Entropy and the Stability of Classical Solutions
It is a tenet of continuum physics that the Second Law of thermodynamics is essentially a statement of stability. In the examples discussed in the previous chapters, the Second Law manifests itself in the presence of companion balance laws, to be satisfied identically, as equalities, by classical solutions, and to be imposed as thermodynamic admissibility inequality constraints on weak solutions of the systems of balance laws. A recurring theme in the exposition of the theory of hyperbolic systems of balance laws in this book will be that companion balance laws induce stability under various guises. Here the reader will get a glimpse of the implications of entropy inequalities on the stability of classical solutions. It will be shown that when the system of balance laws is endowed with a companion balance law induced by a convex entropy, the initial value problem is locally well-posed in the context of classical solutions: Sufficiently smooth initial data generate a classical solution defined on a maximal time interval, typically of finite duration. However, in the presence of damping induced by relaxation mechanisms, and when the initial data are sufficiently small, the classical solution exists globally in time. Classical solutions are unique and depend continuously on their initial values, not only within the class of classical solutions but even within the broader class of weak solutions that satisfy the companion balance law as an inequality admissibility constraint. Similar existence and stability results will be established, even when the entropy fails to be convex, in the following two situations: (a) the entropy is convex only in the direction of a certain cone in state space but the system is equipped with special companion balance laws, called involutions, whose presence compensates for the lack of convexity in complementary directions; or (b) the system is endowed with complementary entropies and the principal entropy is polyconvex. This structure arises in elastodynamics and electromagnetism.
KeywordsWeak Solution Cauchy Problem Compact Subset Classical Solution Hyperbolic System
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