Maximal regularity of the spatially periodic Stokes operator and application to nematic liquid crystal flows

Jonas Sauer

Czechoslovak Mathematical Journal (2016)

  • Volume: 66, Issue: 1, page 41-55
  • ISSN: 0011-4642

Abstract

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We consider the dynamics of spatially periodic nematic liquid crystal flows in the whole space and prove existence and uniqueness of local-in-time strong solutions using maximal L p -regularity of the periodic Laplace and Stokes operators and a local-in-time existence theorem for quasilinear parabolic equations à la Clément-Li (1993). Maximal regularity of the Laplace and the Stokes operator is obtained using an extrapolation theorem on the locally compact abelian group G : = ℝ n - 1 × ℝ / L ℤ to obtain an ℛ -bound for the resolvent estimate. Then, Weis’ theorem connecting ℛ -boundedness of the resolvent with maximal L p regularity of a sectorial operator applies.

How to cite

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Sauer, Jonas. "Maximal regularity of the spatially periodic Stokes operator and application to nematic liquid crystal flows." Czechoslovak Mathematical Journal 66.1 (2016): 41-55. <http://eudml.org/doc/276818>.

@article{Sauer2016,
abstract = {We consider the dynamics of spatially periodic nematic liquid crystal flows in the whole space and prove existence and uniqueness of local-in-time strong solutions using maximal $L^p$-regularity of the periodic Laplace and Stokes operators and a local-in-time existence theorem for quasilinear parabolic equations à la Clément-Li (1993). Maximal regularity of the Laplace and the Stokes operator is obtained using an extrapolation theorem on the locally compact abelian group $G:=\mathbb \{R\}^\{n-1\}\times \mathbb \{R\} / L \mathbb \{Z\}$ to obtain an $\mathcal \{R\}$-bound for the resolvent estimate. Then, Weis’ theorem connecting $\mathcal \{R\}$-boundedness of the resolvent with maximal $L^p$ regularity of a sectorial operator applies.},
author = {Sauer, Jonas},
journal = {Czechoslovak Mathematical Journal},
keywords = {Stokes operator; spatially periodic problem; maximal $L^p$ regularity; nematic liquid crystal flow; quasilinear parabolic equations},
language = {eng},
number = {1},
pages = {41-55},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Maximal regularity of the spatially periodic Stokes operator and application to nematic liquid crystal flows},
url = {http://eudml.org/doc/276818},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Sauer, Jonas
TI - Maximal regularity of the spatially periodic Stokes operator and application to nematic liquid crystal flows
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 1
SP - 41
EP - 55
AB - We consider the dynamics of spatially periodic nematic liquid crystal flows in the whole space and prove existence and uniqueness of local-in-time strong solutions using maximal $L^p$-regularity of the periodic Laplace and Stokes operators and a local-in-time existence theorem for quasilinear parabolic equations à la Clément-Li (1993). Maximal regularity of the Laplace and the Stokes operator is obtained using an extrapolation theorem on the locally compact abelian group $G:=\mathbb {R}^{n-1}\times \mathbb {R} / L \mathbb {Z}$ to obtain an $\mathcal {R}$-bound for the resolvent estimate. Then, Weis’ theorem connecting $\mathcal {R}$-boundedness of the resolvent with maximal $L^p$ regularity of a sectorial operator applies.
LA - eng
KW - Stokes operator; spatially periodic problem; maximal $L^p$ regularity; nematic liquid crystal flow; quasilinear parabolic equations
UR - http://eudml.org/doc/276818
ER -

References

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