Gui-Qiang G Chen – författare

This book offers a survey of recent developments in the analysis of shock reflection-diffraction, a detailed presentation of original mathematical proofs of von Neumann''s conjectures for potential flow, and a collection of related results and new techniques in the analysis of partial differential equations (PDEs), as well as a set of fundamental open problems for further development.Shock waves are fundamental in nature. They are governed by the Euler equations or their variants, generally in the form of nonlinear conservation laws—PDEs of divergence form. When a shock hits an obstacle, shock reflection-diffraction configurations take shape. To understand the fundamental issues involved, such as the structure and transition criteria of different configuration patterns, it is essential to establish the global existence, regularity, and structural stability of shock reflection-diffraction solutions. This involves dealing with several core difficulties in the analysis of nonlinear PDEs—mixed type, free boundaries, and corner singularities—that also arise in fundamental problems in diverse areas such as continuum mechanics, differential geometry, mathematical physics, and materials science. Presenting recently developed approaches and techniques, which will be useful for solving problems with similar difficulties, this book opens up new research opportunities.

This book examines the exciting interface between differential geometry and continuum mechanics, now recognised as being of increasing technological significance. Topics discussed include isometric embeddings in differential geometry and the relation with microstructure in nonlinear elasticity, the use of manifolds in the description of microstructure in continuum mechanics, experimental measurement of microstructure, defects, dislocations, surface energies, and nematic liquid crystals. Compensated compactness in partial differential equations is also treated.

The volume is intended for specialists and non-specialists in pure and applied geometry, continuum mechanics, theoretical physics, materials and engineering sciences, and partial differential equations. It will also be of interest to postdoctoral scientists and advanced postgraduate research students.

These proceedings include revised written versions of the majority of papers presented by leading experts at the ICMS Edinburgh Workshop on Differential Geometry and Continuum Mechanics held in June 2013. All papers have been peer reviewed.

This book presents thirteen papers, representing the most significant advances and current trends in nonlinear hyperbolic conservation laws and related analysis with applications. Topics covered include a survey on multidimensional systems of conservation laws as well as novel results  on liquid crystals, conservation laws with discontinuous flux functions, and applications to sedimentation.  Also included are articles on recent advances in the Euler equations and the Navier-Stokes-Fourier-Poisson system, in addition to new results on collective phenomena described by the Cucker-Smale model.   

The Workshop on Hyperbolic Conservation Laws and Related Analysis with Applications at the International Centre for Mathematical Sciences (Edinburgh, UK) held in Edinburgh, September 2011, produced this fine collection of original research and survey articles. Many leading mathematicians attended the event and submitted their contributions for this volume. It is addressed to researchers and graduate students interested in partial differential equations and related analysis with applications.