Hyperbolic Systems of Balance Laws : Lectures given at the C. I. M. E. Summer School held in Cetraro, Italy, July 14-21 2003

by Bressan, Alberto; Serre, Denis; Williams, Mark; Zumbrun, Kevin; Marcati, Pierangelo

• ISBN13: 9783540721864
• ISBN10: 354072186X
• Format: Paperback
• Copyright: 2007-09-01
• Publisher: Springer Verlag

Summary

The present Cime volume includes 4 lecture courses by Bressan, Serre, Zumbrun and Williams and a Tutorial by Bressan on the Center Manifold Theorem. Bressan's notes start with an extensive review of hyperbolic conservation laws. Then he introduces the vanishing viscosity approach and explains clearly the building blocks of the theory in particular the crucial role of the decomposition by travelling waves. Serre focuses on existence and stability for discrete shock profiles, he reviews the existence both in the rational and in the irrational cases and gives a concise introduction to the use of spectral methods for stability analysis. Finally the lectures by Williams and Zumbrun deal with the stability of multidimensional fronts. Williams' lectures describe the stability of multidimensional viscous shocks. Zumbrun discusses planar stability for viscous shocks with a realistic physical viscosity, and necessary and sufficient conditions for nonlinear stability.

Table of Contents

BV Solutions to Hyperbolic Systems by Vanishing Viscosity	p. 1
Introduction	p. 1
Review of Hyperbolic Conservation Laws	p. 6
Centered Rarefaction Waves	p. 7
Shocks and Admissibility Conditions	p. 8
Solution of the Riemann Problem	p. 11
Glimm and Front Tracking Approximations	p. 12
A Semigroup of Solutions	p. 15
Uniqueness and Characterization of Entropy Weak Solutions	p. 17
The Vanishing Viscosity Approach	p. 19
Parabolic Estimates	p. 25
Decomposition by Traveling Wave Profiles	p. 34
Interaction of Viscous Waves	p. 48
Stability of Viscous Solutions	p. 67
The Vanishing Viscosity Limit	p. 70
References	p. 76
Discrete Shock Profiles: Existence and Stability	p. 79
Introduction	p. 81
Existence Theory Rational Case	p. 86
Steady Lax Shocks	p. 87
More Complex Situations	p. 91
Other Rational Values of ¿	p. 92
Explicit Profiles for the Godunov Scheme	p. 92
DSPs for Strong Steady Shocks Under the Lax-Wendroff Scheme	p. 95
Scalar Shocks Under Monotone Schemes	p. 97
Under-Compressive Shocks	p. 98
An Example from Reaction-Diffusion	p. 99
Homoclinic and Chaotic Orbits	p. 101
Exponentially Small Splitting	p. 102
Conclusions	p. 103
Existence Theory the Irrational Case	p. 104
Obstructions	p. 105
The Small Divisors Problem	p. 105
The Function Y	p. 106
Counter-Examples to (2.9)	p. 108
The Lax-Friedrichs Scheme with an Almost Linear Flux	p. 111
The Scalar Case	p. 113
The Approach by Liu and Yu	p. 116
Semi-Discrete vs Discrete Traveling Waves	p. 117
Semi-Discrete Profiles	p. 118
A Strategy Towards Fully Discrete Traveling Waves	p. 118
Sketch of Proof of Theorem 3.1	p. 120
The Richness of Discrete Dynamics	p. 123
Stability Analysis: The Evans Function	p. 125
Spectral Stability	p. 126
The Essential Spectrum of L	p. 127
Construction of the Evans Function	p. 131
The Gap Lemma	p. 132
The Geometric Separation	p. 133
Stability Analysis: Calculations	p. 135
Calculations with Lax Shocks	p. 137
The Homotopy from ¿ =1 to ∞	p. 139
The Large Wave-Length Analysis	p. 140
Conclusions	p. 141
