Journal of the Mathematical Society of Japan

Maximal regularity of the time-periodic Stokes operator on unbounded and bounded domains

Yasunori MAEKAWA and Jonas SAUER

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We investigate the time-periodic Stokes equations with non-homogeneous divergence data in the whole space, the half space, bent half spaces and bounded domains. The solutions decompose into a well-studied stationary part and a purely periodic part, for which we establish $\mathrm{L}^{p}$ estimates. For the whole space and the half space case we use a reduction of the Stokes equations to $(n-1)$ heat equations. Perturbation and localisation methods yield the result on bent half spaces and bounded domains. A one-to-one correspondence between maximal regularity for the initial value problem and time periodic maximal regularity is proven, providing a short proof for the maximal regularity of the Stokes operator avoiding the notion of $\mathcal{R}$-boundedness. The results are applied to a quasilinear model governing the flow of nematic liquid crystals.

Article information

J. Math. Soc. Japan, Volume 69, Number 4 (2017), 1403-1429.

First available in Project Euclid: 25 October 2017

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Zentralblatt MATH identifier

Primary: 35B10: Periodic solutions 76D03: Existence, uniqueness, and regularity theory [See also 35Q30]
Secondary: 35B45: A priori estimates 35K59: Quasilinear parabolic equations 76D07: Stokes and related (Oseen, etc.) flows

Stokes operator time-periodic maximal regularity


MAEKAWA, Yasunori; SAUER, Jonas. Maximal regularity of the time-periodic Stokes operator on unbounded and bounded domains. J. Math. Soc. Japan 69 (2017), no. 4, 1403--1429. doi:10.2969/jmsj/06941403.

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