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Path-connectedness of incompressible Euler solutions

Philippe Anjolras

TL;DR

This work proves that the set of weak solutions to the incompressible Euler equations is path-connected by constructing Hölder-$\tfrac{1}{2}$-regular paths in $C^0_t L^2_x$ between any two weak solutions. The authors extend the De Lellis–Székelyhidi convex integration framework to generic convex targets, replacing balls by arbitrary convex compact sets and introducing a gauge-based geometry via $j_{\mathcal{K}}$, $r_{min}$, and $r_{max}$. A core contribution is the identification of the $\Lambda$-convex hull with the convex hull and the development of good oscillation directions, enabling localized oscillations that drive the construction. The main result is complemented by a detailed iterative scheme (subsolutions, improvement steps, and dyadic iteration) that yields a continuous path entirely within the solution set, with implications for non-uniqueness and the topological structure of weak solutions.

Abstract

We study the incompressible Euler equation and prove that the set of weak solutions is path-connected. More precisely, we construct paths of Hölder regularity $C^{1/2}$, valued in $C^0_{t, loc} L^2_x$ endowed with the strong topology. The main result relies on a convex integration construction adapted from the seminal work of De Lellis and Székelyhidi [14, The Euler equations as a differential inclusion], extending it to a more broader geometric framework, replacing balls with arbitrary convex compact sets.

Path-connectedness of incompressible Euler solutions

TL;DR

This work proves that the set of weak solutions to the incompressible Euler equations is path-connected by constructing Hölder--regular paths in between any two weak solutions. The authors extend the De Lellis–Székelyhidi convex integration framework to generic convex targets, replacing balls by arbitrary convex compact sets and introducing a gauge-based geometry via , , and . A core contribution is the identification of the -convex hull with the convex hull and the development of good oscillation directions, enabling localized oscillations that drive the construction. The main result is complemented by a detailed iterative scheme (subsolutions, improvement steps, and dyadic iteration) that yields a continuous path entirely within the solution set, with implications for non-uniqueness and the topological structure of weak solutions.

Abstract

We study the incompressible Euler equation and prove that the set of weak solutions is path-connected. More precisely, we construct paths of Hölder regularity , valued in endowed with the strong topology. The main result relies on a convex integration construction adapted from the seminal work of De Lellis and Székelyhidi [14, The Euler equations as a differential inclusion], extending it to a more broader geometric framework, replacing balls with arbitrary convex compact sets.
Paper Structure (24 sections, 22 theorems, 182 equations, 1 figure)

This paper contains 24 sections, 22 theorems, 182 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Comparison with earlier works
  3. Outline of the article
  4. Convex integration with generic convex sets
  5. Notations and statement of the main result
  6. Elementary properties of compact sets and their gauges
  7. Geometric analysis
  8. Differential inclusion
  9. Tartar's wave cone
  10. Lambda-convex hull of K
  11. Step 1 : $C_{\mathcal{K}}$ is convex.
  12. Step 2 : $\overline{\mathcal{K}}$ contains all extremal points of $C_{\mathcal{K}}$.
  13. Step 3 : Identification of the convex hull.
  14. Step 4 : Identification of the $\Lambda$-convex hull.
  15. Step 5 : Good oscillation directions.
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $(u_0, u_1) \in C^0_t L^2_x$ be two weak solutions of the incompressible Euler equation equ-Euler-incomp. There exists a path $\gamma : [0, 1] \to C^0_t L^2_x$ valued in the set of weak solutions of the incompressible Euler equation, such that $\gamma(0) = u_0$, $\gamma(1) = u_1$, and having loc

Figures (1)

  • Figure 1: The convex of lemma \ref{['lem-calcul-aK']}

Theorems & Definitions (51)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3: DeLellisSzekelyhidioriginal
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Lemma 2.5
  • Proof
  • Remark 2.6
  • ...and 41 more