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On the Stokes problem in exterior domains: The maximum modulus theorem

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  • We study the Stokes initial boundary value problem, in $(0,T) \times Ω$, where $Ω \subseteq \mathbb{R}^n$, $n\geq3$, is an exterior domain, assuming that the initial data belongs to $L^\infty(Ω)$ and has null divergence in weak sense. We prove the maximum modulus theorem for the corresponding solutions. Crucial for the proof of this result is the analogous one proved by Abe-Giga for bounded domains. Our proof is developed by duality arguments and employing the semigroup properties of the resolving operator defined on $L^1(Ω)$. Our results are similar to the ones proved by Solonnikov by means of the potential theory.
    Mathematics Subject Classification: Primary: 35Q35, Secondary: 35Q30, 76D07.


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