Solutions of the Navier‐Stokes initial value problem in weighted L q ‐spaces

The problem of strong solvability of the nonstationary Navier‐Stokes equations is considered in weighted L q ‐spaces L q ω (Ω), where the domain Ω ⊂ ℝ n is the half space ℝ n + or a bounded domain with boundary of class C 1,1 and the weight ω belongs to the Muckenhoupt class A q . We give general conditions on the weight function ensuring the existence of a unique strong solution at least locally in time. In particular, these conditions admit weight functions ω ∈ A q , which become singular at the boundary or, in the case Ω = ℝ n + , grow for | x | →∞. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)