Calculations with Under-Compressive Shocks	p. 144
Results for the Godunov Scheme	p. 145
The Caseof Perfect Gases	p. 150
The Role of the Functional Y in the Nonlinear Stability	p. 151
References	p. 156
Stability of Multidimensional Viscous Shocks	p. 159
Lecture One: The Small Viscosity Limit: Introduction, Approximate Solution	p. 160
Approximate Solution	p. 162
Summary	p. 166
Lecture Two: Full Linearization, Reduction to ODEs, Conjugation to a Limiting Problem	p. 167
Full Versus Partial Linearization	p. 167
The Extra Boundary Condition	p. 169
Corner Compatible Initial Data and Reduction to a Forward
Problem	p. 170
Principal Parts, Exponential Weights	p. 171
Some Difficulties	p. 172
Semiclassical Form	p. 173
Frozen Coefficients; ODEs Depending on Frequencies as Parameters	p. 174
Three Frequency Regimes	p. 175
First-Order System	p. 175
Conjugation	p. 176
Conjugation to HP Form	p. 178
Lecture Three: Evans Functions, Lopatinski Determinants, Removing the Translational Degeneracy	p. 178
Evans Functions, Instabilities, the Zumbrun-Serre Result	p. 179
The Evans Function as a Lopatinski Determinant	p. 182
Doubling	p. 182
Slow Modes and Fast Modes	p. 183
Removing the Translational Degeneracy	p. 184
Lecture Four: Block Structure, Symmetrizers, Estimates	p. 187
The MF Regime	p. 187
The SF Regime	p. 190
The Sign Condition	p. 192
Glancing Blocks and Glancing Modes	p. 193
Auxiliary Hypothesis for Lecture 5	p. 195
The SF Estimate	p. 196
The HF Regime	p. 198
Summary of Estimates	p. 198
Lecture Five: Long Time Stability via Degenerate Symmetrizers	p. 200
Nonlinear Stability	p. 201
L1 - L2 Estimates	p. 202
Proof of Proposition 5.1	p. 204
The Dual Problem	p. 205
Decomposition of UH&plusmm;	p. 206
Interior Estimates	p. 208
L∞ Estimates	p. 211
Nonlinear Stability Results	p. 211
Appendix A: The Uniform Stability Determinant	p. 212
Appendix B: Continuity of Decaying Eigenspaces	p. 213
Appendix C: Limits as z → ±∞ of Slow Modes at Zero Frequency	p. 215
Appendix D: Evans ↠ Transversality + Uniform Stability	p. 216
Appendix E: Proofs in Lecture 3	p. 219
Construction of R	p. 219
Propositions 3.4 and 3.5	p. 220
Appendix F: The HF Estimate	p. 221
Block Stucture	p. 223
Symmetrizer and Estimate	p. 223
Appendix G: Transition to PDE Estimates	p. 225
References	p. 226
Planar Stability Criteria for Viscous Shock Waves of Systems with Real Viscosity	p. 229
Introduction: Structure of Physical Equations	p. 230
Description of Results	p. 242
Analytical Preliminaries	p. 251
Reduction to Low Frequency	p. 266
Low Frequency Analysis/Completion of Proofs	p. 284
Appendices	p. 305
Appendix A: Semigroup Facts	p. 305
Appendix B: Proof of Proposition 1.21	p. 315
Appendix C: Proof of Proposition 5.15	p. 317
References	p. 320
Tutorial on the Center Manifold Theorem	p. 327
Review of Linear O. D.E's	p. 327
Statement of the Center Manifold Theorem	p. 329
Proof of the Center Manifold Theorem	p. 331
Reduction to the Case of a Compact Perturbation	p. 331
Characterization of the Global Center Manifold	p. 332
Construction of the Center Manifold	p. 333
Proof of the Invariance Property (ii)	p. 335
Proof of (iv)	p. 335
Proof of the Tangency Property (iii)	p. 335
Proof of the Asymptotic Approximation Property (v)	p. 336
Smoothness of the Center Manifold	p. 337
The Contraction Mapping Theorem	p. 342
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