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

Siran Li, Ya-Guang Wang

TL;DR

This work establishes the interior double Hölder regularity of the hydrodynamic pressure $p$ for weak solutions of the Euler equations in bounded $C^2$ domains with velocity $u$ in $C^{0,eta}$ for $0<\beta<1/2$, proving $p\in C^{0,2\beta}_{int}$. The authors present an elementary, PDE-based approach that relies on a modified pressure framed by two cutoff functions and a Neumann Green function, avoiding pseudodifferential calculus. Key contributions include a careful localization near the boundary, explicit control of boundary-layer terms, and a quantitative bound $|p(x_1)-p(x_2)|\le C|x_1-x_2|^{2\beta}$ with constants depending on dimension, geometry, and $\|u\|_{C^{0,\beta}}^2$, plus detailed boundary behavior. This provides a robust interior regularity mechanism relevant to turbulence analysis and Onsager-type questions in bounded domains, complementing prior results in full space and with higher-regularity domains.

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 $C^2$-domain $Ω\subset \mathbb{R}^d$; $d\geq 3$. That is, for velocity $u \in C^{0,γ}(Ω;\mathbb{R}^d)$ with some $0<γ<1/2$, we show that the pressure $p \in C^{0,2γ}_{\rm int}(Ω)$. 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 Anal. 249 (2025), 28] and the potential estimates in [L. Silvestre, unpublished notes].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 $\partialΩ$.

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

TL;DR

This work establishes the interior double Hölder regularity of the hydrodynamic pressure for weak solutions of the Euler equations in bounded domains with velocity in for , proving . The authors present an elementary, PDE-based approach that relies on a modified pressure framed by two cutoff functions and a Neumann Green function, avoiding pseudodifferential calculus. Key contributions include a careful localization near the boundary, explicit control of boundary-layer terms, and a quantitative bound with constants depending on dimension, geometry, and , plus detailed boundary behavior. This provides a robust interior regularity mechanism relevant to turbulence analysis and Onsager-type questions in bounded domains, complementing prior results in full space and with higher-regularity domains.

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 -domain ; . That is, for velocity with some , we show that the pressure . 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 -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 .
Paper Structure (13 sections, 3 theorems, 83 equations)

This paper contains 13 sections, 3 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Main Theorem
  3. Notations
  4. Organisation
  5. The modified pressure
  6. Proof of Theorem \ref{['thm']}
  7. The second cutoff function
  8. Estimate for $I_1$
  9. Estimate for $I_2$
  10. Proof of Theorem \ref{['thm']}
  11. Concluding remarks
  12. Two-dimensional case
  13. Unbounded domains

Key Result

Theorem 1.1

Let $(u,p)$ be a weak solution of the Euler equation euler in the bounded $C^2$-domain $\Omega \subset \mathbb{R}^d$ for $d \geq 3$. Assume that $u$ is of Hölder regularity $C^{0,\gamma}(\Omega;\mathbb{R}^d)$ in the spatial variable for some $0<\gamma<1/2$. Then the hydrodynamic pressure $p$ is loca with the constant $C=C\left(d,\gamma,{\widetilde{\Omega}}\right)\cdot \|u\|_{C^{0,\gamma}(\Omega)}

Theorems & Definitions (8)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 2.1
  • Lemma 3.1
  • proof : Proof of Lemma \ref{['lemma: I1']}
  • Lemma 3.2
  • proof : Proof of Lemma \ref{['lemma: I2']}