Large Time Behavior of Solutions to 3-D MHD System with Initial Data Near Equilibrium

Large Time Behavior of Solutions to 3-D MHD System with Initial Data Near Equilibrium Califano and Chiuderi (Phys Rev E 60 (PartB):4701–4707, 1999) conjectured that the energy of an incompressible Magnetic hydrodynamical system is dissipated at a rate that is independent of the ohmic resistivity. The goal of this paper is to mathematically justify this conjecture in three space dimensions provided that the initial magnetic field and velocity is a small perturbation of the equilibrium state (e 3, 0). In particular, we prove that for such data, a 3-D incompressible MHD system without magnetic diffusion has a unique global solution. Furthermore, the velocity field and the difference between the magnetic field and e 3 decay to zero in both L ∞ and L 2 norms with explicit rates. We point out that the decay rate in the L 2 norm is optimal in sense that this rate coincides with that of the linear system. The main idea of the proof is to exploit Hörmander’s version of the Nash–Moser iteration scheme, which is very much motivated by the seminar papers by Klainerman (Commun Pure Appl Math 33:43–101, 1980, Arch Ration Mech Anal 78:73–98, 1982, Long time behaviour of solutions to nonlinear wave equations. PWN, Warsaw, pp 1209–1215, 1984) on the long time behavior to the evolution equations. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archive for Rational Mechanics and Analysis Springer Journals

Large Time Behavior of Solutions to 3-D MHD System with Initial Data Near Equilibrium

Loading next page...
 
/lp/springer_journal/large-time-behavior-of-solutions-to-3-d-mhd-system-with-initial-data-ohgLM8uKS8
Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer-Verlag GmbH Germany, part of Springer Nature
Subject
Physics; Classical Mechanics; Physics, general; Theoretical, Mathematical and Computational Physics; Complex Systems; Fluid- and Aerodynamics
ISSN
0003-9527
eISSN
1432-0673
D.O.I.
10.1007/s00205-018-1265-x
Publisher site
See Article on Publisher Site

Abstract

Califano and Chiuderi (Phys Rev E 60 (PartB):4701–4707, 1999) conjectured that the energy of an incompressible Magnetic hydrodynamical system is dissipated at a rate that is independent of the ohmic resistivity. The goal of this paper is to mathematically justify this conjecture in three space dimensions provided that the initial magnetic field and velocity is a small perturbation of the equilibrium state (e 3, 0). In particular, we prove that for such data, a 3-D incompressible MHD system without magnetic diffusion has a unique global solution. Furthermore, the velocity field and the difference between the magnetic field and e 3 decay to zero in both L ∞ and L 2 norms with explicit rates. We point out that the decay rate in the L 2 norm is optimal in sense that this rate coincides with that of the linear system. The main idea of the proof is to exploit Hörmander’s version of the Nash–Moser iteration scheme, which is very much motivated by the seminar papers by Klainerman (Commun Pure Appl Math 33:43–101, 1980, Arch Ration Mech Anal 78:73–98, 1982, Long time behaviour of solutions to nonlinear wave equations. PWN, Warsaw, pp 1209–1215, 1984) on the long time behavior to the evolution equations.

Journal

Archive for Rational Mechanics and AnalysisSpringer Journals

Published: Jun 2, 2018

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off