Multidimensional Conservation Laws: Overview, Problems, and Perspective

TL;DR

This paper reviews recent advances, discusses open problems, and offers perspectives on the mathematical theory of multidimensional conservation laws, focusing on phenomena, models, and analytical frameworks for entropy solutions.

Contribution

It provides a comprehensive overview of recent developments, highlights open problems, and discusses analytical methods and models in the study of multidimensional conservation laws.

Findings

Analysis of multidimensional steady supersonic and transonic problems

Discussion of shock reflection-diffraction phenomena

Introduction of divergence-measure vector fields for entropy solutions

Abstract

Some of recent important developments are overviewed, several longstanding open problems are discussed, and a perspective is presented for the mathematical theory of multidimensional conservation laws. Some basic features and phenomena of multidimensional hyperbolic conservation laws are revealed, and some samples of multidimensional systems/models and related important problems are presented and analyzed with emphasis on the prototypes that have been solved or may be expected to be solved rigorously at least for some cases. In particular, multidimensional steady supersonic problems and transonic problems, shock reflection-diffraction problems, and related effective nonlinear approaches are analyzed. A theory of divergence-measure vector fields and related analytical frameworks for the analysis of entropy solutions are discussed.