Remarks on Onsager's Conjecture and Anomalous Dissipation on domains with and without boundaries
Speaker: Theodore Drivas, Princeton University
Location: Warren Weaver Hall 1302
Date: Thursday, April 5, 2018, 11 a.m.
We first discuss the inviscid limit of the global energy dissipation of
Leray solutions of incompressible Navier-Stokes on the torus. Assuming
that the solutions have Besov norms bounded uniformly in viscosity, we
establish an upper bound on energy dissipation. As a consequence, Onsager-type
"quasi-singularities" are required in the Leray solutions, even if the total
energy dissipation is o(ν) in the limit ν → 0. Next, we discuss an extension
of Onsager's conjecture for domains with solid boundaries. We give a localized
regularity condition for energy conservation of weak solutions of the Euler
equations assuming Besov regularity of the velocity with σ>1/3 for any U⋐Ω
and, on an arbitrary thin layer around the boundary, boundedness of velocity,
pressure and continuity of the wall-normal velocity. We also prove that the global
viscous dissipation vanishes in the inviscid limit for Leray-Hopf solutions of the
Navier-Stokes equations under the similar assumptions, but holding uniformly in a
vanishingly thin viscous boundary layer. Finally, if a strong Euler solution exists,
we show that equicontinuity at the boundary within a O(ν) strip alone suffices to
conclude the absence of anomalous dissipation.
The first part of the talk concerns joint work with G. Eyink, the second with H.Q. Nguyen.