Non-Algebraic Decay for Solutions to the Navier-Stokes Equations

TL;DR

This paper extends Wiegner's classical results by establishing non-algebraic decay rates for solutions of the Navier-Stokes equations in two dimensions, filling a gap in the existing theory.

Contribution

It provides a rigorous proof of non-algebraic decay rates for 2D Navier-Stokes solutions, complementing and completing Wiegner's original algebraic decay results.

Findings

Established non-algebraic decay rates for 2D Navier-Stokes solutions.

Filled a theoretical gap in the asymptotic behavior analysis of solutions.

Extended the understanding of long-time behavior beyond algebraic decay.

Abstract

Around forty years ago, Michael Wiegner provided, in a seminal paper, sharp algebraic decay rates for solutions of the Navier--Stokes equations, showing that these solutions behave asymptotically like the solutions of the heat equation with the same data as $t\to+\infty$, in the $L^2$-norm, up to some critical decay rate. In the present paper, we close a gap that appears in the conclusion of Wiegner's theorem in the 2D case, for solutions with non-algebraic decay rate.