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On measure‐valued solutions to a two‐dimensional gravity‐driven avalanche flow model
Author(s) -
Gwiazda Piotr
Publication year - 2005
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.660
Subject(s) - mathematics , lipschitz continuity , monotone polygon , measure (data warehouse) , flow (mathematics) , compressibility , relation (database) , euler equations , mathematical analysis , pure mathematics , geometry , mechanics , physics , database , computer science
This paper concerns measure‐valued solutions for the two‐dimensional granular avalanche flow model introduced by Savage and Hutter. The system is similar to the isentropic compressible Euler equations, except for a Coulomb–Mohr friction law in the source term. We will partially follow the study of measure‐valued solutions given by DiPerna and Majda. However, due to the multi‐valued nature of the friction law, new more sensitive measures must be introduced. The main idea is to consider the class of x ‐dependent maximal monotone graphs of non‐single‐valued operators and their relation with 1‐Lipschitz, Carathéodory functions. This relation allows to introduce generalized Young measures for x ‐dependent maximal monotone graph. Copyright © 2005 John Wiley & Sons, Ltd.