On approximate solutions of the equations of incompressible magnetohydrodynamics

Autor: Livio Pizzocchero, Emanuele Tassi
Přispěvatelé: Dipartimento de Matematica [Milano], Università degli Studi di Milano [Milano] (UNIMI), Joseph Louis LAGRANGE (LAGRANGE), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Observatoire de la Côte d'Azur, COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Université Côte d'Azur (UCA)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS), Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E7 Systèmes dynamiques : théories et applications, Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Università degli Studi di Milano = University of Milan (UNIMI), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Institut national des sciences de l'Univers (INSU - CNRS)-Observatoire de la Côte d'Azur, COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Université Côte d'Azur (UCA)-Université Côte d'Azur (UCA)-Centre National de la Recherche Scientifique (CNRS)
Rok vydání: 2020
Předmět:
Zdroj: Nonlinear Analysis: Theory, Methods and Applications
Nonlinear Analysis: Theory, Methods and Applications, Elsevier, 2020, 195, pp.111726. ⟨10.1016/j.na.2019.111726⟩
Nonlinear Analysis: Theory, Methods and Applications, 2020, 195, pp.111726. ⟨10.1016/j.na.2019.111726⟩
ISSN: 0362-546X
Popis: Inspired by an approach proposed previously for the incompressible Navier-Stokes (NS) equations, we present a general framework for the a posteriori analysis of the equations of incompressible magnetohydrodynamics (MHD) on a torus of arbitrary dimension d; this setting involves a Sobolev space of infinite order, made of C^infinity vector fields (with vanishing divergence and mean) on the torus. Given any approximate solution of the MHD Cauchy problem, its a posteriori analysis with the method of the present work allows to infer a lower bound on the time of existence of the exact solution, and to bound from above the Sobolev distance of any order between the exact and the approximate solution. In certain cases the above mentioned lower bound on the time of existence is found to be infinite, so one infers the global existence of the exact MHD solution. We present some applications of this general scheme; the most sophisticated one lives in dimension d=3, with the ABC flow (perturbed magnetically) as an initial datum, and uses for the Cauchy problem a Galerkin approximate solution in 124 Fourier modes. We illustrate the conclusions arising in this case from the a posteriori analysis of the Galerkin approximant; these include the derivation of global existence of the exact MHD solution with the ABC datum, when the dimensionless viscosity and resistivity are equal and stay above an explicitly given threshold value.
Author's note. Some overlaps with previous works by one of us on Navier-Stokes equations, namely: arXiv:1511.00533, arXiv:1405.3421, arXiv:1402.0487, arXiv:1310.5642, arXiv:1304.2972, arXiv:1203.6865, arXiv:1104.3832, arXiv:1009.2051, arXiv:1007.4412, arXiv:0909.3707, arXiv:0709.1670. The present work has a different subject (MHD), and the overlaps are to make the paper self-contained
Databáze: OpenAIRE
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