Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $\dot{F}^{-\alpha,r}_{q_\alpha}(\mathbb{R}^3)$ 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 $L^{q_\alpha}_x L^2_T$ with the characterization of
Triebel-Lizorkin space via fractional semigroup, we prove the well-posedness of
the gNS in $\dot{F}^{-\alpha,r}_{q_\alpha}(\mathbb{R}^3)$ 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 $\dot{F}^{-\alpha,r}_{q_\alpha}(\mathbb{R}^3)$ 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 $BMO^{-1}$ well-posed work and Bourgain-Pavlovi\'c's
$\dot{B}_\infty^{-1,\infty}$ ill-posed work.

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The categories of this paper are: cond-mat
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-th
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The categories of this paper are: adap-org
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The categories of this paper are: plasm-ph
31
The categories of this paper are: ao-sci
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The categories of this paper are: eess
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The categories of this paper are: q-bio
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The categories of this paper are: astro-ph
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The categories of this paper are: nlin
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The categories of this paper are: cs
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The categories of this paper are: mtrl-th
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The categories of this paper are: econ
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The categories of this paper are: hep-lat
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-ex