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# Nontrivial absolutely continuous part of anomalous dissipation measures in time

Mar 2023

We positively answer Question 2.2 and Question 2.3 in [Bru\`e, De Lellis,2023] in dimension $4$ by building new examples of solutions to the forced $4d$incompressible Navier-Stokes equations, which exhibit anomalous dissipation,related to the zeroth law of turbulence \cite{kolmo}. We also prove that theunique smooth solution $v_\nu$ of the $4d$ Navier--Stokes equations withtime-independent body forces is $L^\infty$-weakly* converging to a solution ofthe forced Euler equations $v_0$ as the viscosity parameter $\nu \to 0$.Furthermore, the sequence $\nu |\nabla v_\nu|^2$ is weakly* converging (up tosubsequences), in the sense of measure, to $\mu \in \mathcal{M} ((0,1) \times\mathbb{T}^4)$ and $\mu_T = \pi_{\#} \mu$ has a non-trivial absolutelycontinuous part where $\pi$ is the projection into the time variable. Moreover,we also show that $\mu$ is close, up to an error measured in $H^{-1}_{t,x}$, tothe Duchon--Robert distribution $\mathcal{D}[v_0]$ of the solution to the $4d$forced Euler equations. Finally, the kinetic energy profile of $v_0$ is smoothin time. Our result relies on a new anomalous dissipation result for theadvection--diffusion equation with a divergence free $3d$ autonomous velocityfield and the study of the $3+\frac{1}{2}$ dimensional incompressibleNavier--Stokes equations. This study motivates some open problems.

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