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On Periodic and Almost Periodic Solutions to Incompressible Viscous Fluid Flow Problems on the Whole Line

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Mathematics for Nonlinear Phenomena — Analysis and Computation (MNP2015 2015)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 215))

Abstract

It is shown that a large class of semilinear evolution equations on the whole line with periodic or almost periodic forces admit periodic or almost periodic mild solutions. The approach presented generalizes the method described in [28] to the case of the whole line and to forces which are almost periodic in the sense of H. Bohr. It relies on interpolation methods and on \(L^p-L^q\)-smoothing properties of the underlying linearized equation. Applied to incompressible fluid flow problems, the approach yields new results on (almost) periodic solutions to the Navier-Stokes-Oseen equations, to the flow past rotating obstacles, to the Navier-Stokes equations in the rotational setting as well as to Ornstein–Uhlenbeck type equations.

Dedicated to Yoshikazu Giga on the occasion of his 60th Birthday.

Anton Seyfert is supported by the DFG International Research Training Group IRTG 1529 on Mathematical Fluid Dynamics at TU Darmstadt.

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Acknowledgements

This work is financially supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED). The work of the second author is also supported by Vietnam Institute for Advanced Study in Mathematics (VIASM).

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Authors and Affiliations

  1. Fachbereich Mathematik, TU Darmstadt, Schlossgartenstr. 7, 64289, Darmstadt, Germany

    Matthias Hieber

  2. 607 Benedum Engineering Hall, University of Pittsburgh, Pittsburgh, PA, 15261, USA

    Matthias Hieber

  3. School of Applied Mathematics and Informatics, Hanoi University of Science and Technology Vien Toan ung dung va Tin hoc, Dai hoc Bach khoa Hanoi, 1 Dai Co Viet, Hanoi, Vietnam

    Thieu Huy Nguyen

  4. Fachbereich Mathematik, TU Darmstadt, Schlossgartenstr. 7, 64289, Darmstadt, Germany

    Anton Seyfert

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  1. Matthias Hieber
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  2. Thieu Huy Nguyen
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  3. Anton Seyfert
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Correspondence to Matthias Hieber .

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Editors and Affiliations

  1. Mathematical Institute, Tohoku University, Sendai, Miyagi, Japan

    Yasunori Maekawa

  2. Department of Mathematics, Hokkaido University, Sapporo, Hokkaido, Japan

    Shuichi Jimbo

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Hieber, M., Nguyen, T.H., Seyfert, A. (2017). On Periodic and Almost Periodic Solutions to Incompressible Viscous Fluid Flow Problems on the Whole Line. In: Maekawa, Y., Jimbo, S. (eds) Mathematics for Nonlinear Phenomena — Analysis and Computation. MNP2015 2015. Springer Proceedings in Mathematics & Statistics, vol 215. Springer, Cham. https://doi.org/10.1007/978-3-319-66764-5_4

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