Stability of Strong Discontinuities in Magnetohydrodynamics and Electrohydrodynamics

By: Alexander Blokhin, Yuri Trakhinin

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Hardcover | 1 December 2002

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This monograph examines multidimensional stability of strong discontinuities (e.g. shock waves) for systems of conservation laws and surveys the author's results for models of ideal magnetohydrodynamics (classical, 'pressure anisotropic', relativistic) and electrohydrodynamics. The primary attention is concentrated on linearised stability analysis, especially on the issue of uniform stability in the sense of the uniform Kreiss-Lopatinski condition. A so-called 'equational' approach based on obtaining, by the dissipative integrals technique, a priori estimates without loss of smoothness for corresponding linearised stability problems in the domains of uniform stability is described. Recent results for ideal models of MHD (classical MHD, 'pressure anisotropic' MHD of Chew, Goldberger and Low, relativistic MHD) and also for a certain non-hyperbolic model are presented as the system of electrohydrodynamics (EHD).

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Non-Fiction Science Physics Classical Mathematics Dynamics & Statics Engineering & Technology Mechanical Engineering & Materials Materials Science Mechanics of Solids Dynamics & Vibration Electricity Science in General

Preface	p. ix
Introduction	p. 1
Stability of strong discontinuities	p. 1
Standard physical approach to the stability problem	p. 2
"Equational" approach	p. 3
Basic steps of the "equational" approach to the stability analysis	p. 7
Symmetrization of quasilinear systems of conservation laws	p. 7
Equations of strong discontinuity	p. 11
Setting of the linearized stability problem	p. 12
Linearization of quasilinear equations and jump conditions	p. 12
Evolutionarity condition as the necessary one for well-posedness	p. 14
Important remarks on evolutionary (classical) and nonevolutionary (nonclassical) discontinuities	p. 15
Classical Lax discontinuities	p. 16
Separating of instability domains	p. 17
Uniform linearized stability	p. 21
Uniform Lopatinski condition	p. 22
Concept of uniform stability	p. 25
A priori estimates for the LSP	p. 26
Passage to nonlinear stability	p. 27
Neutral stability	p. 28
Method of dissipative energy integrals	p. 32
Advantages of the "equational" approach	p. 34
Equations of ideal magneto- and electrohydrodynamics	p. 37
Equations of magnetohydrodynamics for an ideal medium	p. 37
Equations of magnetohydrodynamics with anisotropic pressure	p. 39
Equations of electrohydrodynamics for an ideal medium	p. 43
Symmetrization of MHD equations	p. 47
Symmetric form of the MHD equations (3.1)-(3.4) and its properties	p. 47
Symmetrization of the MHD equations (3.1)-(3.4)	p. 47
Local estimates for smooth periodic solutions to the Cauchy problem	p. 53
Symmetrization of MHD equations with anisotropic pressure	p. 60
Strong discontinuities in magneto- and electrohydrodynamics	p. 69
Strong discontinuities in MHD	p. 69
Equations of MHD strong discontinuities	p. 69
Classification of MHD strong discontinuities	p. 70
Strong discontinuities in plasma with anisotropic pressure	p. 73
Equations of strong discontinuities in MHD CGL	p. 73
Classification of strong discontinuities in MHD CGL	p. 74
Deduction of the analog of the Hugoniot adiabat for MHD CGL	p. 76
Strong discontinuities in EHD	p. 79
Setting of main linearized stability problems in magneto- and electrohydrodynamics	p. 81
Setting of the main LSP for MHD strong discontinuities	p. 81
Magnetoacoustic system	p. 81
The main LSP for MHD strong discontinuities	p. 84
Setting of the main LSP for strong discontinuities in MHD CGL	p. 86
Linear system of MHD with anisotropic pressure	p. 86
The main LSP for strong discontinuities in MHD CGL	p. 90
Setting of the main LSP for EHD shock waves	p. 93
Linear EHD system for a polytropic gas	p. 93
The main LSP for EHD shock waves	p. 95
Evolutionarity analysis for shock waves in magneto- and electrohydrodynamics	p. 99
Evolutionary shock waves in MHD	p. 99
Fast MHD shock wave	p. 101
Slow MHD shock wave	p. 105
Evolutionary shock waves in MHD CGL	p. 107
Fast shock wave in MHD CGL	p. 110
Slow shock wave in MHD CGL	p. 114
EHD shock waves	p. 115
Stability of fast MHD shock waves	p. 119
Setting of the LSP for fast MHD shock waves	p. 121
Uniform stability of fast MHD shocks under a weak magnetic field	p. 127
On 3-D stability	p. 137
Uniform stability condition for fast parallel MHD shock waves	p. 145
The LSP for fast parallel shocks	p. 145
The LC and the ULC for hyperbolic IBVP's with the 1-shock property	p. 147
The LC for Problem 8.4.1	p. 150
The ULC for Problem 8.4.1	p. 152
A complete 2-D stability analysis of fast MHD shocks in a polytropic gas	p. 155
Numerical testing of the LC and the ULC	p. 155
Numerical investigation of the stability of fast MHD shocks	p. 159
Concluding remarks	p. 163
Stability of slow MHD shock waves	p. 167
Setting of the LSP for slow MHD shock waves	p. 167
Instability of slow MHD shocks under a strong magnetic field	p. 170
Stability of MHD contact discontinuity	p. 179
Setting of the LSP for contact discontinuity	p. 179
Uniform stability of MHD contact discontinuity	p. 184
Stability of rotational discontinuity in MHD	p. 187
Setting of the LSP for rotational discontinuity	p. 188
Equivalent statements of Problem 11.1.1	p. 193
Instability of rotational discontinuity under a strong magnetic field	p. 198
Stability of MHD tangential discontinuity	p. 203
Setting of the LSP for tangential discontinuity	p. 204
Ill-posedness of Problem 12.1.1	p. 214
Stability of MHD shock waves in a collisionless magnetized plasma	p. 221
Setting of the LSP for fast MHD CGL shock waves	p. 222
Uniform stability of the fast parallel MHD CGL shock wave	p. 227
Weak stability of the fast perpendicular MHD CGL shock wave. Sufficient condition for uniform stability	p. 229
Setting of the LSP for slow MHD CGL shock waves	p. 231
Instability of the slow parallel MHD CGL shock wave in a cold plasma	p. 236
Stability of rotational discontinuity in MHD CGL	p. 241
Setting of the LSP for rotational discontinuity in MHD CGL	p. 241
Equivalent statements of Problem 14.1.1	p. 249
Instability of rotational discontinuity in a cold plasma	p. 254
Stability of EHD shock waves	p. 259
The LSP for EHD shock waves	p. 259
Uniform stability of EHD shock waves under a small surface charge	p. 262
Shock waves in relativistic MHD	p. 275
Equations of relativistic MHD	p. 276
Symmetric form and hyperbolicity conditions	p. 277
Shock waves in relativistic MHD: the LSP for parallel shocks	p. 279
A Complete stability analysis of fast parallel shocks for an arbitrary state equation	p. 284
Instability domain for fast parallel relativistic MHD shock waves	p. 284
Uniform stability domain for fast parallel relativistic MHD shock waves	p. 288
References	p. 293
Index	p. 305
Table of Contents provided by Ingram. All Rights Reserved.

Stability of Strong Discontinuities in Magnetohydrodynamics and Electrohydrodynamics

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