This book are notes prepared for the PhD courses that the author has been teaching during the last 10 years. The material available in the already existing literature (papers and essays) has been collected in this unique text, presenting the results with all the details for the reader’s convenience, fixing a unified notation, and providing a consistent framework for the subject. These notes cover many of the arguments that usually can be found in high level essays, where the proofs are simply sketched, and in papers, which are not easily available and not always self-contained. This book is intended for 1. PhD students in Mathematics, Physics and Mechanical Engineering in order to learn the basic features of nonlinear scalar equations, 2. researchers interested in nonlinear hyperbolic PDEs in order to learn the details behind some known and deep results on nonlinear scalar equations, 3. teachers of courses on nonlinear PDEs. The readers are expected to know the basic measure theory and Sobolev spaces. Preface References Acknowledgements Contents 1 Introduction References 2 The Method of Characteristics 2.1 Hyperbolic Equations with Constant Coefficients 2.2 Hyperbolic Equations with Variable Coefficients 2.3 Hyperbolic Equations with Nonlinear Coefficients 2.4 Exercises Reference 3 Entropy Solutions 3.1 Weak Solutions 3.2 Rankine-Hugoniot Condition 3.3 Nonuniqueness of Weak Solutions 3.4 Entropy Conditions 3.5 Entropic Shocks 3.6 Change of Coordinates 3.7 Uniqueness and Stability of Entropy Solutions 3.8 Exercises References 4 Riemann Problem 4.1 Strictly Convex Fluxes 4.2 General Fluxes 4.3 Exercises Reference 5 Functions with Bounded Variation 5.1 Definition and Structural Properties 5.2 Compactness 5.3 Exercises References 6 Wave Front Tracking 6.1 Piecewise Constant Approximations 6.2 Global Existence of BV Solutions 6.3 Exercises References 7 Vanishing Viscosity 7.1 A Priori Estimates, Compactness, and Convergence 7.2 Error Estimate 7.3 Exercises References 8 Compensated Compactness 8.1 Young Measures 8.2 Murat Compact Embedding and Div-Curl Lemma 8.3 Scalar Conservation Laws 8.4 Global Existence of Bounded Solutions References 9 Periodic Solutions 9.1 Well-Posedness of Periodic Entropy Solutions 9.2 Asymptotic Behavior Reference 10 Oleinik Estimate 10.1 Oleinik Estimate 10.2 Uniqueness 10.3 Exercises References 11 Lax-Oleinik Formula 11.1 Legendre Transform 11.2 Lax–Oleinik Formula 11.3 Exercises References Index
