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Entropy and Partial Differential Equations

Entropy and Partial Di erential Equations Lawrence C. Evans Department of Mathematics, UC Berkeley Inspiring Quotations A good many times I have been present at gatherings of people who, by the standards of traditional culture, are thought highly educated and who have with considerable gusto been expressing their incredulity at the illiteracy of scientists. Once or twice I have been provoked and have asked the company how many of them could describe the Second Law of Thermodynamics. The response was cold: it was also negative. Yet I was asking something which is about the scienti c equivalent of: Have you read a work of Shakespeare's? C. P. Snow, The Two Cultures and the Scienti c Revolution .. C. P. Snow relates that he occasionally became so provoked at literary colleagues who scorned the restricted reading habits of scientists that he would challenge them to explain the second law of thermodynamics. The response was invariably a cold negative silence.

A. Entropy and elliptic equations 1. Definitions 2. Estimates for equilibrium entropy production a. A capacity estimate b. A pointwise bound 3. Harnack’s inequality B. Entropy and parabolic equations 1. Definitions 2. Evolution of entropy a. Entropy increase b. Second derivatives in time c. A differential form of Harnack’s inequality 3 ...

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