Criteria of local in time regularity of the Navier-Stokes equations beyond Serrin’s condition. (English) Zbl 1154.35416

Farwig, Reinhard; Kozono, Hideo; Sohr, Hermann

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

Keywords:

nonstationary Navier-Stokes equations; local in time regularity; Serrin’s condition