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