On the Decay of Infinite Energy Solutions to the Navier-Stokes Equations in the Plane

TL;DR

This paper investigates the long-term decay behavior of infinite energy solutions to the 2D Navier-Stokes equations, showing that their finite energy components decay algebraically over time and describing the asymptotic behavior of such solutions.

Contribution

It proves the boundedness and algebraic decay of finite energy parts of infinite energy solutions, extending understanding of their asymptotic behavior in 2D Navier-Stokes equations.

Findings

Finite energy parts are bounded for all time.

Finite energy parts decay algebraically over time.

Asymptotic behavior of infinite energy solutions is characterized.

Abstract

Infinite energy solutions to the Navier-Stokes equations in $\mathbb{R}^2$ may be constructed by decomposing the initial data into a finite energy piece and an infinite energy piece, which are then treated separately. We prove that the finite energy part of such solutions is bounded for all time and decays algebraically in time when the same can be said of heat energy starting from the same data. As a consequence, we describe the asymptotic behavior of the infinite energy solutions. Specifically, we consider the solutions of Gallagher and Planchon [5] as well as solutions constructed from a "radial energy decomposition". Our proof uses the Fourier Splitting technique of M. E. Schonbek.