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Systems of Conservation Laws 1 : Hyperbolicity, Entropies, Shock Waves.

By: Contributor(s): Publisher: Cambridge : Cambridge University Press, 1999Copyright date: ©1999Description: 1 online resource (287 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9780511151286
Subject(s): Genre/Form: Additional physical formats: Print version:: Systems of Conservation Laws 1 : Hyperbolicity, Entropies, Shock WavesDDC classification:
  • 515.353
LOC classification:
  • QA377 .S46413 1999
Online resources:
Contents:
Cover -- Half-title -- Title -- Copyright -- Dedication -- Contents -- Acknowledgments -- Introduction -- 1 Some models -- 1.1 Gas dynamics in eulerian variables -- The law of a perfect gas -- The Euler equations -- The entropy -- Barotropic models -- 1.2 Gas dynamics in lagrangian variables -- Criticism of the change of variables -- 1.3 The equation of road traffic -- 1.4 Electromagnetism -- Maxwell's equations -- Plane waves -- 1.5 Magneto-hydrodynamics -- Plane waves in M.H.D. -- A simplified model of waves -- 1.6 Hyperelastic materials -- Strings and membranes -- 1.7 Singular limits of dispersive equations -- 1.8 Electrophoresis -- 2 Scalar equations in dimension d = 1 -- 2.1 Classical solutions of the Cauchy problem -- The linear case -- Non-linear case. The method of characteristics -- Blow-up in finite time -- 2.2 Weak solutions, non-uniqueness -- The Rankine-Hugoniot condition -- Non-uniqueness for the Cauchy problem -- 2.3 Entropy solutions, the Kružkov existence theorem -- Approximate solutions -- entropy inequalities -- Irreversibility -- Existence and uniqueness for the Cauchy problem -- Application: admissible discontinuities -- Piecewise smooth entropy solutions -- Oleinik's condition -- Shocks -- 2.4 The Riemann problem -- Self-similar solutions. Rarefactions -- The solution of the Riemann problem -- 2.5 The case of f convex. The Lax formula -- The Hamilton-Jacobi equation -- A dual formula to Lax's -- 2.6 Proof of Theorem 2.3.5: existence -- The approach by semi-groups -- Accretivity of A -- Passage to the limit -- The general case -- 2.7 Proof of Theorem 2.3.5: uniqueness -- An inequality for two entropy solutions -- Integration of the inequality (2.19) -- End of the proof of Proposition 2.3.6 -- 2.8 Comments -- Oleinik's inequality -- Initial datum with bounded total variation -- Uniqueness: the duality method -- 2.9 Exercises.
3 Linear and quasi-linear systems -- 3.1 Linear hyperbolic systems -- Fourier analysis -- Geometric conditions of hyperbolicity -- Plane waves -- Exercises -- 3.2 Quasi-linear hyperbolic systems -- 3.3 Conservative systems -- 3.4 Entropies, convexity and hyperbolicity -- Physical systems -- Proof of theorem -- Exercises -- 3.5 Weak solutions and entropy solutions -- The Rankine-Hugoniot condition -- Reversibility -- 3.6 Local existence of smooth solutions -- Indications about the proof -- A priori estimate of… -- Proof of Lemma 3.6.2 -- Convergence of the iterative scheme -- 3.7 The wave equation -- Huygens' principle -- Conservation and decay -- 4 Dimension d = 1, the Riemann problem -- 4.1 Generalities on the Riemann problem -- 4.2 The Hugoniot locus -- Local description of the Hugoniot locus -- Exercises -- Some symmetric functions -- Proof of Theorem 4.2.1 -- 4.3 Shock waves -- Entropy balance -- Proof of Lemmas 4.3.2 and 4.3.3 -- Genuinely non-linear characteristic fields -- Exercise -- 4.4 Contact discontinuities -- Riemann invariants -- Exercises -- 4.5 Rarefaction waves. Wave curves -- Parametrisation of wave curves -- 4.6 Lax's theorem -- The form of the solution of the Riemann problem -- Local existence of the solution of the Riemann problem -- 4.7 The solution of the Riemann problem for the p-system -- Hypotheses -- Rarefaction waves -- Shocks -- Wave curves -- The solution of the Riemann problem -- 4.8 The solution of the Riemann problem for gas dynamics -- Hypotheses -- The rarefaction waves -- The shocks -- The 1-shock-waves -- The 3-shock-waves -- Parametrisation of shock curves -- Wave curves -- The solution of the Riemann problem -- The case of a perfect gas -- 4.9 Exercises -- 5 The Glimm scheme -- 5.1 Functions of bounded variation -- 5.2 Description of the scheme -- 5.3 Consistency -- 5.4 Convergence -- Compactness.
