Optimal Time Decay of Navier-Stokes Equations With Low Regularity Initial Data

TL;DR

This paper establishes the optimal decay rates for solutions to the Navier-Stokes equations with low regularity initial data by combining negative Besov space estimates and energy methods, relaxing previous smallness conditions.

Contribution

It introduces a novel approach using negative Besov space estimates to achieve optimal decay rates with less restrictive initial data assumptions.

Findings

Achieves optimal decay rates with initial data small in certain negative Besov spaces.

Extends previous results by reducing regularity requirements for initial data.

Combines recent techniques to improve understanding of decay in Navier-Stokes equations.

Abstract

In this paper, we study the optimal time decay rate of isentropic Navier-Stokes equations under the low regularity assumptions about initial data. In the previous works about optimal time decay rate, the initial data need to be small in $H^{[N/2]+2}(\mathbb{R}^{N})$. Our work combined negative Besov space estimates and the conventional energy estimates in Besov space framework which is developed by R. Danchin. Though our methods, we can get optimal time decay rate with initial data just small in $\dot{B}^{N/2-1, N/2+1} \cap \dot{B}^{N/2-1, N/2}$ and belong to some negative Besov space(need not to be small). Finally, combining the recent results in \cite{zhang2014} with our methods, we can only need the initial data to be small in homogeneous Besov space $\dot{B}^{N/2-2, N/2} \cap \dot{B}^{N/2-1}$ to get the optimal time decay rate in space $L^{2}$.