ENTROPY AND PARTIAL DIFFERENTIAL EQUATIONS
INTRODUCTION 1 1 ENTROPY AND EQUILIBRIUM 2 ENTROPY AND IRREVERSIBILITY 25 3 CONTINUUM THERMODYNAMICS 59 4 ELLIPTIC AND PARABOLIC EQUATIONS 75 5 CONSERVATION LAWS AND KINETIC EQUATIONS 102 6 HAMILTON JACOBI AND RELATED EQUATIONS 139 7 ENTROPY AND UNCERTAINTY 149 8 PROBABILITY AND DIFFERENTIAL EQUATIONS 166 CONTENTS CHAPTER CHAPTER CHAPTER CHAPTER CHAPTER CHAPTER CHAPTER CHAPTER 3 INTRODUCTION A OVERVIEW 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 I WILL ESPECIALLY IN CHAPTERS II AND III FOLLOW THE MATHEMATICAL DERIVATION OF ENTROPY PRO VIDED BY MODERN RATIONAL THERMODYNAMICS THEREBY AVOIDING MANY CUSTOMARY PHYSICAL AR GUMENTS 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 I AS WELL DRAW CONNECTIONS WITH HARNACK INEQUALITIES IN CHAPTER V CONSERVA TION LAWS AND CHAPTER VI HAMILTON JACOBI EQUATIONS I REVIEW THE PROPER NOTIONS OF WEAK SOLUTIONS ILLUSTRATING THAT THE INEQUALITIES INHERENT IN THE DE NITIONS 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 I WILL EXPLAIN SOME CONNECTIONS WITH ENTROPY AND DEMONSTRATE VARIOUS PDE APPLICATIONS B THEMES IN SPITE OF THE LONGISH TIME SPENT IN CHAPTERS I III VII REVIEWING PHYSICS THIS IS A MATHEMATICS COURSE ON PARTIAL DI ERENTIAL EQUATIONS MY MAIN CONCERN IS PDE AND HOW VARIOUS NOTIONS INVOLVING ENTROPY HAVE IN UENCED OUR UNDERSTANDING OF PDE AS WE WILL COVER A LOT OF MATERIAL FROM MANY SOURCES LET ME EXPLICITLY WRITE OUT HERE SOME UNIFYING THEMES I THE USE OF ENTROPY IN DERIVING VARIOUS PHYSICAL PDE II THE USE OF ENTROPY TO CHARACTERIZE IRREVERSIBILITY IN PDE EVOLVING IN TIME AND 1 III THE USE OF ENTROPY IN PROVIDING VARIATIONAL PRINCIPLES ANOTHER ONGOING ISSUE WILL BE IV UNDERSTANDING THE RELATIONSHIPS BETWEEN ENTROPY AND CONVEXITY I AM AS USUAL VERY GRATEFUL TO F YEAGER FOR HER QUICK AND ACCURATE TYPING OF THESE NOTES ENTROPY AND PARTIAL DIFFERENTIAL EQUATIONS 2 CHAPTER 1 ENTROPY AND EQUILIBRIUM A THERMAL SYSTEMS IN EQUILIBRIUM WE START FOLLOWING CALLEN C AND WIGHTMAN W BY INTRODUCING A SIMPLE MATHEMATICAL STRUCTURE WHICH WE WILL LATER INTERPRET AS MODELING EQUILIBRIA OF THERMAL SYSTEMS NOTATION WE DENOTE BY X0 X1 XM A TYPICAL POINT OF RM 1 AND HEREAFTER WRITE E X0 A MODEL FOR A THERMAL SYSTEM IN EQUILIBRIUM LET US SUPPOSE WE ARE GIVEN A AN OPEN CONVEX SUBSET OF RM 1 AND B A C1 FUNCTION S R 1 SUCH THAT I S IS CONCAVE II S E 0 III S IS POSITIVELY HOMOGENEOUS OF DEGREE 1 2 WE CALL THE STATE SPACE AND S THE ENTROPY OF OUR SYSTEM S S E X1 XM 3 HERE AND AFTERWARDS WE ASSUME WITHOUT FURTHER COMMENT THAT S AND OTHER FUNCTIONS DERIVED FROM S ARE EVALUATED ONLY IN OPEN CONVEX REGIONS WHERE THE VARIOUS FUNCTIONS MAKE SENSE IN PARTICULAR WHEN WE NOTE THAT 2 III MEANS S E X1 XM S E X1 XM 0 4 WE AUTOMATICALLY CONSIDER IN 4 ONLY THOSE STATES FOR WHICH BOTH SIDES OF 4 ARE DE NED OWING TO 2 II WE CAN SOLVE 3 FOR E AS A C1 FUNCTION OF S X1 XM E E S X1 XM 5 WE CALL THE FUNCTION E THE INTERNAL ENERGY DE NITIONS T E S TEMPERATURE PK E XK KTH GENERALIZED FORCE OR PRESSURE 6 3