Besov Space Regularity Conditions for Weak Solutions of the Navier–Stokes Equations

Abstract

Consider a bounded domain${{\Omega \subseteq \mathbb{R}^3}}$ with smooth boundary, some initial value ${{u_0 \in L^2_{\sigma}(\Omega )}}$, and a weak solution u of the Navier–Stokes system in ${{[0,T) \times\Omega,\,0 < T \le \infty}}$. Our aim is to develop regularity and uniqueness conditions for u which are based on the Besov space

$$B^{q,s}(\Omega ):=\left\{v\in L^2_{\sigma}(\Omega ); \|v\|_{B^{q,s}(\Omega )} := \left(\int\limits^{\infty}_0 \left\|e^{-\tau A}v\right\|^s_q {\rm d} \tau\right)^{1/s}<\infty \right\}$$

with ${{2 < s < \infty,\,3 < q <\infty,\,\frac2{s}+\frac{3}{q} = 1}}$; here A denotes the Stokes operator. This space, introduced by Farwig et al. (Ann. Univ. Ferrara 55:89–110, 2009 and J. Math. Fluid Mech. 14: 529–540, 2012), is a subspace of the well known Besov space ${{{\mathbb{B}}^{-2/s}_{q,s}(\Omega )}}$, see Amann (Nonhomogeneous Navier–Stokes Equations with Integrable Low-Regularity Data. Int. Math. Ser. pp. 1–28. Kluwer/Plenum, New York, 2002). Our main results on the regularity of u exploits a variant of the space ${{B^{q,s}(\Omega )}}$ in which the integral in time has to be considered only on finite intervals (0, δ ) with ${{\delta \to 0}}$. Further we discuss several criteria for uniqueness and local right-hand regularity, in particular, if u satisfies Serrin’s limit condition ${{u\in L^{\infty}_{\text{loc}}([0,T);L^3_{\sigma}(\Omega ))}}$. Finally, we obtain a large class of regular weak solutions u defined by a smallness condition ${{\|u_0\|_{B^{q,s}(\Omega )} \le K}}$ with some constant ${{K=K(\Omega, q)>0}}$.