open access

https://doi.org/10.14943/83791

https://hdl.handle.net/2115/69445

The Stokes operator with Robin boundary conditions in solenoidal subspaces of L 1(R n +) and L ∞(R n +)

英語

2004

Department of Mathematics, Hokkaido University

Hokkaido University Preprint Series in Mathematics

638

2-27

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.

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