On the Onsager conjecture in two dimensions

2015/09/10 by Cheskidov, A., Filho, M. C. Lopes, Lopes, H. J. Nussenzveig +1
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1509.03213

Abstract

This note addresses the question of energy conservation for the 2D Euler system with an L^p-control on vorticity. We provide a direct argument, based on a mollification in physical space, to show that the energy of a weak solution is conserved if ω= ∇ × u ∈ L^\frac32. An example of a 2D field in the class ω∈ L^{\frac32 - ε} for any ε>0, and u∈ B^1/3_3,∞ (Onsager critical space) is constructed with non-vanishing energy flux. This demonstrates sharpness of the kinematic argument. Finally we prove that any solution to the Euler equation produced via a vanishing viscosity limit from Navier-Stokes, with ω∈ L^p, for p>1, conserves energy. This is an Onsager-supercritical condition under which the energy is still conserved, pointing to a new mechanism of energy balance restoration.

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