LAUSR

Consider the unforced incompressible homogeneous Navier–Stokes equations on the d-torus $${\mathbb{T}^d}$$Td where $${d \geq 4}$$d≥4 is the space dimension. It is shown that there exist nontrivial steady-state weak solutions $${u \in L^{2} (\mathbb{T}^d)}$$u∈L2(Td). The result implies the nonuniqueness of finite energy weak solutions for the Navier–Stokes equations in dimensions $${d \geq 4}$$d≥4; it also suggests that the uniqueness of forced stationary problem is likely to fail however smooth the given force is.