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

TL;DR

This paper constructs smooth solutions to the Navier--Stokes equations with linear growth at infinity that blow up in finite time, highlighting potential mechanisms for singularity formation without contradicting global regularity.

Contribution

It reduces the Navier--Stokes evolution to a matrix ODE and demonstrates finite-time blowup for solutions with linear growth at infinity, providing insights into possible blowup mechanisms.

Findings

Existence of smooth solutions with linear growth that blow up in finite time

Blowup solutions exhibit unbounded planar stretching

Vorticity aligns with the middle eigenvector of the strain matrix

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.