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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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A kinetic approach to general first order quasilinear equations
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by Yoshikazu Giga, Tetsuro Miyakawa and Shinnosuke Oharu PDF
Trans. Amer. Math. Soc. 287 (1985), 723-743 Request permission

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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Additional Information
  • © Copyright 1985 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 287 (1985), 723-743
  • MSC: Primary 35L60; Secondary 35Q20, 47H20, 76N15
  • DOI: https://doi.org/10.1090/S0002-9947-1985-0768737-9
  • MathSciNet review: 768737