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

In this paper, we study the Cauchy problem of the 3-dimensional (3D) generalized incompressible Navier-Stokes equations (gNS) in Triebel-Lizorkin space $\dot{F}^{-α,r}_{q_α}(\mathbb{R}^3)$ with $(α,r)\in(1,5/4)\times[2,\infty]$ and $q_α=\frac{3}{α-1}$. 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 $L^{q_α}_x L^2_T$ with the characterization of Triebel-Lizorkin space via fractional semigroup, we prove the well-posedness of the gNS in $\dot{F}^{-α,r}_{q_α}(\mathbb{R}^3)$ 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 $\dot{F}^{-α,r}_{q_α}(\mathbb{R}^3)$ 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.\,\,$α=1$. Thus the Triebel-Lizorkin space framework naturally provides better connection between the well-known Koch-Tataru's $BMO^{-1}$ well-posed work and Bourgain-Pavlović's $\dot{B}_\infty^{-1,\infty}$ ill-posed work.