To solve the (Navier-)Stokes equations in general smooth domains Ως R n , the spaces ~L q (Ω) defined as L q nL 2 when 2 ≤ q < ∞ and L q +L 2 when1 < q < 2 have shown to be a successful strategy. First, the main properties of the spaces ~L q (Ω) and related concepts for solenoidal subspaces, Sobolev spaces, Bochner spaces, and the corresponding Helmholtz projection and Stokes operator will be discussed. Then these concepts are used to construct and analyze very weak, weak, mild, and strong solutions to the instationary (Navier-)Stokes equations in general domains. In particular, the strategy allows to find weak solutions of the (Navier-)Stokes system satisfying the localized energy inequality and the strong energy inequality which are important in the context of Leray structure theorem and partial regularity results.
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