The optimal time-decay estimates for 2-D inhomogeneous Navier-Stokes equations

TL;DR

This paper establishes the optimal decay rates over time for solutions to the 2-D inhomogeneous Navier-Stokes equations, providing a more direct method that extends to various equations and $L^p$ spaces.

Contribution

It introduces a new, more direct approach to derive optimal time-decay estimates for 2-D inhomogeneous Navier-Stokes equations, adaptable to other equations and $L^p$ spaces.

Findings

Proves optimal decay rate $ orm{u(t)}_{ ext{Besov}} = O(t^{1/p - 3/2 - heta/2})$ as $t o \infty$.

Method is more direct than Fourier splitting, applicable to various equations.

Works in $L^p$-based spaces, broadening applicability.

Abstract

In this paper, we derive the optimal time-decay estimates for 2-D inhomogeneous Navier-Stokes equations. In particular, we prove that $\|u(t)\|_{\dot{B}^{\theta}_{p,1}({\mathop{\mathbb R\kern 0pt}\nolimits}^2)}={\mathcal O} (t^{\frac1p-\frac32-\frac{\theta}2})$ as $t\rightarrow\infty$ for any $p\in[2,\infty[,~\theta\in [0,2]$ if initially $\rho_0u_0\in \dot{B}^{-2}_{2,\infty}({\mathop{\mathbb R\kern 0pt}\nolimits}^2)$. This is optimal even for the classical homogeneous Navier-Stokes equations. Different with Schonbek and Wiegner's Fourier splitting device, our method here seems more direct, and can adapt to many other equations as well. Moreover, our method allows us to work in the $L^p$-based spaces.