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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Abstract

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

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

Dates
First available in Project Euclid: 25 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1508918562

Digital Object Identifier
doi:10.2969/jmsj/06941403

Mathematical Reviews number (MathSciNet)
MR3715809

Zentralblatt MATH identifier
1380.35044

Subjects
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

Keywords
Stokes operator time-periodic maximal regularity

Citation

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. https://projecteuclid.org/euclid.jmsj/1508918562


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References

  • H. Abels, Nonstationary Stokes System with Variable Viscosity in Bounded and Unbounded Domains, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 141–157.
  • W. Arendt, Ch. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace transforms and Cauchy problems, Monographs in Mathematics, 96, Birkhäuser Verlag, Basel, 2001.
  • W. Arendt and S. Bu, The operator-valued Marcinkiewicz multiplier theorem and maximal regularity, Math. Z., 240 (2002), 311–343.
  • O. V. Besov, The Littlewood-Paley theorem for a mixed norm, Trudy Mat. Inst. Steklov., 170 (1984), 31–36, 274.
  • F. Bruhat, Distributions sur un groupe localement compact et applications à l'étude des représentations des groupes $\wp $-adiques, Bull. Soc. Math. France, 89 (1961), 43–75.
  • L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Sem. Mat. Univ. Padova, 31 (1961), 308–340.
  • S. Chandrasekhar, Liquid Crystals, Cambridge University Press, Cambridge, paperback edition, 1993.
  • K. de Leeuw, On $L^p$ multipliers, Ann. of Math. (2), 81 (1965), 364–379.
  • R. Denk, J. Saal and J. Seiler, Inhomogeneous symbols, the Newton polygon, and maximal $L^p$-regularity, Russ. J. Math. Phys., 15 (2008), 171–191.
  • G. Dore, $L^p$ regularity for abstract differential equations, In Functional analysis and related topics, 1991 (Kyoto), Lecture Notes in Math., 1540, Springer, Berlin, 1993, 25–38.
  • G. Dore and A. Venni, On the closedness of the sum of two closed operators, Math. Z., 196 (1987), 189–201.
  • K.-J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Math., 194, Springer-Verlag, New York, 2000.
  • Theory and applications of liquid crystals, papers from the IMA workshop held in Minneapolis, Minn., January 1985, (eds. J. L. Ericksen and D. Kinderlehrer), The IMA Volumes in Mathematics and its Applications, 5, Springer-Verlag, New York, 1987.
  • R. Farwig and J. Sauer, Very weak solutions of the stationary Stokes equations in unbounded domains of half space type, Math. Bohem., 140 (2015), 81–109.
  • R. Farwig and H. Sohr, Generalized resolvent estimates for the Stokes system in bounded and unbouded domains, J. Math. Soc. Japan, 46 (1994), 607–643.
  • R. Farwig and H. Sohr, The stationary and nonstationary Stoke problem in exterior domains with nonzero divergence and nonzero boundary data, Math. Meth. Appl. Sci., 17 (1994), 269–291.
  • N. Filonov and T. Shilkin, On the Stokes problem with nonzero divergence, J. Math. Sci. (N.Y.), 166 (2010), 106–117.
  • G. P. Galdi, An introduction to the mathematical theory of the Navier–Stokes equations, Steady-state problems, Springer, New York, second edition, 2011.
  • G. P. Galdi, On time-periodic flow of a viscous liquid past a moving cylinder, Arch. Ration. Mech. Anal., 210 (2013), 451–498.
  • Y. Giga and H. Sohr, Abstract $L^p$ estimates for the Cauchy problem with applications to the Navier–Stokes equations in exterior domains, J. Funct. Anal., 102 (1991), 72–94.
  • M. Hieber, M. Nesensohn, J. Prüß and K. Schade, Dynamics of Nematic Liquid Crystal Flows: The Quasilinear Approach, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 397–408.
  • M. Kyed, Maximal regularity of the time-periodic linearized Navier–Stokes system, J. Math. Fluid Mech., 16 (2014), 523–538.
  • M. Kyed and J. Sauer, A Method for Time-Periodic $L^p$ Estimates, J. Differential Equations, 262 (2017), 633–652.
  • F.-H. Lin, Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena, Comm. Pure Appl. Math., 42 (1989), 789–814.
  • F.-H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501–537.
  • A. Lunardi, Interpolation theory, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie), Lecture Notes. Scuola Normale Superiore di Pisa (New Series), Edizioni della Normale, Pisa, second edition, 2009.
  • Y. Maekawa and H. Miura, On isomorphism for the space of solenoidal vector fields and its application to the Stokes problem, preprint, 2015.
  • M. S. Osborne, On the Schwartz–Bruhat space and the Paley–Wiener theorem for locally compact abelian groups, J. Functional Analysis, 19 (1975), 40–49.
  • Y. Shibata and S. Shimizu, A decay property of the Fourier transform and its application to the Stokes problem, J. Math. Fluid Mech., 3 (2001), 213–230.
  • V. A. Solonnikov, Estimates for solutions of nonstationary Navier–Stokes equations, Zap. Nauch. Sem. Len. Otdel. Mat. Inst. Steklov (LOMI), 38 (1973), 155–231, (English Transl.: J. Soviet Math., 8 (1977), 467–528).
  • R. Temam, Navier–Stokes equations, Theory and numerical analysis, North-Holland Publishing Co., Amsterdam, 1977.
  • L. Weis, A new approach to maximal $L_p$-regularity, In Evolution equations and their applications in physical and life sciences (Bad Herrenalb, 1998), Lecture Notes in Pure and Appl. Math., 215, Dekker, New York, 2001, 195–214.