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Mathematical Methods for the Magnetohydrodynamics of Liquid Metals$
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Jean-Frédéric Gerbeau, Claude Le Bris, and Tony Lelièvre

Print publication date: 2006

Print ISBN-13: 9780198566656

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198566656.001.0001

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MATHEMATICAL ANALYSIS OF TWO-FLUID PROBLEMS

MATHEMATICAL ANALYSIS OF TWO-FLUID PROBLEMS

Chapter:
(p.145) 4 MATHEMATICAL ANALYSIS OF TWO-FLUID PROBLEMS
Source:
Mathematical Methods for the Magnetohydrodynamics of Liquid Metals
Author(s):

Jean-Frédéric Gerbeau

Claude Le Bris

Tony Lelièvre

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198566656.003.0004

This chapter deals with the theoretical aspects of multifluid magnetohydrodynamics problems. In addition to the coupling between hydrodynamics and electromagnetics examined in Chapter 2, the high nonlinearity that now needs to be addressed is geometrical in nature. It results from the presence of one (or many) free interface(s) separating the fluids, assumed non-miscible. The mathematical analysis is substantially more intricate, and a long list of simple, however open, problems can be drawn up. Throughout the chapter, the equation that plays a crucial role is the equation of the conservation of mass. Owing to an argument based on the theory of renormalized solutions, a global-in-time existence result of weak solution is proved. The long-time behaviour of sufficiently regular solutions is also investigated.

Keywords:   multifluid flows, renormalized solutions, weak solutions, long-time behaviour, transport equation, energy inequalities, a priori estimates

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