IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2021i5p461-d504994.html
   My bibliography  Save this article

Local Well-Posedness for Free Boundary Problem of Viscous Incompressible Magnetohydrodynamics

Author

Listed:
  • Kenta Oishi

    (Department of Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, Aichi 464-8602, Japan)

  • Yoshihiro Shibata

    (Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, PA 15260, USA
    Department of Mathematics, Waseda University, Ohkubo 3-4-1, Shinjuku-ku, Tokyo 169-8555, Japan
    Partially supported by Top Global University Project, JSPS Grant-in-aid for Scientific Research (A) 17H0109, and Toyota Central Research Institute Joint Research Fund.)

Abstract

In this paper, we consider the motion of incompressible magnetohydrodynamics (MHD) with resistivity in a domain bounded by a free surface. An electromagnetic field generated by some currents in an external domain keeps an MHD flow in a bounded domain. On the free surface, free boundary conditions for MHD flow and transmission conditions for electromagnetic fields are imposed. We proved the local well-posedness in the general setting of domains from a mathematical point of view. The solutions are obtained in an anisotropic space H p 1 ( ( 0 , T ) , H q 1 ) ∩ L p ( ( 0 , T ) , H q 3 ) for the velocity field and in an anisotropic space H p 1 ( ( 0 , T ) , L q ) ∩ L p ( ( 0 , T ) , H q 2 ) for the magnetic fields with 2 < p < ∞ , N < q < ∞ and 2 / p + N / q < 1 . To prove our main result, we used the L p - L q maximal regularity theorem for the Stokes equations with free boundary conditions and for the magnetic field equations with transmission conditions, which have been obtained by Frolova and the second author.

Suggested Citation

  • Kenta Oishi & Yoshihiro Shibata, 2021. "Local Well-Posedness for Free Boundary Problem of Viscous Incompressible Magnetohydrodynamics," Mathematics, MDPI, vol. 9(5), pages 1-33, February.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:5:p:461-:d:504994
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/5/461/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/9/5/461/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(4), pages 691-705, August.
    2. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(5), pages 879-883, October.
    3. ,, 2000. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 16(2), pages 287-299, April.
    4. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(6), pages 1195-1198, December.
    5. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(1), pages 225-228, February.
    6. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(2), pages 411-413, April.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Minghe Sun, 2005. "Warm-Start Routines for Solving Augmented Weighted Tchebycheff Network Programs in Multiple-Objective Network Programming," INFORMS Journal on Computing, INFORMS, vol. 17(4), pages 422-437, November.
    2. Wong, Patricia J.Y., 2015. "Eigenvalues of a general class of boundary value problem with derivative-dependent nonlinearity," Applied Mathematics and Computation, Elsevier, vol. 259(C), pages 908-930.
    3. José Miquel Cabeças, 2008. "Characterization of cleaners accidents in the Portuguese service sector," Enterprise and Work Innovation Studies, Universidade Nova de Lisboa, IET/CICS.NOVA-Interdisciplinary Centre on Social Sciences, Faculty of Science and Technology, vol. 4(4), pages 139-155, November.
    4. Y. Zhang, 2007. "General Robust-Optimization Formulation for Nonlinear Programming," Journal of Optimization Theory and Applications, Springer, vol. 132(1), pages 111-124, January.
    5. Masatoshi Sakawa & Hideki Katagiri, 2012. "Stackelberg solutions for fuzzy random two-level linear programming through level sets and fractile criterion optimization," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 20(1), pages 101-117, March.
    6. Yakut, Oguz, 2021. "Implementation of hydraulically driven barrel shooting control by utilizing artificial neural networks," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 1206-1223.
    7. X. Qin & G. Huang, 2009. "An Inexact Chance-constrained Quadratic Programming Model for Stream Water Quality Management," Water Resources Management: An International Journal, Published for the European Water Resources Association (EWRA), Springer;European Water Resources Association (EWRA), vol. 23(4), pages 661-695, March.
    8. Md. Yousuf Gazi & Khandakar Tahmida Tafhim, 2019. "Investigation of Heavy-mineral Deposits Using Multispectral Satellite Imagery in the Eastern Coastal Margin of Bangladesh," Earth Sciences Malaysia (ESMY), Zibeline International Publishing, vol. 3(2), pages 16-22, October.
    9. Billionnet, Alain, 2011. "Solving the probabilistic reserve selection problem," Ecological Modelling, Elsevier, vol. 222(3), pages 546-554.
    10. François Clautiaux & Cláudio Alves & José Valério de Carvalho & Jürgen Rietz, 2011. "New Stabilization Procedures for the Cutting Stock Problem," INFORMS Journal on Computing, INFORMS, vol. 23(4), pages 530-545, November.
    11. Eichengreen, Barry & Kletzer, Kenneth & Mody, Ashoka, 2003. "Crisis Resolution: Next Steps," Santa Cruz Center for International Economics, Working Paper Series qt4cj974r4, Center for International Economics, UC Santa Cruz.
    12. Tansel, Aysit & Karao?lan, Deniz, 2016. "The Causal Effect of Education on Health Behaviors: Evidence from Turkey," IZA Discussion Papers 10020, Institute of Labor Economics (IZA).
    13. Di Feng & Bettina Klaus, 2022. "Preference revelation games and strict cores of multiple‐type housing market problems," International Journal of Economic Theory, The International Society for Economic Theory, vol. 18(1), pages 61-76, March.
    14. Anna Scherbina, 2021. "Assessing the Optimality of a COVID Lockdown in the United States," Economics of Disasters and Climate Change, Springer, vol. 5(2), pages 177-201, July.
    15. John McKay, 2005. "How Significant and Effective are North Korea's "Market Reforms"?," Global Economic Review, Taylor & Francis Journals, vol. 34(1), pages 83-97.
    16. Timothy K.M. Beatty & Erling Røed Larsen & Dag Einar Sommervoll, 2005. "Measuring the Price of Housing Consumption for Owners in the CPI," Discussion Papers 427, Statistics Norway, Research Department.
    17. Marco Bianchi & Carlos Tapia & Ikerne del Valle, 2020. "Monitoring domestic material consumption at lower territorial levels: A novel data downscaling method," Journal of Industrial Ecology, Yale University, vol. 24(5), pages 1074-1087, October.
    18. Sonmez, Tayfun & Utku Unver, M., 2005. "House allocation with existing tenants: an equivalence," Games and Economic Behavior, Elsevier, vol. 52(1), pages 153-185, July.
    19. Juarez, Ruben, 2013. "Group strategyproof cost sharing: The role of indifferences," Games and Economic Behavior, Elsevier, vol. 82(C), pages 218-239.
    20. Velloso, Helvia & Vézina, François & Bustillo, Inés, 2006. "The Canadian retirement income system," Documentos de Proyectos 3682, Naciones Unidas Comisión Económica para América Latina y el Caribe (CEPAL).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:9:y:2021:i:5:p:461-:d:504994. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.