A note for double Hölder regularity of the hydrodynamic pressure for weak solutions of Euler equations

2024/09/14 by Siran Li, Li, Siran, Ya-Guang Wang +1 · 1 citation
Engineering · Mathematics · #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.2409.09433

openalex publication_date 2024/09/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give an elementary proof for the interior double Hölder regularity of the hydrodynamic pressure for weak solutions of the Euler Equations in a bounded C²-domain Ω⊂ ℝ^d; d≥ 3. That is, for velocity u ∈ C^0,γ(Ω;ℝ^d) with some 0<γ<1/2, we show that the pressure p ∈ C^0,2γ_{\rm int}(Ω). This is motivated by the studies of turbulence and anomalous dissipation in mathematical hydrodynamics and, recently, has been established in [L. De Rosa, M. Latocca, and G. Stefani, Int. Math. Res. Not. 2024.3 (2024), 2511--2560] over C^2,1-domains by means of pseudodifferential calculus. Our approach involves only standard elliptic PDE techniques, and relies on a variant of the modified pressure introduced in [C. W. Bardos, D. W. Boutros, and E. S. Titi, Hölder regularity of the pressure for weak solutions of the 3D Euler equations in bounded domains, Arch. Rational Mech. Anal. 249 (2025), 28] and the potential estimates in [L. Silvestre, unpublished notes]. The key novel ingredient of our proof is the introduction of two cutoff functions whose localisation parameters are carefully chosen as a power of the distance to ∂Ω.

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