Entropy and Convexity for Nonlinear Partial Differential Equations
16-17 June 2011, Kavli Royal Society International Centre
Entropy methods and notions of convexity have been playing increasingly important roles in the analysis of nonlinear PDEs. For discontinuous solutions to nonlinear conservation laws, the notions of entropy solutions are based on entropy conditions involving convexity, motivated by and consistent with the second law of thermodynamics; entropy methods have become one of the most efficient methods in the analysis of physical relevant, correct discontinuous solutions. The notions of convexity appropriate for multi-dimensional problems, such as polyconvexity, quasiconvexity, and rank-one convexity, are responsible for several recent major advances. In the last three decades, various nonlinear methods involving entropy/convexity to deal with discontinuous/singular solutions have been developed in different areas of PDEs, especially in nonlinear conservation laws and the calculus of variations.
In the meeting we will discuss recent developments in nonlinear methods via entropy/convexity, explore their underlying connections, and develop new unifying methods/ideas involving entropy/convexity for important multidimensional dynamical problems in fluid and solid mechanics. We aim to gain a deep understanding of previously developed methods involving entropy/convexity and their underlying connections, and to develop new ideas and unifying insights so as to help address several longstanding and challenging problems in the governing nonlinear partial differential equations in fluid and solid mechanics.