2023/03/16 by Carl Johan Peter Johansson, Massimo Sorella, Johansson, Carl Johan Peter +1
Economics, Econometrics and Finance · Engineering · Mathematics · #35Q31 #76D05 #76F02 #76F10 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics and Turbulent Flows #Mathematical Physics (math-ph) #Navier-Stokes equation solutions #Stochastic processes and financial applications
paper · doi:10.48550/arxiv.2303.09486
openalex publication_date 2023/03/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
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 vν of the 4d Navier--Stokes equations with time-independent body forces is L^∞-weakly* converging to a solution of the forced Euler equations v0 as the viscosity parameter ν→ 0. Furthermore, the sequence ν|∇ vν|2 is weakly* converging (up to subsequences), in the sense of measure, to μ∈ M ((0,1) × \mathbbT4) and μT = 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-1t,x, to the Duchon--Robert distribution D[v0] of the solution to the 4d forced Euler equations. Finally, the kinetic energy profile of v0 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+(1)/(2) dimensional incompressible Navier--Stokes equations. This study motivates some open problems.