Optimal initial value conditions for the existence of local strong solutions of the Navier–Stokes equations

Abstract

Consider the instationary Navier–Stokes system in a smooth bounded domain ${\Omega\subset \mathbb {R}^3}$ with vanishing force and initial value ${u_0\in L^2_\sigma(\Omega)}$ . Since the work of Kiselev and Ladyzhenskaya (Am. Math. Soc. Transl. Ser. 2 24:79–106, 1963) there have been found several conditions on u ₀ to prove the existence of a unique strong solution ${u\in L^s\left( 0,T;L^q(\Omega)\right)}$ with u(0) = u ₀ in some time interval [0, T), 0 < T ≤ ∞, where the exponents 2 < s < ∞, 3 < q < ∞ satisfy ${\frac{2}{s} + \frac{3}{q} = 1}$ . Indeed, such conditions could be weakened step by step, thus enlarging the corresponding solution classes. Our aim is to prove the following optimal result with the weakest possible initial value condition and the largest possible solution class: Given u ₀, q, s as above and the Stokes operator A ₂, we prove that the condition ${\int_0^\infty \| e^{-tA_2}u_0\|_q^s\, dt < \infty}$ is necessary and sufficient for the existence of such a local strong solution u. The proof rests on arguments from the recently developed theory of very weak solutions.