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Criteria of local in time regularity of the Navier-Stokes equations beyond Serrin’s condition. (English) Zbl 1154.35416
Rencławowicz, Joanna (ed.) et al., Parabolic and Navier-Stokes equations. Part 1. Proceedings of the confererence, Bȩdlewo, Poland, September 10–17, 2006. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 81, Pt. 1, 175-184 (2008).
Summary: Let $$u$$ be a weak solution of the Navier-Stokes equations in a smooth bounded domain $$\Omega \subseteq {\mathbb R}^3$$ and time interval $$[0,T)$$, $$0<T\leq \infty$$, with initial value $$u_0$$, external force $$f=\text{div} F$$, and viscosity $$\nu>0$$. As it is well known, global regularity of $$u$$ for general $$u_0$$ and $$f$$ is an unsolved problem unless we pose additional assumptions on $$u_0$$ or on the solution $$u$$ itself such as Serrin’s condition $$\| u \|_{L^s(0,T;L^q(\Omega))} < \infty$$, where $$2/s + 3/q = 1$$. In the present paper we prove several local and global regularity properties by using assumptions beyond Serrin’s condition e.g. as follows: If the norm $$\| u\|_{L^r(0,T;L^q(\Omega))}$$ and a certain norm of $$F$$ satisfy a $$\nu$$-dependent smallness condition, where Serrin’s number $$2/r + 3/q > 1$$, or if $$u$$ satisfies a local leftward $$L^s$$-$$L^q$$-condition for every $$t\in(0,T)$$, then $$u$$ is regular in $$(0,T)$$.
For the entire collection see [Zbl 1147.35005].

##### MSC:
 35Q30 Navier-Stokes equations 35D10 Regularity of generalized solutions of PDE (MSC2000) 35B65 Smoothness and regularity of solutions to PDEs 76D05 Navier-Stokes equations for incompressible viscous fluids 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
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