´ Franc ´ Ho Chi Minh City Universite ¸ ois Rabelais Tours / Universite M2 report in mathematics 2014/2015
Entropy Solutions to Conservation Laws Vincent PERROLLAZ,
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Summary. Partial Differential Equations of the form : ∂t u + ∂x (f (u)) = 0, occur naturally in the modeling of many physical phenomena : gas dynamics, traffic flows, petroleum reservoir engineering, electro magnetism, magneto hydrodynamics, shallow water theory... While this type of equation is well posed in small time for regular data (say C 1 ), solutions generally develop singularities in finite time, this physically correspond to the apparition of shock waves in gas of breaking waves for shallow water equation, and mathematically to spontaneous discontinuities. One therefore needs to explore a more general concept of solution which turns out to be the entropy framework in BV space. We will study this setting in the case where u is a scalar function. After which depending on the time left and on the taste of the student, we may turn to problems of control theory, to numerical simulations or to the mathematical theory of systems of conservation laws. References. Bressan A. Hyperbolic systems of conservation laws. The one-dimensional Cauchy problem, Oxford Lecture Series in Mathematics and its Applications. 20. Oxford : Oxford University Press. xii, 250 p. Coron J.-M. Control and Nonlinearity , Mathematical Surveys and Monographs 136. Providence, RI : American Mathematical Society (AMS), xiv, 426 p. Leveque, R. Numerical Methods for Conservation Laws, Lectures in Mathematics, ETH Zrich. Basel etc. : Birkhuser Verlag. 214 p.
1. Vincent PERROLLAZ, LMPT, UMR 7350, Facult´e des Sciences et Techniques, Parc de Grandmont, 37200 Tours. mail :
[email protected] 1
