On Double Hölder regularity of the hydrodynamic pressure in bounded domains

• Published in: Calculus of variations and partial differential equations Vol. 62; no. 3
• Main Authors: De Rosa, Luigi, Latocca, Mickaël, Stefani, Giorgio
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.04.2023; Springer Nature B.V; Springer Verlag
• Subjects: Analysis; Analysis of PDEs; Boundary conditions; Calculus of Variations and Optimal Control; Optimization; Control; Domains; Fluid flow; Hydrodynamic pressure; Incompressible flow; Incompressible fluids; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Regularity; Systems Theory; Theoretical; Toruses; 35D30; 35J15; 35J25; 76B03; hydrodynamic pressure; boundary regularity; Schauder estimates; Incompressible fluids
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-023-02432-7

We prove that the hydrodynamic pressure p associated to the velocity u ∈ C θ ( Ω ) , θ ∈ ( 0 , 1 ) , of an inviscid incompressible fluid in a bounded and simply connected domain Ω ⊂ R d with C 2 + boundary satisfies p ∈ C θ ( Ω ) for θ ≤ 1 2 and p ∈ C 1 , 2 θ - 1 ( Ω ) for θ > 1 2 . Moreover, when ∂ Ω ∈ C 3 + , we prove that an almost double Hölder regularity p ∈ C 2 θ - ( Ω ) holds even for θ < 1 2 . This extends and improves the recent result of Bardos and Titi (Philos Trans R Soc A, 2022) obtained in the planar case to every dimension d ≥ 2 and it also doubles the pressure regularity. Differently from Bardos and Titi (2022), we do not introduce a new boundary condition for the pressure, but instead work with the natural one. In the boundary-free case of the d -dimensional torus, we show that the double regularity of the pressure can be actually achieved under the weaker assumption that the divergence of the velocity is sufficiently regular, thus not necessarily zero.