Entropy and Convexity for Nonlinear Partial Differential Equations: Introduction

Abstract

Partial differential equations are ubiquitous in almost all applications of mathematics, where they provide a natural mathematical description of many phenomena involving change in physical, chemical, biological, and social processes. The concept of entropy originated in thermodynamics and statistical physics during the 19 th century to describe the heat exchanges that occur in the thermal processes in a thermodynamic system, while the original notion of convexity is for sets and functions in mathematics. Since then, entropy and convexity have become two of the most important concepts in mathematics. In particular, nonlinear methods via entropy and convexity have been playing an increasingly important role in the analysis of nonlinear partial differential equations in recent decades. This opening article of the theme issue is intended to provide an introduction to entropy, convexity, and related nonlinear methods for the analysis of nonlinear partial differential equations. We also provide a brief discussion about the content and contributions of the papers that make up this theme issue.Partial differential equations (PDEs) are ubiquitous in almost all applications of mathematics, where they provide a natural mathematical description of many phenomena involving change in physical, chemical, biological, and social processes. The behaviour of every material object, with length scales ranging from sub-atomic to astronomical and timescales ranging from picoseconds to millennia, can be modelled by PDEs or by equations having similar features.The concept of entropy originated in thermodynamics and statistical physics during the 19 th century to describe the heat exchanges that occur in the thermal processes in a thermodynamic system, while the original notion of convexity is for sets and functions in mathematics. Since then, entropy and convexity have become two of the most important concepts in mathematics. In particular, nonlinear methods via entropy and convexity have been playing an increasingly important role in the analysis of nonlinear PDEs in recent decades. For example, for discontinuous or singular solutions to nonlinear conservation laws which may contain shock waves and concentrations, the notion of entropy solutions is based on entropy conditions involving convexity, motivated by and consistent with the Second Law of Thermodynamics. In addition, entropy methods have become one of the most efficient methods in the analysis of physically relevant discontinuous or singular 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 and convexity have been developed to deal with discontinuous and singular solutions in different areas of PDEs, especially in nonlinear conservation laws, the calculus of variations, and gradient flows.