Estimate of the error -- Conclusion -- Entropy inequalities -- 5.5 Stability -- Supplements apropos of the local Riemann problem -- A linear functional -- A quadratic functional -- The induction -- 5.6 The example of Nishida -- Hypotheses and theorem -- A distance in U -- Stability -- The isothermal model of gas dynamics -- 5.7 2 × 2 Systems with diminishing total variation -- Description -- Stability -- 5.8 Technical lemmas -- Proof of Lemma 5.5.2 -- Proof of Theorem 5.1.3 -- 5.9 Supplementary remarks -- Other numerical schemes -- The rich case -- 'Continuous' Glimm functional -- 5.10 Exercises -- 6 Second order perturbations -- 6.1 Dissipation by viscosity -- Non-dissipative case -- Dissipation or production of entropy -- Partially hyperbolic systems -- 6.2 Global existence in the strictly dissipative case -- Local existence in L -- Norms -- Hypotheses -- Estimate of the derivatives -- Proof of lemma -- Extension of the solution… -- Existence with a small diffusion -- 6.3 Smooth convergence as… -- The energy estimate -- Two most favourable cases -- Uniformity of the existence times -- 6.4 Scalar case. Accuracy of approximation -- Proof of the lemma -- Proof of Theorem 6.4.2 -- 6.5 Exercises -- 7 Viscosity profiles for shock waves -- 7.1 Typical example of a limit of viscosity solutions -- Profiles vs. Lax's entropy condition -- Profile vs. Lax's shock condition -- 7.2 Existence of the viscosity profile for a weak shock -- The scalar case -- The case of weak shocks with B = b(u)I -- Extensions of Theorem 7.2.1… -- 7.3 Profiles for gas dynamics -- Isentropic fluid with viscosity -- 7.4 Asymptotic stability -- Generalities on the stability of profiles -- The scalar case -- 7.5 Stability of the profile for a Lax shock -- Transport vs. diffusion -- Non-linear diffusion waves -- The rôle of the terms… -- Calculation of the diffusion waves.
Liu's theorem -- 7.6 Influence of the diffusion tensor -- Example: gas dynamics -- 7.7 Case of over-compressive shocks -- Example 7.7.2 -- Over-compressive shocks -- Instability of the over-compressive shock -- 7.8 Exercises -- Bibliography -- Index.
Summary: Graduate text on mathematical theory of conservation laws and partial differential equations.
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Cover -- Half-title -- Title -- Copyright -- Dedication -- Contents -- Acknowledgments -- Introduction -- 1 Some models -- 1.1 Gas dynamics in eulerian variables -- The law of a perfect gas -- The Euler equations -- The entropy -- Barotropic models -- 1.2 Gas dynamics in lagrangian variables -- Criticism of the change of variables -- 1.3 The equation of road traffic -- 1.4 Electromagnetism -- Maxwell's equations -- Plane waves -- 1.5 Magneto-hydrodynamics -- Plane waves in M.H.D. -- A simplified model of waves -- 1.6 Hyperelastic materials -- Strings and membranes -- 1.7 Singular limits of dispersive equations -- 1.8 Electrophoresis -- 2 Scalar equations in dimension d = 1 -- 2.1 Classical solutions of the Cauchy problem -- The linear case -- Non-linear case. The method of characteristics -- Blow-up in finite time -- 2.2 Weak solutions, non-uniqueness -- The Rankine-Hugoniot condition -- Non-uniqueness for the Cauchy problem -- 2.3 Entropy solutions, the Kružkov existence theorem -- Approximate solutions -- entropy inequalities -- Irreversibility -- Existence and uniqueness for the Cauchy problem -- Application: admissible discontinuities -- Piecewise smooth entropy solutions -- Oleinik's condition -- Shocks -- 2.4 The Riemann problem -- Self-similar solutions. Rarefactions -- The solution of the Riemann problem -- 2.5 The case of f convex. The Lax formula -- The Hamilton-Jacobi equation -- A dual formula to Lax's -- 2.6 Proof of Theorem 2.3.5: existence -- The approach by semi-groups -- Accretivity of A -- Passage to the limit -- The general case -- 2.7 Proof of Theorem 2.3.5: uniqueness -- An inequality for two entropy solutions -- Integration of the inequality (2.19) -- End of the proof of Proposition 2.3.6 -- 2.8 Comments -- Oleinik's inequality -- Initial datum with bounded total variation -- Uniqueness: the duality method -- 2.9 Exercises.

