**Editors: John M. Ball and Gui-Qiang G. Chen**

Partial differential equations (PDEs) are ubiquitous in almost all applications of mathematics, where they provide a natural mathematical description of many phenomena for 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 19th 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.

This theme issue is devoted to fundamental questions concerning entropy, convexity, and related nonlinear methods designed to help understand the very difficult problems posed by multi-dimensional, nonlinear PDE problems. In particular, it includes the discussion of several recent developments in nonlinear methods via entropy and convexity, the exploration of their underlying connections, and the development of new unifying methods, ideas, and insights involving entropy and convexity for important multi-dimensional PDE problems in fluid/solid mechanics and other areas. These new developments are at the forefront of current research.

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