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Higher-Order Global Regularity of an Inviscid Voigt-Regularization of the Three-Dimensional Inviscid Resistive Magnetohydrodynamic Equations

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Abstract

We prove existence, uniqueness, and higher-order global regularity of strong solutions to a particular Voigt-regularization of the three-dimensional inviscid resistive magnetohydrodynamic (MHD) equations. Specifically, the coupling of a resistive magnetic field to the Euler-Voigt model is introduced to form an inviscid regularization of the inviscid resistive MHD system. The results hold in both the whole space \({\mathbb{R}^3}\) and in the context of periodic boundary conditions. Weak solutions for this regularized model are also considered, and proven to exist globally in time, but the question of uniqueness for weak solutions is still open. Furthermore, we show that the solutions of the Voigt regularized system converge, as the regularization parameter \({\alpha \rightarrow 0}\), to strong solutions of the original inviscid resistive MHD, on the corresponding time interval of existence of the latter. Moreover, we also establish a new criterion for blow-up of solutions to the original MHD system inspired by this Voigt regularization.

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Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, College Station, TX, 77843, USA

    Adam Larios

  2. Department of Mathematics, and Department of Mechanical and Aero-space Engineering, University of California, Irvine, CA, 92697-3875, USA

    Edriss S. Titi

  3. Department of Mechanical and Aero-space Engineering, University of California, Irvine, CA, 92697-3875, USA

    Edriss S. Titi

  4. Department of Computer Science and Applied Mathematics, The Weizmann Institute of Science, Rehovot, 76100, Israel

    Edriss S. Titi

  5. Fellow of the Center of Smart Interfaces Technische, Universität Darmstadt, Darmstadt, Germany

    Edriss S. Titi

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  1. Adam Larios
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Correspondence to Adam Larios.

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Communicated by H. Beirão da Veiga

The authors are thankful for the warm hospitality of the Institute for Mathematics and its Applications (IMA), University of Minnesota, where part of this work was completed. This work was supported in part by the NSF grants no. DMS-0708832, DMS-1009950. E.S.T. also acknowledges the kind hospitality of the Freie Universität - Berlin, and the support of the Minerva Stiftung/Foundation.

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Larios, A., Titi, E.S. Higher-Order Global Regularity of an Inviscid Voigt-Regularization of the Three-Dimensional Inviscid Resistive Magnetohydrodynamic Equations. J. Math. Fluid Mech. 16, 59–76 (2014). https://doi.org/10.1007/s00021-013-0136-3

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