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Transactions of the American Mathematical Society

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A kinetic approach to general first order quasilinear equations

Authors: Yoshikazu Giga, Tetsuro Miyakawa and Shinnosuke Oharu
Journal: Trans. Amer. Math. Soc. 287 (1985), 723-743
MSC: Primary 35L60; Secondary 35Q20, 47H20, 76N15
MathSciNet review: 768737
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Abstract: This paper presents a new method for constructing entropy solutions of first order quasilinear equations of conservation type, which is illustrated in terms of the kinetic theory of gases. Regarding a quasilinear equation as a model of macroscopic conservation laws in gas dynamics, we introduce as the corresponding microscopic model an auxiliary linear equation involving a real parameter $ \xi $ which plays the role of the velocity argument. Approximate solutions for the quasilinear equation are then obtained by integrating solutions of the linear equation with respect to the parameter $ \xi $. All of these equations are treated in the Fréchet space $ L_{{\text{loc}}}^1({R^n})$, and a convergence theorem for such approximate solutions to the entropy solutions is established with the aid of nonlinear semigroup theory.

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Keywords: First order quasilinear equations, conservation laws, entropy condition, nonlinear semigroups
Article copyright: © Copyright 1985 American Mathematical Society