  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 into 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.
