On uniqueness of symmetric Navier-Stokes flows around a body in the plane

2013/10/21 by Tomoyuki Nakatsuka, Nakatsuka, Tomoyuki
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations #math.AP

paper · pdf · doi:10.48550/arxiv.1310.5506

arxiv created 2013/10/21 · openalex publication_date 2013/10/21 · arxiv updated 2013/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We investigate the uniqueness of symmetric weak solutions to the stationary Navier-Stokes equation in a two-dimensional exterior domain Ω. It is known that, under suitable symmetry condition on the domain and the data, the problem admits at least one symmetric weak solution tending to zero at infinity. Given two symmetric weak solutions u and v, we show that if u satisfies the energy inequality ‖ ∇ u ‖_{L² (Ω)}² ≤ (f,u) and sup_{x ∈ Ω} (|x|+1)|v(x)| is sufficiently small, then u=v. The proof relies upon a density property for the solenoidal vector field and the Hardy inequality for symmetric functions.

Related

• A remark on weak-strong uniqueness for suitable weak solutions of the Navier-Stokes equations
• Remarks on the weak formulation of the Navier-Stokes equations on the 2D hyperbolic space
• Existence of weak solutions to p-Navier-Stokes equations
• Uniqueness of the solution to the 2D Vlasov-Navier-Stokes system
• On Nonhomogeneous Boundary Value Problems for the Stationary Navier--Stokes Equations in 2D Symmetric Semi-Infinite Outlets