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Bilinear estimates in homogeneous Triebel-Lizorkin spaces and the Navier-Stokes equations. (English) Zbl 1078.35087

The authors consider the Navier-Stokes equations in ${ℝ}^{n}$, i.e.

$\begin{array}{cc}\hfill {u}_{t}-{\Delta }u+\left(u·\nabla \right)u+\nabla p& =0\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{ℝ}^{n}×\left(0,T\right),\hfill \\ \hfill \text{div}\phantom{\rule{0.166667em}{0ex}}u& =0\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{ℝ}^{n}×\left(0,T\right),\hfill \\ \hfill u\left(0\right)& =a\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{ℝ}^{n}·\hfill \end{array}$

They prove that a strong solution which satisfies

${\int }_{0}^{T}{\parallel u\left(t\right)\parallel }_{{\stackrel{˙}{F}}_{\infty ,\infty }^{-\alpha }}^{\frac{2}{1-\alpha }}dt<\infty \phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{some}\phantom{\rule{4.pt}{0ex}}0<\alpha <1\phantom{\rule{2.em}{0ex}}\left(1\right)$

can be continued beyond $T$. Here, ${\stackrel{˙}{F}}_{p,q}^{s}$ is a homogeneous Triebel-Lizorkin space. Furthermore, they show that a weak solution satisfying (1) is strong in $\left(\epsilon ,T\right)$ provided that $n\le 4$. The main tool in proving the above results is a Hölder type inequality in Triebel-Lizorkin spaces.

MSC:
 35Q30 Stokes and Navier-Stokes equations 42B25 Maximal functions, Littlewood-Paley theory 76D05 Navier-Stokes equations (fluid dynamics)