Stationary Solutions and Nonuniqueness of Weak Solutions for the Navier–Stokes Equations in High Dimensions

2018/07/31 by Xiaoyutao Luo · 36 citations
Engineering · Mathematics · #Compressibility #Energy (signal processing) #Homogeneous #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations #Nonlinear system #Space (punctuation) #Stability and Controllability of Differential Equations #Stationary solution #Uniqueness #Weak solution #math.AP

paper · pdf · doi:10.1007/s00205-019-01366-9

published in Archive for Rational Mechanics and Analysis 233(2), 701-747 (Springer Science+Business Media) · 31 pages; removed an incorrect statement in proposition 3.4 and modified to accommodate the change

openalex created_date 2018/08/03 · arxiv created 2019/01/22 · openalex publication_date 2019/03/06 · arxiv updated 2019/03/27 · openalex updated_date 2026/08/05

Abstract

Consider the unforced incompressible homogeneous Navier-Stokes equations on the d-torus \mathbbT^d where d≥ 4 is the space dimension. It is shown that there exist nontrivial steady-state weak solutions u∈ L²(\mathbbT^d). The result implies the nonuniqueness of finite energy weak solutions for the Navier-Stokes equations in dimensions d ≥ 4. And it also suggests that the uniqueness of forced stationary problem is likely to fail however smooth the given force is.

Citations

• Sur le mouvement d'un liquide visqueux emplissant l'espace
• L 3,∞ -solutions of the Navier-Stokes equations and backward uniqueness

Cited by

• Non-uniqueness of mild solutions for 2d-heat equations with singular initial data
• On Onsager-type conjecture for the Elsässer energies of the ideal MHD equations