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On Computability of Navier-Stokes’ Equation

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Evolving Computability (CiE 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9136))

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Abstract

We approach the question of whether the Navier-Stokes Equation admits recursive solutions in the sense of Weihrauch’s Type-2 Theory of Effectivity: A suitable encoding (“representation”) is carefully constructed for the space of solenoidal vector fields in the \(L_q\) sense over the \(d\)-dimensional open unit cube with zero boundary condition. This is shown to render both the Helmholtz projection and the semigroup generated by the Stokes operator uniformly computable in the case \(q=2\).

The first author is partially supported by the National Science Foundation under grant No. DMS-1210979; the last author acknowledges support from German Research Foundation (DFG) under grant Zi 1009/4-1 and from IRTG 1529.

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Notes

  1. 1.

    We use \(q\in [1,\infty ]\) to denote the norm index, \(P\) for the pressure field, \(p\) for polynomials, \(\mathcal {P}\) for sets of (tuples of) the latter, and \(\mathbb {P}\) for the Helmholtz Projection.

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Author information

Authors and Affiliations

  1. Virginia Tech, Blacksburg, USA

    Shu Ming Sun

  2. University of Cincinnati, Cincinnati, USA

    Ning Zhong

  3. TU Darmstadt, Darmstadt, Germany

    Martin Ziegler

Authors
  1. Shu Ming Sun
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  2. Ning Zhong
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  3. Martin Ziegler
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Corresponding author

Correspondence to Martin Ziegler .

Editor information

Editors and Affiliations

  1. Department of Computer Science, Swansea University, Swansea, United Kingdom

    Arnold Beckmann

  2. University of Bucharest, Bucharest, Romania

    Victor Mitrana

  3. Sofia University, Sofia, Bulgaria

    Mariya Soskova

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Sun, S.M., Zhong, N., Ziegler, M. (2015). On Computability of Navier-Stokes’ Equation. In: Beckmann, A., Mitrana, V., Soskova, M. (eds) Evolving Computability. CiE 2015. Lecture Notes in Computer Science(), vol 9136. Springer, Cham. https://doi.org/10.1007/978-3-319-20028-6_34

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  • DOI: https://doi.org/10.1007/978-3-319-20028-6_34

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-20027-9

  • Online ISBN: 978-3-319-20028-6

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