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Let  u  be a weak solution of the Navier–Stokes equations in an exterior domain Ω ⊆ ℝ 3  and a time interval [0, T [, 0 &lt;  T  ≤ ∞, with initial value  u  0  and external force  f  = div  F . We address the problem to find the optimal (weakest possible) initial value condition in order to obtain a strong solution  u  ∈  L   s  (0, T ;  L   q  (Ω)) in some time interval [0, T [,  0 &lt;  T  &lt; ∞, where  s,q  with 3 &lt;  q  &lt; ∞ and 2/ s  + 3/ q  = 1 are so-called Serrin exponents. Our main result states, for Serrin exponents  s,q  with  3 &lt;  q  ≤ 8, a smallness condition on ∫ 0   T   ||  e  - ν  τ A  u  0  ||  q    s   d  τ  to imply existence of a strong solution  u  ∈  L   s  (0, T ;  L   q  (Ω)); here  A  denotes the Stokes operator. Moreover, when 3 &lt;  q  &lt; ∞, we will prove the necessity of the condition ∫ 0  ∞  ||  e  - ντ  A   u  0 ||  q    s   d  τ  &lt; ∞ to get a strong solution  u  on [0, T [, 0 &lt;  T  ≤ ∞.

Optimal initial value conditions for local strong solutions of the Navier–Stokes equations in exterior domains