Asymptotic methods for magnetohydrodynamic instability
Authors:
Misha Vishik and Susan Friedlander
Journal:
Quart. Appl. Math. 56 (1998), 377-398
MSC:
Primary 76E25; Secondary 35Q35, 76W05
DOI:
https://doi.org/10.1090/qam/1622515
MathSciNet review:
MR1622515
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Abstract: A sufficient condition for instability is obtained for the linearized equations of ideal magnetohydrodynamics. The results are proved by studying the full asymptotic expansion for the evolution operator using oscillatory integrals. It is shown that the growth rate of the evolution operator is bounded from below by the growth rate of an operator given by a system of local hyperbolic PDEs.
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L. Boutet de Monvel, Hypoelliptic operators with double characteristics, Comm. Pure Appl. Math. 27, 585–639 (1974)
A. Calderon and R. Vaillancourt, A class of bounded pseudodifferential operators, Proc. Nat. Acad. Sci. USA 69, 1185–1187 (1972)
S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover, 1981
J. Chazarain and A. Piriou, Introduction to the Theory of Linear Partial Differential Equations, North-Holland Publishing Co., Amsterdam, New York, 1982
R. Coifman and Y. Meyer, Au delá des opérateurs pseudodifferentials, Asterisque 57, 1–185 (1978)
K. S. Eckhoff, Linear waves and stability in ideal magnetohydrodynamics, Phys. Fluids 30, no. 12, 3673–3685 (1987)
F. G. Friedlander, Sound Pulses, Cambridge University Press, 1958
S. Friedlander and M. Vishik, Instability criteria for steady flows of a perfect fluid, Chaos 2, no. 3, 455–460 (1992)
S. Friedlander and M. Vishik, On stability and instability criteria for magnetohydrodynamics, Chaos 5, no. 2, 416–423 (1995)
E. Hameiri, On the essential spectrum of ideal MHD, Comm. Pure Appl. Math. 38, 43–66 (1985)
J. Hounie, On the $L^{2}$-continuity of pseudodifferential operators, Comm. Partial Differential Equations 11, no. 7, 765–778 (1986)
A. E. Lifschitz, Continuous spectrum in general toroidal systems (ballooning and Alfven modes), Phys. Lett. 122A, 350–356 (1987)
D. Ludwig, Exact and asymptotic solution of the Cauchy problem, Comm. Pure Appl. Math. 13, 473–508 (1960)
E. Stein, Harmonic Analysis, Princeton University Press, 1994
F. Treves, Introduction to Pseudodifferential and Fourier Integral Operators, Plenum Press, New York, London, 1980
M. Vishik, Spectrum of small oscillations of an ideal fluid and Lyapunov exponents, J. Math. Pures Appl. 75, 531–557 (1996)
M. Vishik and S. Friedlander, Dynamo theory methods for hydrodynamic stability, J. Math. Pures Appl. 72, 145–180 (1993)
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© Copyright 1998
American Mathematical Society
