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Viscosity solutions with shocks
Author(s) -
Giga Yoshikazu
Publication year - 2002
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3015
Subject(s) - classification of discontinuities , monotone polygon , mathematics , conservation law , jump , viscosity , viscosity solution , convergence (economics) , nonlinear system , hausdorff space , graph , mathematical analysis , pure mathematics , geometry , discrete mathematics , physics , quantum mechanics , economics , economic growth
A solution of single nonlinear first order equations may develop jump discontinuities even if initial data is smooth. Typical examples include a crude model equation describing some bunching phenomena observed in epitaxial growth of crystals as well as conservation laws where jump discontinuities are called shocks. Conventional theory of viscosity solutions does not apply. We introduce a notion of proper (viscosity) solutions to track whole evolutions for such equations in multi‐dimensional spaces. We establish several versions of comparison principles. We also study the vanishing viscosity method to construct a unique global proper solution at least when the evolution is monotone in time or the initial data is monotone in some sense under additional technical assumptions. In fact, we prove that the graph of approximate solutions converges to that of a proper solution in the Hausdorff distance topology. Such a convergence is also established for conservation laws with monotone data. In particular, local uniform convergence outside shocks is proved. © 2001 John Wiley & Sons, Inc.