1,271 research outputs found
Multidimensional Conservation Laws: Overview, Problems, and Perspective
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.Comment: 43 pages, 3 figure
user's guide to viscosity solutions of second order partial differential equations
The notion of viscosity solutions of scalar fully nonlinear partial
differential equations of second order provides a framework in which startling
comparison and uniqueness theorems, existence theorems, and theorems about
continuous dependence may now be proved by very efficient and striking
arguments. The range of important applications of these results is enormous.
This article is a self-contained exposition of the basic theory of viscosity
solutions.Comment: 67 page
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