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 $d\ge 4$ 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 $d\ge 4$. And it also suggests that the uniqueness of forced stationary problem is likely to fail however smooth the given force is.

31 pages; removed an incorrect statement in proposition 3.4 and modified to accommodate the change