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

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⊂Rd; d ≥ 3. More precisely, for velocity u∈C0,γ(Ω;Rd) with 0 < γ < 1/2, we show that the pressure p∈Cint0,2γ(Ω). This is motivated by the studies of turbulence and anomalous dissipation in mathematical hydrodynamics. Our approach involves only standard elliptic PDE techniques and relies on a variant of the modified pressure introduced in Bardos et al. [Arch. Ration. Mech. Anal. 249, 28 (2025)] and the potential estimates in L. Silvestre (unpublished notes). The key novel ingredient of our proof is the introduction of two cutoff functions whose localization parameters are carefully chosen as a power of the distance to ∂Ω. Quantitative estimates for the C0,2γ-norm of p near the boundary ∂Ω are also obtained.