Nontrivial absolutely continuous part of anomalous dissipation measures in time

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 [K41]. We also prove that the unique smooth solution $v_\nu$ of the $4d$ Navier--Stokes equations with time-independent body forces is $L^\infty$-weakly* converging to a solution of the forced Euler equations $v_0$ as the viscosity parameter $\nu \to 0$. Furthermore, the sequence $\nu |\nabla v_\nu|^2$ is weakly* converging (up to subsequences), 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 absolutely continuous part where $\pi$ is the projection onto the time variable. Moreover, we also show that $\mu$ is close, up to an error measured in $H^{-1}_{t,x}$, to the 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 smooth in time. Our result relies on a new anomalous dissipation result for the advection--diffusion equation with a divergence free $3d$ autonomous velocity field and the study of the $3+\frac{1}{2}$ dimensional incompressible Navier--Stokes equations. This study motivates some open problems.