Absence of anomalous dissipation for vortex sheets

Tarek Elgindi; Milton Lopes Filho; Helena Nussenzveig Lopes

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arXiv · 2504.18523

Absence of anomalous dissipation for vortex sheets

Tarek Elgindi·Milton Lopes Filho·Helena Nussenzveig Lopes

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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.

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Tarek Elgindi, Milton Lopes Filho, Helena Nussenzveig Lopes. 2025-04-25. Absence of anomalous dissipation for vortex sheets. https://arxiv.org/abs/2504.18523

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