Necessary and sufficient conditions for local strong solvability of the Navier–Stokes equations in exterior domains

Abstract

We consider the instationary Navier–Stokes equations in a smooth exterior domain ${\Omega \subseteq \mathbb{R}^3}$ with initial value u ₀, external force f = div  F and viscosity ν. It is an important question to characterize the class of initial values ${u_0\in L^2_{\sigma}(\Omega)}$ that allow a strong solution ${u \in L^s(0,T; L^q(\Omega))}$ in some interval ${[0,T[ \, , 0 < T \leq \infty}$ where s, q with 3 < q < ∞ and ${\frac{2}{s} + \frac{3}{q} =1}$ are so-called Serrin exponents. In Farwig and Komo (Analysis (Munich) 33:101–119, 2013) it is proved that ${\int_0^{\infty} \| e^{-\nu t A} u_0 \|_q^{s} \, {d}t < \infty}$ is necessary and sufficient for the existence of a strong solution ${u \in L^s(0,T ; L^q(\Omega)) \, , 0 < T \leq \infty}$, if additionally 3 < q ≤ 8; here, A denotes the Stokes operator. In this paper, we will show that this result remains true if q > 8, and consequently, ${\int_0^{\infty} \| e^{-\nu t A} u_0 \|_q^{s} \, {d}t < \infty}$ is the optimal initial value condition to obtain such a strong solution for all possible Serrin exponents s, q.