Path-connectedness of incompressible Euler solutions

2025/04/29 by Philippe Anjolras, Anjolras, Philippe
Earth and Planetary Sciences · Mathematics · #Analysis of PDEs (math.AP) #Aquatic and Environmental Studies #FOS: Mathematics #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.2504.20737

openalex publication_date 2025/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 C⁰_{t, loc} L²_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.

Citations

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• The Euler equations as a differential inclusion
• Statistical hydrodynamics

Related

• Sharp energy regularity and typicality results for Hölder solutions of incompressible Euler equations
• Weak solutions to the steady incompressible Euler equations with source terms
• On the interior regularity of weak solutions to the 2-D incompressible Euler equations
• A sufficient condition for uniqueness of weak solutions of the incompressible Euler system
• The 2D Euler equation on singular domains