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Well‐posedness theory for hyperbolic conservation laws
Author(s) -
Liu TaiPing,
Yang Tong
Publication year - 1999
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/(sici)1097-0312(199912)52:12<1553::aid-cpa3>3.0.co;2-s
Subject(s) - conservation law , mathematics , nonlinear system , superposition principle , term (time) , quadratic equation , mathematical analysis , hyperbolic partial differential equation , entropy (arrow of time) , property (philosophy) , initial value problem , partial differential equation , geometry , physics , philosophy , epistemology , quantum mechanics
The paper presents a well‐posedness theory for the initial value problem for a general system of hyperbolic conservation laws. We will start with the refinement of Glimm's existence theory and discuss the principle of nonlinear through wave tracing. Our main goal is to introduce a nonlinear functional for two solutions with the property that it is equivalent to the L 1 ( x ) distance between the two solutions and is time‐decreasing. Moreover, the functional is constructed explicitly in terms of the wave patterns of the solutions through the nonlinear superposition. It consists of a linear term measuring the L 1 ( x ) distance, a quadratic term measuring the coupling of waves and distance, and a generalized entropy functional. © 1999 John Wiley & Sons, Inc.