Well-posedness and ill-posedness for the 3D generalized Navier-Stokes equations in $\dot{F}^{-\alpha,r}_{\frac{3}{\alpha-1}}$

Chao Deng; Xiaohua Yao

doi:10.3934/dcds.2014.34.437

Article Contents

2014, Volume 34, Issue 2: 437-459. Doi: 10.3934/dcds.2014.34.437

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Well-posedness and ill-posedness for the 3D generalized Navier-Stokes equations in $\dot{F}^{-\alpha,r}_{\frac{3}{\alpha-1}}$

Chao Deng^1, and
Xiaohua Yao^2,

1.
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu 221116
2.
School of Mathematics and Statistics, Central China Normal University, Wuhan, Hubei 430079

Received: February 2013

Revised: March 2013

Published: February 2014

Abstract / Introduction Related Papers Cited by

Abstract

In this paper, we study the Cauchy problem of the 3-dimensional (3D) generalized Navier-Stokes equations (gNS) in the Triebel-Lizorkin spaces $\dot{F}^{-\alpha,r}_{q_\alpha}$ with $(\alpha,r)\in(1,\frac{5}{4})\times[1,\infty]$ and $q_\alpha=\frac{3}{\alpha-1}$. Our work establishes a dichotomy of well-posedness and ill-posedness depending on $r$. Specifically, by combining the new endpoint bilinear estimates in $L^{\!q_\alpha}_x\!L^2_T$ and $L^\infty_T\dot{F}^{-\alpha,1}_{q_\alpha}$ and characterization of the Triebel-Lizorkin spaces via fractional semigroup, we prove well-posedness of the gNS in $\dot{F}^{-\alpha,r}_{q_\alpha}$ for $r\in[1,2]$. Meanwhile, for any $r\in(2,\infty]$, we show that the solution to the gNS can develop norm inflation in the sense that arbitrarily small initial data in $\dot{F}^{-\alpha,r}_{q_\alpha}$ can produce arbitrarily large solution after arbitrarily short time.

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Mathematics Subject Classification: 76D03, 35Q35.

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