Absence of anomalous dissipation for vortex sheets
Tarek Elgindi, Milton Lopes Filho, Helena Nussenzveig Lopes
TL;DR
This work proves the absence of anomalous dissipation in the vanishing viscosity limit for 2D Navier–Stokes on the torus, allowing forcing and initial vorticities that are sums of a nonnegative bounded Radon measure and an $L^1$ component. Central to the result is a novel refinement of Nash’s inequality in 2D, developed via a convolution/interpolation framework and extended to measure-valued vorticities; this yields a robust control of the viscous dissipation term $\zeta^\nu = \nu \iint |\omega^\nu|^2$. Under uniform bounds on initial data, forcing, and $\omega^\nu$, the authors show $\limsup_{\nu\to 0^+} \zeta^\nu(T)=0$, and the vanishing-viscosity limit produces a physically realizable weak Euler solution with energy balance. The results extend to vortex-sheet initial data and provide explicit dissipation-rate estimates in terms of the measure data, offering a rigorous link between measure-valued vorticities and dissipation in 2D turbulence, with comparisons to related recent work.
Abstract
A family of solutions of the incompressible Navier-Stokes equations is said to present anomalous dissipation if energy dissipation due to viscosity does not vanish in the limit of small viscosity. In this article we present a proof of absence of anomalous dissipation for 2D flows on the torus, with an arbitrary non-negative measure plus an integrable function as initial vorticity and square-integrable initial velocity. Our result applies to flows with forcing and provides an explicit estimate for the dissipation at small viscosity. The proof relies on a new refinement of a classical inequality due to J. Nash.