# The Stokes operator with Robin boundary conditions in solenoidal subspaces of L^1({\mathbb R}^n_+) and L^\infty({\mathbb R}^n_+)

Preprint Series # 638
Saal, Juergen The Stokes operator with Robin boundary conditions in solenoidal subspaces of L^1({\mathbb R}^n_+) and L^\infty({\mathbb R}^n_+). (2004);

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## Abstract

We prove that the Stokes operator with Robin boundary conditions is the generator of a bounded holomorphic semigroup on L^\infty_\sigma({\mathbb R}^n_+), which is even strongly continuous on the space \BUC_\sigma({\mathbb R}^n_+). Contrary to that result it is also proved that there exists no Stokes semigroup on L^1_\sigma({\mathbb R}^n_+), except if we assume the special case of Neumann boundary conditions. Nevertheless, we also obtain gradient estimates for the solution of the Stokes equations in L^1_\sigma({\mathbb R}^n_+) for all types of Robin boundary conditions.

Item Type: Preprint 35-xx PARTIAL DIFFERENTIAL EQUATIONS 128