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Zbl 1078.35087
Kozono, Hideo; Shimada, Yukihiro
Bilinear estimates in homogeneous Triebel-Lizorkin spaces and the Navier-Stokes equations.
(English)
[J] Math. Nachr. 276, 63-74 (2004). ISSN 0025-584X; ISSN 1522-2616/e

The authors consider the Navier-Stokes equations in $\bbfR^n$, i.e. \align u_t-\Delta u+ (u\cdot\nabla)u+\nabla p &= 0\quad\text{in }\bbfR^n\times (0,T),\\ \text{div\,}u &= 0\quad\text{in }\bbfR^n\times (0,T),\\ u(0) &= a\quad\text{in }\bbfR^n.\endalign They prove that a strong solution which satisfies $$\int^T_0\Vert u(t)\Vert^{{2\over 1-\alpha}}_{\dot F^{-\alpha}_{\infty,\infty}} dt<\infty\text{ for some }0<\alpha< 1\tag1$$ can be continued beyond $T$. Here, $\dot F^s_{p,q}$ is a homogeneous Triebel-Lizorkin space. Furthermore, they show that a weak solution satisfying (1) is strong in $(\varepsilon, T)$ provided that $n\le 4$. The main tool in proving the above results is a Hölder type inequality in Triebel-Lizorkin spaces.
[Klaus Deckelnick (Magdeburg)]
MSC 2000:
*35Q30 Stokes and Navier-Stokes equations
42B25 Maximal functions
76D05 Navier-Stokes equations (fluid dynamics)

Keywords: Navier-Stokes equations; Triebel-Lizorkin space; Littlewood-Paley decomposition

Cited in: Zbl 1189.35225

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