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 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.