On anomalous dissipation induced by transport noise

TL;DR

This paper demonstrates that transport noise can induce anomalous dissipation of enstrophy and energy in fluid equations across all dimensions, even at arbitrarily small noise intensities, with implications for understanding turbulence.

Contribution

It introduces Meyers' type estimates for SPDEs with transport noise and establishes uniform-in-time convergence, showing anomalous dissipation can occur with minimal noise.

Findings

Transport noise causes anomalous dissipation in 2D Navier-Stokes and diffusion equations.

Uniform-in-time convergence in smooth spaces is achieved for scaling limits.

Anomalous dissipation can occur even with very small transport noise.

Abstract

In this paper, we show that suitable transport noises produce anomalous dissipation of both enstrophy of solutions to 2D Navier-Stokes equations and of energy of solutions to diffusion equations in all dimensions. The key ingredients are Meyers' type estimates for SPDEs with transport noise, which are combined with recent scaling limits for such SPDEs. The former enables us to establish, for the first time, uniform-in-time convergence in a space of positive smoothness for such scaling limits. Compared to previous work, one of the main novelties is that anomalous dissipation might take place even in the presence of a transport noise of arbitrarily small intensity. Physical interpretations of our results are also discussed.