Entropy and Partial Differential Equations by Lawrence C. Evans
This course surveys various uses of “entropy” concepts in the study of PDE, both linear and nonlinear. We will begin in Chapters I–III with a recounting of entropy in physics, with particular emphasis on axiomatic approaches to entropy as
- (i) characterizing equilibrium states (Chapter I),
- (ii) characterizing irreversibility for processes (Chapter II), and
- (iii) characterizing continuum thermodynamics (Chapter III). Later we will discuss probabilistic theories for entropy as
- (iv) characterizing uncertainty (Chapter VII).
Iwill, especially in Chapters IIand III, follow the mathematical derivation of entropy provided by modern rational thermodynamics, thereby avoiding many customary physical arguments. The main references here will be Callen [C], Owen [O], and Coleman–Noll [C-N]. In Chapter IV I follow Day [D] by demonstrating for certain linear second-order elliptic and parabolic PDE that various estimates are analogues of entropy concepts (e.g. the Clausius inequality). Ias well draw connections with Harnack inequalities. In Chapter V (conservation laws) and Chapter VI(Hamilton–Jacobi equations) Ireview the proper notions of weak solutions, illustrating that the inequalities inherent in the definitions can be interpreted as irreversibility conditions. Chapter VII introduces the probabilistic interpretation of entropy and Chapter VIII concerns the related theory of large deviations. Following Varadhan [V] and Rezakhanlou [R], Iwill explain some connections with entropy, and demonstrate various PDE applications.
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