2023/12/16 by Jingwen Han, Yun Wang, Han, Jingwen +3
Engineering · Mathematics · #Advanced Differential Equations and Dynamical Systems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2312.10382
openalex publication_date 2023/12/16 · openalex created_date 2023/12/20 · openalex updated_date 2026/07/28
In this paper, the Liouville-type theorems for the steady Navier-Stokes system are investigated. First, we prove that any bounded smooth helically symmetric solution in ℝ3 must be a constant vector. Second, for steady Navier-Stokes system in a slab supplemented with Navier boundary conditions, we prove that any bounded smooth solution must be zero if either the swirl or radial velocity is axisymmetric, or rur decays to zero as r tends to infinity. Finally, when the velocity is not big in L∞-space, the general three-dimensional steady Navier-Stokes flow in a slab with the Navier boundary conditions must be a Poiseuille type flow. The key idea of the proof is to establish Saint-Venant type estimates that characterize the growth of Dirichlet integral of nontrivial solutions.