Algebraic decay rates for 3D Navier-Stokes and Navier-Stokes-Coriolis equations in $ \dot{H}^{\frac{1}{2}}$

TL;DR

This paper derives algebraic decay rates for solutions to 3D Navier-Stokes and Navier-Stokes-Coriolis equations in the critical space, using Fourier analysis to connect initial data decay to solution behavior.

Contribution

It introduces a method to estimate decay rates in the critical space, highlighting the roles of linear and nonlinear dynamics in solution decay.

Findings

Solutions exhibit algebraic decay depending on initial data decay character.

The Fourier Splitting Method effectively estimates decay rates in the critical space.

Linear and nonlinear effects are explicitly characterized in decay behavior.

Abstract

An algebraic upper bound for the decay rate of solutions to the Navier-Stokes and Navier-Stokes-Coriolis equations in the critical space $\dot{H} ^{\frac{1}{2}} (\mathbb{R} ^3)$ is derived using the Fourier Splitting Method. Estimates are framed in terms of the decay character of initial data, leading to solutions with algebraic decay and showing in detail the roles played by the linear and nonlinear parts.