Hölder regularity of the pressure for weak solutions of the 3D Euler equations in bounded domains

Abstract

We consider the three-dimensional incompressible Euler equations on a bounded domain $\Omega$ with $C^4$ boundary. We prove that if the velocity field $u\in C^{0,\alpha }(\Omega )$ with $\alpha >0$ (where we are omitting the time dependence), it follows that the corresponding pressure $p$ of a weak solution to the Euler equations belongs to the Hölder space $C^{0,\alpha }(\Omega )$. We also prove that away from the boundary $p$ has $C^{0,2\alpha }$ regularity. In order to prove these results we use a local parametrisation of the boundary and a very weak formulation of the boundary condition for the pressure of the weak solution, as was introduced in [C. Bardos and E.S. Titi, Philos. Trans. Royal Soc. A, 380 (2022), 20210073], which is different than the commonly used boundary condition for classical solutions of the Euler equations. Moreover, we provide an explicit example illustrating the necessity of this new very weak formulation of the boundary condition for the pressure. Furthermore, we also provide a rigorous derivation of this new formulation of the boundary condition for weak solutions of the Euler equations. This result is of importance for the proof of the first half of the Onsager Conjecture, the sufficient conditions for energy conservation of weak solutions to the three-dimensional incompressible Euler equations in bounded domains. In particular, the results in this paper remove the need for separate regularity assumptions on the pressure in the proof of the Onsager conjecture.

arXiv comment59 pages

References in corpus (18)

• On admissibility criteria for weak solutions of the Euler equations
• Dissipative continuous Euler flows
• Euler Equations of Incompressible Ideal Fluids
• Non-uniqueness and h-principle for Hölder-continuous weak solutions of the Euler equations
• Onsager's Conjecture for the Incompressible Euler Equations in Bounded Domains
• Mathematics and Turbulence: where do we stand?
• Onsager's Conjecture with Physical Boundaries and an Application to the Vanishing Viscosity Limit
• Onsager's conjecture and anomalous dissipation on domains with boundary
• Remarks on the emergence of weak Euler solutions in the vanishing viscosity limit
• Remarks on high Reynolds numbers hydrodynamics and the inviscid limit
• Onsager's conjecture on the energy conservation for solutions of Euler equations in bounded domains
• Non-uniqueness for the Euler equations: the effect of the boundary
• Regularity results for rough solutions of the incompressible Euler equations via interpolation methods
• Boundary Vorticity Estimates for Navier-Stokes and Application to the Inviscid Limit
• Local formulas for the hydrodynamic pressure and applications
• $C^{0,\alpha }$ boundary regularity for the pressure in weak solutions of the $2d$ Euler equations
• On Double Hölder Regularity of the Hydrodynamic Pressure in Bounded Domains
• Full double Hölder regularity of the pressure in bounded domains