Finite-time blowup for smooth solutions of the Navier--Stokes equations on the whole space with linear growth at infinity

Abstract

In this paper we consider smooth solutions of the Navier--Stokes equations with a linear dependence on the spatial variable. We reduce the evolution of these solutions to a matrix ODE, and show that there are such solutions that blowup in finite-time. Note that because these solutions have linear growth at infinity, this blowup is not a counterexample disproving the global regularity of strong solutions of the Navier--Stokes equations, as strong solutions must have sufficient decay at infinity. This paper does not resolve the Millennium Problem. Nonetheless, these solutions do exhibit several properties that are closely related to the problem of blowup for strong solutions of Navier--Stokes equations, including the presence of unbounded planar stretching, and the alignment of the vorticity with the middle eigenvector of the strain matrix.