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Consider a smooth bounded domain Ω ⊆ R 3 , and the Navier-Stokes system in [0, ∞) × Ω with initial value u 0 ∈ L 2 σ (Ω) and external force f = div F, F ∈ L 2 (0, ∞; L 2 (Ω))∩L s /2 (0, ∞; L q /2 (Ω)) where 2 < s < ∞, 3 < q < ∞, 2 s + 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 ∈ L s (0, T ; L q (Ω)) in some initial interval [0, T ), 0 < T ≤ ∞. 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 ∞ 0 ||e −t A u 0 || s q dt < ∞, A denotes the Stokes operator, is sufficient and necessary for the existence of such a strong solution u. In particular, if ∞ 0 ||e −t A u 0 || s q dt = ∞, u 0 ∈ L 2 σ (Ω), then any weak solution u in the usual sense does not satisfy Serrin's condition u ∈ L s (0, T ; L q (Ω)) for each 0 < T ≤ ∞.

We construct a solution for the boundary value problems of the Stokes resolvent system in bounded and exterior domains of R n (n ≥ 2) with prescribed Dirichlet-and Neumann boundary data. The construction is based on the explicit form of the corresponding fundamental and double layer tensor and uses hydrodynamical potential theory and the method of integral equations.

The motion of a viscous incompressible fluid flow in bounded domains with a smooth boundary can be described by the nonlinear Navier-Stokes equations. This description corresponds to the so-called Eulerian approach. We develop a new approximation method for the Navier-Stokes equations in both the stationary and the non-stationary case by a suitable coupling of the Eulerian and the Lagrangian representation of the flow, where the latter is defined by the trajectories of the particles of the fluid. The method leads to a sequence of uniquely determined approximate solutions with a high degree of regularity containing a convergent subsequence with limit function v such that v is a weak solution of the Navier-Stokes equations. AMS-MSC (2000): 35B65, 35D05, 76D05