##### Department of Mathematics,

University of California San Diego

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### Analysis Seminar

## Theodore Drivas

#### Princeton University

## Remarks on Onsager's Conjecture and Anomalous Dissipation on domains with and without boundaries.

##### Abstract:

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 vanishes in the limit v $\rightarrow$ 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 (local) Besov regularity of the velocity with exponent >1/3 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(v) strip alone suffices to conclude the absence of anomalous dissipation. The talk concerns joint work with G. Eyink and H.Q. Nguyen.

Host: Tarek Elgindi

### May 22, 2018

### 9:45 AM

### AP&M 7321

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