A note for double Hölder regularity of the hydrodynamic pressure for weak solutions of Euler equations
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| Format: | Preprint |
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2024
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| author | Li, Siran Wang, Ya-Guang |
| author_facet | Li, Siran Wang, Ya-Guang |
| contents | 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 Anal. 249 (2025), 28] and the potential estimates in [L. Silvestre, unpublished notes].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 $\partialΩ$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_09433 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note for double Hölder regularity of the hydrodynamic pressure for weak solutions of Euler equations Li, Siran Wang, Ya-Guang Analysis of PDEs 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 $\partialΩ$. |
| title | A note for double Hölder regularity of the hydrodynamic pressure for weak solutions of Euler equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2409.09433 |