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

TL;DR

This paper demonstrates the existence of nontrivial stationary weak solutions to the Navier-Stokes equations in dimensions four and higher, indicating nonuniqueness of solutions and potential failure of uniqueness even with smooth forcing.

Contribution

It establishes the existence of stationary solutions and nonuniqueness for the Navier-Stokes equations in high dimensions ($d \\geq 4$), extending understanding of solution behavior.

Findings

Existence of nontrivial steady-state weak solutions in $d \\geq 4$ dimensions.

Nonuniqueness of finite energy weak solutions in high dimensions.

Implication that uniqueness of stationary solutions may fail even with smooth forcing.

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.