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Zbl 1126.35047
Chen, Qionglei; Zhang, Zhifei
Regularity criterion via the pressure on weak solutions to the 3D Navier-Stokes equations. (English)
[J] Proc. Am. Math. Soc. 135, No. 6, 1829-1837 (2007). ISSN 0002-9939; ISSN 1088-6826/e

The authors consider a Leray-Hopf weak solution $(u,p)$ of the Navier-Stokes equations in $\bbfR^3\times(0,T)$. They prove that this solution is regular provided that the initial velocity $u_0$ belongs to $L^2 (\bbfR^3)\cap L^q (\bbfR^3)$ for some $q>3$ and that the pressure satisfies $$\int^T_0\|p(t) \|_{\dot B^0_{\infty,\infty}}\,dt<\infty.$$ The main tool in obtaining this result is an a-priori-estimate of the form $$\sup_{0\le t\le T}\|u(t)\|_{L^s}\le C(\| u_0\|_{L^s}+(CT)^{\frac{1} {s}}+e)^{\exp(C\int^T_0\|p(t)\|_{\dot B^0_{\infty, \infty}}\,dt)},\quad 3<s\le 4,$$ which is derived with the help of the Paley-Littlewood decomposition.
[Klaus Deckelnick (Magdeburg)]

MSC 2000:: *35Q30 Stokes and Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory
35B65 Smoothness of solutions of PDE

Keywords: Leray-Hopf solution; Besov spaces; Paley-Littlewood decomposition

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