor within this weak-solution setting. The result provides a rigorous universality principle for singular limits and offers a robust framework for understanding the incompressible limit through weak solution theory, independent of small-scale oscillations modeled by convex integration.

Abstract

We study the low Mach number limit of the compressible Euler equations through the lens of convex integration. For any prescribed 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.
Paper Structure (4 sections, 7 theorems, 102 equations)

This paper contains 4 sections, 7 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Construction of subsolutions to the compressible Euler equations
  3. Convex integration
  4. The proof of the Main result

Key Result

Theorem 1.1

Let $n = 2$ or $3$ and $\gamma > 1$. For any given $T > 0$, consider any weak solution ${\bf u}\in L^\infty(0,T;L^2(\mathbb{T}^n))$ to the incompressible Euler equation incom Euler with initial data ${\bf u}^0\in L^2(\mathbb{T}^n)$, as defined in Definition definition of weak solution incom. Then fo with the following properties.

Theorems & Definitions (16)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.1
  • Remark 1.1: On the adiabatic exponent $\gamma$
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1: $L^1$-coercivity
  • proof
  • ...and 6 more