Stationary solutions and nonuniqueness of weak solutions for the Navier-Stokes equations in high dimensions

Abstract

Consider the unforced incompressible homogeneous Navier-Stokes equations on the $d$-torus $\mathbb{T}^d$ where $d\geq 4$ is the space dimension. It is shown that there exist nontrivial steady-state weak solutions $u\in L^{2}(\mathbb{T}^d)$. The result implies the nonuniqueness of finite energy weak solutions for the Navier-Stokes equations in dimensions $d \geq 4$. And it also suggests that the uniqueness of forced stationary problem is likely to fail however smooth the given force is.