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 $C^2$-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 Anal. 249 (2025), 28] and the potential estimates in [L. Silvestre, unpublished notes].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 $\partialΩ$.

💡 Research Summary

The paper addresses the interior regularity of the hydrodynamic pressure associated with weak solutions of the incompressible, inviscid Euler equations in a bounded C² domain Ω⊂ℝᵈ with d≥3. The authors assume that the velocity field u is spatially Hölder continuous with exponent γ∈(0,½), i.e. u∈C^{0,γ}(Ω;ℝᵈ). Their main result is that the pressure p enjoys a “double‑Hölder” regularity: p∈C^{0,2γ}_loc(Ω). This improves upon earlier works that required higher boundary regularity (C^{2,1} or C^{3,α}) and relied on sophisticated pseudodifferential calculus, Littlewood–Paley theory, or microlocal analysis.

The novelty lies in providing an elementary proof that uses only standard elliptic PDE tools: Green’s functions for the Neumann problem, integration by parts, and elementary potential estimates. The proof proceeds through several carefully designed steps.

1. First modified pressure.
   A smooth cutoff φ_δ depending on the distance to the boundary is introduced. Setting P = p + φ_δ (u·ν)² yields a function that satisfies a Neumann boundary condition involving the second fundamental form II of ∂Ω: ∂_νP = II(u^⊤,u^⊤). However, the term Q = φ_δ (u·ν)² does not have a well‑defined normal derivative on ∂Ω, which prevents a direct integration‑by‑parts argument.

2. Second cutoff and homogeneous Neumann pressure.
   To eliminate the problematic boundary term, a second cutoff η_δ is defined, vanishing in a thin inner collar (distance ≤δ/2) and equal to one outside a slightly larger collar (distance ≥δ). The authors also introduce a correction term R = dist·φ_δ·II(u^⊤,u^⊤) evaluated at the nearest boundary point. The final modified pressure ℘ = P + R = p + Q + R satisfies a homogeneous Neumann condition ∂_ν℘ = 0, which allows the use of the Neumann Green function K_N.

3. Potential representation.
   For any two points x₁, x₂∈Ω, the authors consider ψ_{x₁,x₂}, the solution of Δψ = δ_{x₁}−δ_{x₂} with Neumann boundary data. By testing the equation for ℘ against ψ_{x₁,x₂} and inserting the second cutoff η_δ, the boundary contributions from Q disappear, and the remaining integrals involve only bulk quantities.

4. Balancing singularities via the choice of δ.
   The derivatives of η_δ scale like δ^{-1} and δ^{-2}, which could potentially dominate the estimates. The authors choose δ as a power of the distance |x₁−x₂|: specifically δ≈|x₁−x₂|^{(d−2)/(d−2+2γ)}. This choice balances the singular factors arising from ∇η_δ and Δη_δ against the decay of the Green function derivatives, yielding an overall bound proportional to |x₁−x₂|^{2γ}.

5. Final estimate and constants.
   The authors obtain the quantitative inequality
   |℘(x₁)−℘(x₂)| ≤ C ‖u‖_{C^{0,γ}}² |x₁−x₂|^{2γ},
   where C depends only on the dimension d, the Hölder exponent γ, and geometric quantities of Ω (the C⁰‑norm of the second fundamental form, the injectivity radius, and the intrinsic diameter). Since ℘ differs from the original pressure p only by the explicitly controlled terms Q+R, the same double‑Hölder bound holds for p. Moreover, the authors give a refined description of how the constant behaves when the points approach the boundary, expressed in terms of κ = min{dist(x₁,∂Ω),dist(x₂,∂Ω)}.

6. Context and implications.
   This result recovers and sharpens the interior regularity statements previously obtained by De Rosa, Latocca, and Stefani (2024) and by Bardos, Boutros, and Titi (2025), but with a much simpler toolbox. The proof avoids pseudodifferential operators, Littlewood–Paley decompositions, and delicate microlocal arguments, making the technique accessible to a broader PDE audience. The explicit dependence of the constant on geometric data also clarifies the role of the boundary layer in turbulence‑related problems, such as the Onsager conjecture and anomalous energy dissipation in bounded domains.

In summary, the paper delivers an elementary yet powerful proof that the pressure associated with a Hölder‑continuous Euler velocity field enjoys interior C^{0,2γ} regularity. The method hinges on a clever double‑cutoff construction, a homogeneous Neumann reformulation, and careful potential estimates, thereby providing a transparent and fully quantitative alternative to earlier sophisticated approaches.