Non-blow-up of the 3D ideal magnetohydrodynamics equations for a class of three-dimensional initial data in cylindrical domains

Alex Mahalov, B. Nicolaenko, F. Golse

Research output: Contribution to journalArticle

Abstract

The non blow-up of the 3D ideal incompressible magnetohydrodynamics (MHD) equations is proved for a class of three-dimensional initial data characterized by uniformly large vorticity and magnetic field in bounded cylindrical domains. There are no conditional assumptions on properties of solutions at later times, nor are the global solutions close to some 2D manifold. The approach of proving regularity is based on investigation of fast, singular, oscillating limits and nonlinear averaging methods in the context of almost periodic functions. We establish the global regularity of the 3D limit resonant MHD equations without any restrictions on the size of the 3D initial data. After establishing the strong convergence to the limit resonant equations, we bootstrap this into the regularity on arbitrarily large time intervals for solutions of the 3D MHD equations with weakly-aligned uniformly large vorticity and magnetic field at t = 0. Bibliography: 36 titles.

Original languageEnglish (US)
Pages (from-to)3768-3777
Number of pages10
JournalJournal of Mathematical Sciences
Volume136
Issue number2
DOIs
StatePublished - Jul 2006

Fingerprint

Magnetohydrodynamic Equations
Magnetohydrodynamics
Vorticity
Three-dimensional
Regularity
Magnetic Field
Magnetic fields
Global Regularity
Almost Periodic Functions
Averaging Method
Bibliographies
Strong Convergence
Global Solution
Bootstrap
Blow-up
Restriction
Interval
Class

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Non-blow-up of the 3D ideal magnetohydrodynamics equations for a class of three-dimensional initial data in cylindrical domains. / Mahalov, Alex; Nicolaenko, B.; Golse, F.

In: Journal of Mathematical Sciences, Vol. 136, No. 2, 07.2006, p. 3768-3777.

Research output: Contribution to journalArticle

@article{4e0790170a264055bca6fdd0fdcc7413,
title = "Non-blow-up of the 3D ideal magnetohydrodynamics equations for a class of three-dimensional initial data in cylindrical domains",
abstract = "The non blow-up of the 3D ideal incompressible magnetohydrodynamics (MHD) equations is proved for a class of three-dimensional initial data characterized by uniformly large vorticity and magnetic field in bounded cylindrical domains. There are no conditional assumptions on properties of solutions at later times, nor are the global solutions close to some 2D manifold. The approach of proving regularity is based on investigation of fast, singular, oscillating limits and nonlinear averaging methods in the context of almost periodic functions. We establish the global regularity of the 3D limit resonant MHD equations without any restrictions on the size of the 3D initial data. After establishing the strong convergence to the limit resonant equations, we bootstrap this into the regularity on arbitrarily large time intervals for solutions of the 3D MHD equations with weakly-aligned uniformly large vorticity and magnetic field at t = 0. Bibliography: 36 titles.",
author = "Alex Mahalov and B. Nicolaenko and F. Golse",
year = "2006",
month = "7",
doi = "10.1007/s10958-006-0198-3",
language = "English (US)",
volume = "136",
pages = "3768--3777",
journal = "Journal of Mathematical Sciences",
issn = "1072-3374",
publisher = "Springer Science and Business Media Deutschland GmbH",
number = "2",

}

TY - JOUR

T1 - Non-blow-up of the 3D ideal magnetohydrodynamics equations for a class of three-dimensional initial data in cylindrical domains

AU - Mahalov, Alex

AU - Nicolaenko, B.

AU - Golse, F.

PY - 2006/7

Y1 - 2006/7

N2 - The non blow-up of the 3D ideal incompressible magnetohydrodynamics (MHD) equations is proved for a class of three-dimensional initial data characterized by uniformly large vorticity and magnetic field in bounded cylindrical domains. There are no conditional assumptions on properties of solutions at later times, nor are the global solutions close to some 2D manifold. The approach of proving regularity is based on investigation of fast, singular, oscillating limits and nonlinear averaging methods in the context of almost periodic functions. We establish the global regularity of the 3D limit resonant MHD equations without any restrictions on the size of the 3D initial data. After establishing the strong convergence to the limit resonant equations, we bootstrap this into the regularity on arbitrarily large time intervals for solutions of the 3D MHD equations with weakly-aligned uniformly large vorticity and magnetic field at t = 0. Bibliography: 36 titles.

AB - The non blow-up of the 3D ideal incompressible magnetohydrodynamics (MHD) equations is proved for a class of three-dimensional initial data characterized by uniformly large vorticity and magnetic field in bounded cylindrical domains. There are no conditional assumptions on properties of solutions at later times, nor are the global solutions close to some 2D manifold. The approach of proving regularity is based on investigation of fast, singular, oscillating limits and nonlinear averaging methods in the context of almost periodic functions. We establish the global regularity of the 3D limit resonant MHD equations without any restrictions on the size of the 3D initial data. After establishing the strong convergence to the limit resonant equations, we bootstrap this into the regularity on arbitrarily large time intervals for solutions of the 3D MHD equations with weakly-aligned uniformly large vorticity and magnetic field at t = 0. Bibliography: 36 titles.

UR - http://www.scopus.com/inward/record.url?scp=33744825254&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33744825254&partnerID=8YFLogxK

U2 - 10.1007/s10958-006-0198-3

DO - 10.1007/s10958-006-0198-3

M3 - Article

AN - SCOPUS:33744825254

VL - 136

SP - 3768

EP - 3777

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 2

ER -