Onsager's conjecture and anomalous dissipation on domains with boundary

Abstract

We give a localized regularity condition for energy conservation of weak solutions of the Euler equations on a domain $\Omega \subset {\mathbb{R}}^d$, $d\ge 2$, with boundary. In the bulk of fluid, we assume Besov regularity of the velocity $u\in L^3(0,T;B_3^{1/3,c_0})$. On an arbitrary thin neighborhood of the boundary, we assume boundedness of velocity and pressure and, at the boundary, we assume continuity of wall-normal velocity. We also prove two theorems which establish that the global viscous dissipation vanishes in the inviscid limit for Leray--Hopf solutions $u^{\nu }$ of the Navier-Stokes equations under the similar assumptions, but holding uniformly in a thin boundary layer of width $O({\nu }^{\min \{1,\frac{1}{2(1-\sigma )}\}})$ when $u\in L^3(0,T;B_3^{\sigma ,c_0})$ in the interior for any $\sigma \in [1/3,1]$. The first theorem assumes continuity of the velocity in the boundary layer whereas the second assumes a condition on the vanishing of energy dissipation within the layer. In both cases, strong $L_t^3{L}_{x,loc}^3$ convergence holds to a weak solution of the Euler equations. Finally, if a strong Euler solution exists in the background, we show that equicontinuity at the boundary within a $O(\nu )$ strip alone suffices to conclude the absence of anomalous dissipation.

23 pages, 1 figure. Theorem 3 added in version 2. Sharpened the interior regularity assumption on the velocity in Theorems 1,2 and 3 in version 3

References in corpus (2)

• Onsager's Conjecture for the Incompressible Euler Equations in Bounded Domains
• Remarks on high Reynolds numbers hydrodynamics and the inviscid limit

Cited by in corpus (15)

• An Onsager Singularity Theorem for Leray Solutions of Incompressible Navier-Stokes
• Remarks on the emergence of weak Euler solutions in the vanishing viscosity limit
• Energy equalities for compressible Navier-Stokes equations
• Energy Conservation for the Compressible Euler and Navier-Stokes Equations with Vacuum
• A physics-inspired alternative to spatial filtering for large-eddy simulations of turbulent flows
• The Josephson-Anderson Relation and the Classical D'Alembert Paradox
• Emergent Universal Drag Law in a Model of Superflow
• Energy equality in compressible fluids with physical boundaries
• Energy balance for forced two-dimensional incompressible ideal fluid flow
• Hölder regularity of the pressure for weak solutions of the 3D Euler equations in bounded domains
• Hodge-theoretic analysis on manifolds with boundary, heatable currents, and Onsager's conjecture in fluid dynamics
• Sufficient conditions for local scaling laws for stationary martingale solutions to the 3D Navier-Stokes equations
• Boundary conditions and polymeric drag reduction for the Navier-Stokes equations
• Construction of the Hodge-Neumann heat kernel, local Bernstein estimates, and Onsager's conjecture in fluid dynamics
• A Kato-type criterion for vanishing viscosity near the Onsager's critical regularity