S.N. Kruzhkov's lectures on first-order quasilinear PDEs
Résumé
The present contribution originates from short notes intended to accompany the lectures of Professor Stanislav Nikola\u{\i}evich Kruzhkov given for the students of the Moscow State Lomonosov University during the years 1994--1997. Since then, they were enriched by many exercises which should allow the reader to assimilate more easily the contents of the lectures and to appropriate the fundamental techniques. This text is prepared for graduate students studying PDEs, but the exposition is elementary, and no previous knowledge of PDEs is required. Yet a command of basic analysis and ODE tools is needed. The text can also be used as an exercise book. The lectures provide an exposition of the nonlocal theory of quasilinear partial differential equations of first order, also called conservation laws. According to S.N.~Kruzhkov's ``ideology'', much attention is paid to the motivation (from both the mathematical viewpoint and the context of applications) of each step in the development of the theory. Also the historical development of the subject is reflected in these notes. We consider questions of local existence of smooth solutions to Cauchy problems for linear and quasilinear equations. We expose a detailed theory of discontinuous weak solutions to quasilinear equations with one spatial variable. We derive the Rankine--Hugoniot condition, motivate in various ways admissibility conditions for generalized (weak) solutions and relate the admissibility issue to the notions of entropy and of energy. We pay special attention to the resolution of the so-called Riemann problem. The lectures contain many original problems and exercises; many aspects of the theory are explained by means of examples. The text is completed by an afterword showing that the theory of conservation laws is yet full of challenging questions and awaiting for new ideas.
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