3 Linear and quasi-linear systems -- 3.1 Linear hyperbolic systems -- Fourier analysis -- Geometric conditions of hyperbolicity -- Plane waves -- Exercises -- 3.2 Quasi-linear hyperbolic systems -- 3.3 Conservative systems -- 3.4 Entropies, convexity and hyperbolicity -- Physical systems -- Proof of theorem -- Exercises -- 3.5 Weak solutions and entropy solutions -- The Rankine-Hugoniot condition -- Reversibility -- 3.6 Local existence of smooth solutions -- Indications about the proof -- A priori estimate of… -- Proof of Lemma 3.6.2 -- Convergence of the iterative scheme -- 3.7 The wave equation -- Huygens' principle -- Conservation and decay -- 4 Dimension d = 1, the Riemann problem -- 4.1 Generalities on the Riemann problem -- 4.2 The Hugoniot locus -- Local description of the Hugoniot locus -- Exercises -- Some symmetric functions -- Proof of Theorem 4.2.1 -- 4.3 Shock waves -- Entropy balance -- Proof of Lemmas 4.3.2 and 4.3.3 -- Genuinely non-linear characteristic fields -- Exercise -- 4.4 Contact discontinuities -- Riemann invariants -- Exercises -- 4.5 Rarefaction waves. Wave curves -- Parametrisation of wave curves -- 4.6 Lax's theorem -- The form of the solution of the Riemann problem -- Local existence of the solution of the Riemann problem -- 4.7 The solution of the Riemann problem for the p-system -- Hypotheses -- Rarefaction waves -- Shocks -- Wave curves -- The solution of the Riemann problem -- 4.8 The solution of the Riemann problem for gas dynamics -- Hypotheses -- The rarefaction waves -- The shocks -- The 1-shock-waves -- The 3-shock-waves -- Parametrisation of shock curves -- Wave curves -- The solution of the Riemann problem -- The case of a perfect gas -- 4.9 Exercises -- 5 The Glimm scheme -- 5.1 Functions of bounded variation -- 5.2 Description of the scheme -- 5.3 Consistency -- 5.4 Convergence -- Compactness.

Estimate of the error -- Conclusion -- Entropy inequalities -- 5.5 Stability -- Supplements apropos of the local Riemann problem -- A linear functional -- A quadratic functional -- The induction -- 5.6 The example of Nishida -- Hypotheses and theorem -- A distance in U -- Stability -- The isothermal model of gas dynamics -- 5.7 2 × 2 Systems with diminishing total variation -- Description -- Stability -- 5.8 Technical lemmas -- Proof of Lemma 5.5.2 -- Proof of Theorem 5.1.3 -- 5.9 Supplementary remarks -- Other numerical schemes -- The rich case -- 'Continuous' Glimm functional -- 5.10 Exercises -- 6 Second order perturbations -- 6.1 Dissipation by viscosity -- Non-dissipative case -- Dissipation or production of entropy -- Partially hyperbolic systems -- 6.2 Global existence in the strictly dissipative case -- Local existence in L -- Norms -- Hypotheses -- Estimate of the derivatives -- Proof of lemma -- Extension of the solution… -- Existence with a small diffusion -- 6.3 Smooth convergence as… -- The energy estimate -- Two most favourable cases -- Uniformity of the existence times -- 6.4 Scalar case. Accuracy of approximation -- Proof of the lemma -- Proof of Theorem 6.4.2 -- 6.5 Exercises -- 7 Viscosity profiles for shock waves -- 7.1 Typical example of a limit of viscosity solutions -- Profiles vs. Lax's entropy condition -- Profile vs. Lax's shock condition -- 7.2 Existence of the viscosity profile for a weak shock -- The scalar case -- The case of weak shocks with B = b(u)I -- Extensions of Theorem 7.2.1… -- 7.3 Profiles for gas dynamics -- Isentropic fluid with viscosity -- 7.4 Asymptotic stability -- Generalities on the stability of profiles -- The scalar case -- 7.5 Stability of the profile for a Lax shock -- Transport vs. diffusion -- Non-linear diffusion waves -- The rôle of the terms… -- Calculation of the diffusion waves.

Liu's theorem -- 7.6 Influence of the diffusion tensor -- Example: gas dynamics -- 7.7 Case of over-compressive shocks -- Example 7.7.2 -- Over-compressive shocks -- Instability of the over-compressive shock -- 7.8 Exercises -- Bibliography -- Index.

Graduate text on mathematical theory of conservation laws and partial differential equations.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2019. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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