Onsager's Conjecture for the Incompressible Euler Equations in Bounded Domains

Abstract

The goal of this note is to show that, also in a bounded domain $\Omega \subset {\mathbb{R}}^n$, with $\partial \Omega \in C^2$, any weak solution, $(u(x,t),p(x,t))$, of the Euler equations of ideal incompressible fluid in $\Omega \times (0,T)\subset {\mathbb{R}}^n\times {\mathbb{R}}_t$, with the impermeability boundary condition: $u\cdot \vec{n}=0$ on $\partial \Omega \times (0,T)$, is of constant energy on the interval $(0,T)$ provided the velocity field $u\in L^3((0,T);C^{0,\alpha }(\overline{\Omega }))$, with $\alpha >\frac{1}{3}.$

References in corpus (4)

• Regularity and Energy Conservation for the Compressible Euler Equations
• Mathematics and Turbulence: where do we stand?
• Energy conservation for the weak solutions of the compressible Navier-Stokes equations
• An Onsager Singularity Theorem for Turbulent Solutions of Compressible Euler Equations

Cited by in corpus (20)

• Onsager's Conjecture with Physical Boundaries and an Application to the Vanishing Viscosity Limit
• Onsager's conjecture and anomalous dissipation on domains with boundary
• An Onsager Singularity Theorem for Leray Solutions of Incompressible Navier-Stokes
• Remarks on the emergence of weak Euler solutions in the vanishing viscosity limit
• Statistical solutions and Onsager's conjecture
• Energy equalities for compressible Navier-Stokes equations
• Energy Conservation for the Compressible Euler and Navier-Stokes Equations with Vacuum
• On the Endpoint Regularity in Onsager's Conjecture
• Onsager's conjecture on the energy conservation for solutions of Euler equations in bounded domains
• The Josephson-Anderson Relation and the Classical D'Alembert Paradox
• On the Extension of Onsager's Conjecture for General Conservation Laws
• $C^{0,\alpha }$ boundary regularity for the pressure in weak solutions of the $2d$ Euler equations
• On Energy Conservation for the Hydrostatic Euler Equations: An Onsager Conjecture
• Onsager's Conjecture for Subgrid Scale $\alpha$-Models of Turbulence
• On Double Hölder Regularity of the Hydrodynamic Pressure in Bounded Domains
• Nonuniqueness of generalised weak solutions to the primitive and Prandtl equations
• Full double Hölder regularity of the pressure in bounded domains
• Vanishing viscosity limits for axisymmetric flows with boundary
• On the conservation of helicity by weak solutions of the 3D Euler and inviscid MHD equations
• Hölder regularity of the pressure for weak solutions of the 3D Euler equations in bounded domains