WIAS Preprint No. 3000, (2023)

Viscous flow past a translating body with oscillating boundary


Authors

  • Eiter, Thomas
    ORCID: 0000-0002-7807-1349
  • Shibata, Yoshihiro

2020 Mathematics Subject Classification

  • 35B40 35Q30 35R37 76D05 76D07

Keywords

  • Time-periodic solutions, moving boundary, exterior domain, maximal regularity, spatial decay

DOI

10.20347/WIAS.PREPRINT.3000

Abstract

We study an incompressible viscous flow around an obstacle with an oscillating boundary that moves by a translational periodic motion, and we show existence of strong time-periodic solutions for small data in different configurations: If the mean velocity of the body is zero, existence of time-periodic solutions is provided within a framework of Sobolev functions with isotropic pointwise decay. If the mean velocity is non-zero, this framework can be adapted, but the spatial behavior of flow requires a setting of anisotropically weighted spaces. In the latter case, we also establish existence of solutions within an alternative framework of homogeneous Sobolev spaces. These results are based on the time-periodic maximal regularity of the associated linearizations, which is derived from suitable R-bounds for the Stokes and Oseen resolvent problems. The pointwise estimates are deduced from the associated time-periodic fundamental solutions.

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