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

Yanlin Liu

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

Analysis of PDEs 2024-07-08 v1

Authors: Yanlin Liu

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.

Keywords

Navier-Stokes equations time decay estimates homogeneous case Fourier analysis Besov space optimal regularity

Cite

@article{arxiv.2407.03752,
  title  = {The optimal time-decay estimates for 2-D inhomogeneous Navier-Stokes equations},
  author = {Yanlin Liu},
  journal= {arXiv preprint arXiv:2407.03752},
  year   = {2024}
}

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

Abstract

Keywords

Cite

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