Advertisement

On Computability of Navier-Stokes’ Equation

  • Shu Ming Sun
  • Ning Zhong
  • Martin Ziegler
Conference paper
First Online:
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9136)

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 \(q=2\).

Keywords

Infinitesimal Generator Analytic Semigroup Stokes Operator Rigorous Framework Recursive Analysis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

References

  1. 1.
    Beggs, E., Costa, J.F., Tucker, J.V.: Axiomatising physical experiments as Oracles to algorithms. Philos. Trans. R. Soc. Math. Phys. Eng. Sci. 370, 3359–3384 (2012)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Giga, Y.: Time and spatial analyticity of solutions of the Navier-Stokes equations. Comm. Partial Differ. Equ. 8, 929–948 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Giga, Y., Miyakawa, T.: Solutions in \(L^r\) of the Navier-Stokes initial value problem. Arch. Ration. Mech. Anal. 89(3) 5.VIII, 267–281 (1985)Google Scholar
  4. 4.
    Girault, V., Raviart, P.-A.: Finite Element Methods for Navier-Stokes Equations. Series in Computational Mathematics, vol. 5. Springer, Heidelberg (1986)zbMATHGoogle Scholar
  5. 5.
    Patel, M.K., Markatos, N.C., Cross, M.: A critical evaluation of seven discretization schemes for convection-diffusion equations. Int. J. Numer. Meth. Fluids 5(3), 225–244 (1985)zbMATHCrossRefGoogle Scholar
  6. 6.
    Pour-El, M.B., Richards, J.I.: The wave equation with computable initial data such that its unique solution is not computable. Adv. Math. 39(4), 215–239 (1981)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Pour-El, M.B., Richards, J.I.: Computability in Analysis and Physics. Springer, Heidelberg (1989)zbMATHCrossRefGoogle Scholar
  8. 8.
    Pour-El, M.B., Zhong, N.: The wave equation with computable initial data whose unique solution is nowhere computable. Math. Logic Q. 43(4), 499–509 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    W.D. Smith: On the uncomputability of hydrodynamics, NEC preprint (2003)Google Scholar
  10. 10.
    Soare, R.I.: Computability and recursion. Bull. Symbolic Logic 2, 284–321 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Tao, T.: Finite time blowup for an averaged three-dimensional Navier-Stokes equation. arXiv:1402.0290 (submitted); http://terrytao.wordpress.com/2014/02/04/
  12. 12.
    Weihrauch, K.: Computable Analysis: An Introduction. Texts in Theoretical Computer Science. An EATCS Series. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  13. 13.
    Weihrauch, K., Zhong, N.: Is wave propagation computable or can wave computers beat the turing machine? Proc. Lond. Math. Soc. 85(2), 312–332 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Weihrauch, K., Zhong, N.: Computing the solution of the Korteweg-de Vries equation with arbitrary precision on turing machines. Theor. Comput. Sci. 332, 337–366 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Weihrauch, K., Zhong, N.: Computing schrödinger propagators on type-2 turing machines. J. Complex. 22(6), 918–935 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Weihrauch, K., Zhong, N.: Computable analysis of the abstract Cauchy problem in Banach spaces and its applications I. Math. Logic Q. 53, 511–531 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Wiegner, M.: The Navier-Stokes equations - a neverending challenge? Jahresbericht der Deutschen Mathematiker Vereinigung (DMV) 101(1), 1–25 (1999). http://dml.math.uni-bielefeld.de/JB_DMV/JB_DMV_101_1.pdf zbMATHMathSciNetGoogle Scholar
  18. 18.
    Ziegler, M.: Physically-relativized church-turing hypotheses: physical foundations of computing and complexity theory of computational physics. Appl. Math. Comput. 215(4), 1431–1447 (2009)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Virginia TechBlacksburgUSA
  2. 2.University of CincinnatiCincinnatiUSA
  3. 3.TU DarmstadtDarmstadtGermany

Personalised recommendations

Cite paper

Buy options