Well-posedness and ill-posedness of the 3D generalized Navier-Stokes equations in Triebel-Lizorkin spaces

Abstract

In this paper, we study the Cauchy problem of the 3-dimensional (3D) generalized incompressible Navier-Stokes equations (gNS) in Triebel-Lizorkin space ${\dot{F}}_{q_{\alpha }}^{-\alpha ,r}({\mathbb{R}}^3)$ with $(\alpha ,r)\in (1,5/4)\times [2,\infty ]$ and $q_{\alpha }=\frac{3}{\alpha -1}$. Our work establishes a {\it dichotomy} of well-posedness and ill-posedness depending on $r=2$ or $r>2$. Specifically, by combining the new endpoint bilinear estimates in $L_x^{q_{\alpha }}{L}_T^2$ with the characterization of Triebel-Lizorkin space via fractional semigroup, we prove the well-posedness of the gNS in ${\dot{F}}_{q_{\alpha }}^{-\alpha ,r}({\mathbb{R}}^3)$ for $r=2$. On the other hand, for any $r>2$, we show that the solution to the gNS can develop {\it norm inflation} in the sense that arbitrarily small initial data in the spaces ${\dot{F}}_{q_{\alpha }}^{-\alpha ,r}({\mathbb{R}}^3)$ can lead the corresponding solution to become arbitrarily large after an arbitrarily short time. In particular, such dichotomy of Triebel-Lizorkin spaces is also true for the classical N-S equations, i.e.\,\,$\alpha =1$. Thus the Triebel-Lizorkin space framework naturally provides better connection between the well-known Koch-Tataru's $BM{O}^{-1}$ well-posed work and Bourgain-Pavlović's ${\dot{B}}_{\infty }^{-1,\infty }$ ill-posed work.

29 pages

References in corpus (1)

• Well-posedness of the Cauchy problem for the fractional power dissipative equations