Stationary solutions and nonuniqueness of weak solutions for the Navier-Stokes equations in high dimensions (Q2418416)

The author discusses the incompressible Navier-Stokes equations (NSE) on the torus \(\mathbb{T}^d= \mathbb{R}^d/\mathbb{Z}^d\) in high dimensions \(d\geq 4\). The author proves that there exists nontrivial steady-state weak solution of NSE in \(L^2(\mathbb{T}^d)\). There exists such a divergence-free initial data \(u_0 \in L^2(\mathbb{T}^d)\) that non-stationary NSE have at least two different weak solutions of finite energy. An analogous result takes place for the non-homogeneous steady-state NSE.