The Navier–Stokes equation in some general Banach spaces

We investigate the Cauchy problem for the incompressible Navier–Stokes equations, focusing on initial data situated within generalized Besov or Triebel–Lizorkin spaces defined by a Banach lattice over [Formula: see text] with [Formula: see text]. The external force term is assumed to belong to a Bochner space associated with another Besov or Triebel–Lizorkin space constructed over the same Banach lattice [Formula: see text]. The conditions imposed on this Banach lattice allow for the applicability of the results to a wide range of spaces.