2023/07/03 by Lucas Huysmans, Huysmans, Lucas, Edriss S. Titi +1
Mathematics · #35D30 #35Q35 #60J60 #76F25 (Primary) 35A02 #76R10 #76R50 (Secondary) #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2307.00809
openalex publication_date 2023/07/03 · openalex created_date 2023/07/05 · openalex updated_date 2026/07/28
We study selection by vanishing viscosity for the transport of a passive scalar f(x,t)∈ℝ advected by a bounded, divergence-free vector field u(x,t)∈ℝ2. This is described by the initial value problem to the PDE (∂ f)/(∂ t) + ∇⋅ (u f) = 0, or with positive viscosity/diffusivity ν>0, to the PDE (∂ f)/(∂ t) + ∇⋅ (u f) -νΔf = 0. We demonstrate the failure of the vanishing viscosity limit to select (a) unique solutions or (b) physically admissible solutions in the sense of non-increasing energy/entropy.