Skip to main content
SpringerLink
Log in
Book cover

Topics in Clifford Analysis pp 173–190Cite as

Birkhäuser

Applications of Parabolic Dirac Operators to the Instationary Viscous MHD Equations on Conformally Flat Manifolds

  • Chapter
  • First Online:

Part of the book series: Trends in Mathematics ((TM))

Abstract

In this paper we apply classical and recent techniques from quaternionic analysis using parabolic Dirac type operators and related Teodorescu and Cauchy-Bitzadse type operators to set up some analytic representation formulas for the solutions to the time dependent incompressible viscous magnetohydrodynamic equations on some conformally flat manifolds, such as cylinders and tori associated with different spinor bundles. Also in this context a special variant of hypercomplex Eisenstein series related to the parabolic Dirac operator serve as kernel functions.

In honor of Professor Sprößig’s 70th birthday

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. S. Bernstein, Factorization of the nonlinear Schrödinger equation and applications. Complex Var. Elliptic Equ. 51(5), 429–452 (2006)

    Article  MathSciNet  Google Scholar 

  2. X. Blanc, B. Ducomet, Weak and strong solutions of equations of compressible magnetohydrodynamics, in Handbook of Mathematical Analysis in Mechanics of Viscous Fluids (Springer, Cham, 2016)

    Google Scholar 

  3. E. Bulla, D. Constales, R.S. Kraußhar, J. Ryan, Dirac type operators for arithmetic subgroups of generalized modular groups. J. Reine Angew. Math. 643, 1–19 (2010)

    Article  MathSciNet  Google Scholar 

  4. M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations, in Handbook of Mathematical Fluid Dynamics, vol. 3, ed. by S. Friedlander, D. Serre (Elsevier, Amsterdam, 2004), pp. 161–244

    Google Scholar 

  5. P. Cerejeiras, U. Kähler, Elliptic boundary value problems of fluid dynamics over unbounded domains. Math. Methods Appl. Sci. 23(1), 81–101 (2000)

    Article  MathSciNet  Google Scholar 

  6. P. Cerejeiras, N. Vieira, Regularization of the non-stationary Schrödinger operator. Math. Methods Appl. Sci. 32(4), 535–555 (2009)

    Article  MathSciNet  Google Scholar 

  7. P. Cerejeiras, U. Kähler, F. Sommen, Parabolic Dirac operators and the Navier-Stokes equations over time-varying domains. Math. Methods Appl. Sci. 28(14), 1715–1724 (2005)

    Article  MathSciNet  Google Scholar 

  8. P. Cerejeiras, U. Kähler, R.S. Kraußhar, Some applications of parabolic Dirac operators to the instationary Navier-Stokes problem on conformally flat cylinders and tori \(\mathbb {R}^3\), in Clifford Analysis and Related Topics, ed. by P. Cerejeiras et al. (Springer, New York, 2018)

    Google Scholar 

  9. Q. Chen, C. Miao, Z. Zhang, On the regularity criterion of weak solution for the 3D viscous magneto-hydrodynamics equations. Comm. Math. Phys. 284, 919–930 (2008)

    Article  MathSciNet  Google Scholar 

  10. D. Constales, R.S. Kraußhar, On the Navier-Stokes equation with Free Convection in three dimensional triangular channels. Math. Methods Appl. Sci. 31(6), 735–751 (2008)

    Article  MathSciNet  Google Scholar 

  11. D. Constales, R.S. Kraußhar, Multiperiodic eigensolutions to the Dirac operator and applications to the generalized Helmholtz equation on flat cylinders and on the n-torus. Math. Methods Appl. Sci. 32(16), 2050–2070 (2009)

    Article  MathSciNet  Google Scholar 

  12. R. Delanghe, F. Sommen, V. Souček, Clifford Algebra and Spinor Valued Functions (Kluwer, Dortrecht, 1992)

    Book  Google Scholar 

  13. N. Faustino, K. Gürlebeck, A. Hommel, U. Kähler, Difference potentials for the Navier-Stokes equations in unbounded domains. J. Differ. Equ. Appl. 12(6), 577–595 (2006)

    Article  MathSciNet  Google Scholar 

  14. S. Gala, Extension criterion on regularity for weak solutions to the 3D MHD equations. Math. Methods Appl. Sci. 32(12), 1496–1503 (2010)

    MathSciNet  MATH  Google Scholar 

  15. Y. Ge, S. Shao, Global solution of 3D incompressible magnetohydrodyanamic equations with finite energy. J. Math. Anal. Appl. 425, 571–578 (2015)

