Journal of the Mathematical Society of Japan

Weak Neumann implies $H^\infty$ for Stokes

Matthias GEIßERT and Peer Christian KUNSTMANN

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Let $\Omega\subset {\mathbb R}^n$ be a domain with uniform $C^3$ boundary and assume that the Helmholtz decomposition exists in ${\mathbb L}^q(\Omega):=L^q(\Omega)^n$ for some $q\in(1,\infty)$. We show that a suitable translate of the Stokes operator admits a bounded ${\cal H}^\infty$-calculus in ${\mathbb L}_\sigma^p(\Omega)$ for $p\in(\min\{q,q'\},\max\{q,q'\})$. For the proof we use a recent maximal regularity result for the Stokes operator on such domains ([GHHS12]) and an abstract result for the ${\cal H}^\infty$-calculus in complemented subspaces ([KKW06], [KW13]).

Article information

J. Math. Soc. Japan Volume 67, Number 1 (2015), 183-193.

First available in Project Euclid: 22 January 2015

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10] 47A60: Functional calculus

Stokes operator $H^\infty$-functional calculus general unbounded domains Helmholtz decomposition fractional powers


GEIßERT, Matthias; KUNSTMANN, Peer Christian. Weak Neumann implies $H^\infty$ for Stokes. J. Math. Soc. Japan 67 (2015), no. 1, 183--193. doi:10.2969/jmsj/06710183.

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