Energy conservation in the limit of filtered solutions for the 2D Euler equations

Abstract

We consider energy conservation in a two-dimensional incompressible and inviscid flow through weak solutions of the filtered-Euler equations, which describe a regularized Euler flow based on a spatial filtering. We show that the energy dissipation rate for the filtered weak solution with vorticity in $L^p$, $p>3/2$ converges to zero in the limit of the filter parameter. Although the energy defined in the whole space is not finite in general, we formally extract a time-dependent part, which is well-defined for filtered solutions, from the energy and define the energy dissipation rate as its time-derivative. Moreover, the limit of the filtered weak solution is a weak solution of the Euler equations and it satisfies a local energy balance in the sense of distributions. For the case of $p=3/2$, we find the same result as $p>3/2$ by assuming Onsager's critical condition for the family of the filtered solutions.

20 pages

References in corpus (4)

• Strong convergence of the vorticity for the 2D Euler Equations in the inviscid limit
• A note on weak solutions of conservation laws and energy/entropy conservation
• Universality of the anomalous enstrophy dissipation at the collapse of three point vortices on Euler-Poincaré models
• Convergence of filtered weak solutions to the 2D Euler equations with vortex sheet initial data