Nontrivial absolutely continuous part of anomalous dissipation measures in time

Abstract

We positively answer Question 2.2 and Question 2.3 in [Bruè, 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 $v_ν$ 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 $ν\to 0$. Furthermore, the sequence $ν|\nabla v_ν|^2$ is weakly* converging (up to subsequences), in the sense of measure, to $μ\in \mathcal{M} ((0,1) \times \mathbb{T}^4)$ and $μ_T = π_{\#} μ$ has a non-trivial absolutely continuous part where $π$ is the projection onto the time variable. Moreover, we also show that $μ$ 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.

Figures (4)

• Figure 1: The figure shows the profile $\theta_{\mathop{\mathrm{in}}\nolimits}$ averaged in $x,y$ depending only on $z$, more precisely on the vertical axis we represent the value $\langle | \theta_{\mathop{\mathrm{in}}\nolimits} | \rangle_{x,y} (z) = \int_{{\mathbb T}^2 } |\theta_{\mathop{\mathrm{in}}\nolimits}(x,y,z)| dx dy$ and on the horizontal axis we have the $z$-variable.
• Figure 2: We represent the functions $\langle |\theta_{0,j} | \rangle_{x,y}$ and $\langle |\theta_{\kappa_q ,j}\rangle_{x,y}$ at a precise time ($t= t_{q,j}$ that will be defined in \ref{['d:time_t_j']}) depending only on $z$ and representing the stability between the two functions at these times $t_{q,j}$, more precisely $\langle |\theta_{0,j} | \rangle_{x,y} (z) = \int_{{\mathbb T}^2} | \theta_{0,j} (t_{q,i} , x,y,z)| dx dy$ and similarly $\langle |\theta_{\kappa_q,j} | \rangle_{x,y} (z) = \int_{{\mathbb T}^2} | \theta_{\kappa ,j} (t_{q,i} , x,y,z)| dx dy$.
• Figure 3: The figure represents the phenomena of "anomalous dissipation" (loss of most of the $L^2$ norm) of the function $\theta_{\kappa_q, i}$. In the figure we draw the function $\langle |\theta_{\kappa_q, i}| \rangle_{x,y} (z)= \int_{{\mathbb T}^2} | \theta_{0,i} (t_{q,i} + \Tilde{t}_q , x,y,z)| dx dy$ depending only on $z$, the horizontal axis, at a certain time ($t= t_{q,i} + \tilde{t}_q$ we will define later in \ref{['d:time_t_j']}).
• Figure 4: The smooth energy profile of the limiting solution: $e(t) = \int_{{\mathbb T}^3} |\theta_{0} (t, x,y,z)|^2 dx dy dz$.

Theorems & Definitions (44)

• Definition 1: Physical solutions
• Remark 1
• Definition 2: Weak physical solutions + anomalous dissipation measure
• Remark 2
• Theorem A
• Remark 3
• Remark 4
• Definition 3
• Theorem B
• Remark 5