Partial Differential Equations of Mixed Type: Analysis and Applications

TL;DR

This paper explores the analysis of nonlinear partial differential equations of mixed type, highlighting their significance in fluid mechanics and differential geometry, and discussing recent developments and challenges in their study.

Contribution

It provides an overview of the current state, historical context, and recent advances in understanding nonlinear PDEs of mixed type, emphasizing their complex classification and applications.

Findings

Many nonlinear PDEs in fluid mechanics and geometry are of mixed type.

Recent developments have advanced the analysis of these complex PDEs.

The paper identifies open problems and future research directions.

Abstract

Partial differential equations (PDEs) are at the heart of many mathematical and scientific advances. While great progress has been made on the theory of PDEs of standard types during the last eight decades, the analysis of nonlinear PDEs of mixed type is still in its infancy. The aim of this expository paper is to show, through several longstanding fundamental problems in fluid mechanics, differential geometry, and other areas, that many nonlinear PDEs arising in these areas are no longer of standard types, but lie at the boundaries of the classification of PDEs or, indeed, go beyond the classification to be of mixed type. Some interrelated connections, historical perspectives, recent developments, and current trends in the analysis of nonlinear PDEs of mixed type are also presented.