On optimal initial value conditions for local strong solutions of the Navier–Stokes equations

Abstract

Consider a smooth bounded domain ${\varOmega\subseteq{\mathbb R}^3}$ , and the Navier–Stokes system in ${[0,\infty)\times\varOmega}$ with initial value ${u_0\in L^2_\sigma(\varOmega)}$ and external force f =  div F, ${F\,{\in} \,L^2(0,\infty; L^2(\varOmega))\cap L^{s/2}(0,\infty; L^{q/2}(\varOmega))}$ where ${2\,< \,s\,< \,\infty, 3\,< \,q\,< \,\infty, \frac{2}{s}+\frac{3}{q} \,{=} \,1}$ , are so-called Serrin exponents. It is an important question what is the optimal (weakest possible) initial value condition in order to obtain a unique strong solution ${u\in L^s(0,T; L^q(\varOmega))}$ in some initial interval [0, T), ${0 < T \leq \infty}$ . Up to now several sufficient conditions on u ₀ are known which need not be necessary. Our main result, see Theorem 1.1, shows that the condition ${\int_0^\infty||e^{-t A}u_0||_q^s {\rm d}t < \infty}$ , A denotes the Stokes operator, is sufficient and necessary for the existence of such a strong solution u. In particular, if ${\int_0^\infty||e^{-t A}u_0||_q^s {\rm d}t = \infty}$ , ${u_0\in L_\sigma^2(\varOmega)}$ , then any weak solution u in the usual sense does not satisfy Serrin’s condition ${u\in L^s(0,T; L^q(\varOmega))}$ for each 0 < T ≤ ∞.