    Article  Google Scholar 

  16. H. Goedbloed, S. Poedts, Advanced Magnetohydrodynamics: With Applications to Laboratory and Astrophysical Plasmas (Cambridge University Press, Cambridge, 2010)

    Book  Google Scholar 

  17. M. Gunzburger, A. Meir, J. Peterson, On the existence, uniqueness and finite element approximation of the equations of stationary, incompressible magnetohydrodynamics. Math. Comput. 56(194), 523–563 (1991)

    Article  MathSciNet  Google Scholar 

  18. K. Gürlebeck, W. Sprößig, Quaternionic Analysis and Elliptic Boundary Value Problems (Birkhäuser, Basel, 1990)

    Google Scholar 

  19. R.S. Kraußhar, On the incompressible viscous MHD equations and explicit solution formulas for some three dimensional radially symmetric domains, in Hypercomplex Analysis and Applications, ed. by I. Sabadini, F. Sommen. Trends in Mathematics (Birkhäuser, Basel, 2011), pp. 125–137

    Google Scholar 

  20. V. Kravchenko, Applied Quaternionic Analysis. Research and Exposition in Mathematics, vol. 28 (Heldermann Verlag, Lemgo, 2003)

    Google Scholar 

  21. N.H. Kuiper, On conformally flat spaces in the large. Ann. Math. 50, 916–924 (1949)

    Article  MathSciNet  Google Scholar 

  22. H.B. Lawson, M.-L. Michelsohn, Spin Geometry (Princeton University Press, New York, 1989)

    MATH  Google Scholar 

  23. Z. Lei, On axially symmetric incompressible magnetohydrodynamics in three dimensions. J. Diff. Equ. 259, 3202–3215 (2015)

    Article  MathSciNet  Google Scholar 

  24. C. Miao, B. Yuan, On well-posedness of the Cauchy problem for MHD systems in Besov spaces. Math. Methods Appl. Sci. 32(1), 53–76 (2010)

    Article  MathSciNet  Google Scholar 

  25. S. Rashidi, J.A. Esfahani, M. Maskaniyan, Applications of magnetohydrodynamics in biological systems-a review on the numerical studies. J. Magn. Magn. Mater. 439, 358–372 (2017)

    Article  Google Scholar 

  26. M. Sermagne, R. Temam, Some mathematical questions related to the MHD equations. Commun. Pure Appl. Math. 6, 635–664 (1983)

    MathSciNet  MATH  Google Scholar 

  27. S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3D. J. Am. Math. Soc. 12, 445–495 (1999)

    Article  Google Scholar 

  28. E. Zeidler, Nonlinear Functional Analysis and ist Applications – IV. Applications to Mathematical Physics (Springer, Berlin, 1988)

    Google Scholar 

  29. X. Zhai, Z. Yin, Global well-posedness for the 3D incompressible inhomogeneous Navier-Stokes equations and MHD equations. J. Diff. Equ. 262, 1359–1412 (2017)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The work of the third author is supported by the project Neue funktionentheoretische Methoden für instationäre PDE, funded by Programme for Cooperation in Science between Portugal and Germany, DAAD-PPP Deutschland-Portugal, Ref: 57340281. The work of the first and second authors is supported via the project New Function Theoretical Methods in Computational Electrodynamics approved under the agreement Ações Integradas Luso-Alemãs DAAD-CRUP, ref. A-15/17, and by Portuguese funds through the CIDMA—Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (“FCT–Fundação para a Ciência e a Tecnologia”), within project UID/MAT/0416/2019.

Author information

Authors and Affiliations

  1. Departamento de Matemática, Universidade de Aveiro, Aveiro, Portugal

    Paula Cerejeiras & Uwe Kähler

  2. Fachgebiet Mathematik, Erziehungswissenschaftliche Fakultät, Universität Erfurt, Erfurt, Germany

    R. Sören Kraußhar

Authors
  1. Paula Cerejeiras
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Uwe Kähler
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. R. Sören Kraußhar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paula Cerejeiras .

Editor information

Editors and Affiliations

  1. Institut für Angewandte Analysis, TU Bergakademie Freiberg, Freiberg, Germany

    Swanhild Bernstein

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Cerejeiras, P., Kähler, U., Kraußhar, R.S. (2019). Applications of Parabolic Dirac Operators to the Instationary Viscous MHD Equations on Conformally Flat Manifolds. In: Bernstein, S. (eds) Topics in Clifford Analysis. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-23854-4_8

Download citation

Publish with us

Policies and ethics

Search

Navigation