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Transactions of the American Mathematical Society

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Asymptotics toward the planar rarefaction wave for viscous conservation law
in two space dimensions


Authors: Masataka Nishikawa and Kenji Nishihara
Journal: Trans. Amer. Math. Soc. 352 (2000), 1203-1215
MSC (1991): Primary 35L65, 35L67, 76L05
DOI: https://doi.org/10.1090/S0002-9947-99-02290-4
Published electronically: September 20, 1999
MathSciNet review: 1491872
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Abstract: This paper is concerned with the asymptotic behavior of the solution toward the planar rarefaction wave $r(\frac{x}{t})$ connecting $u_{+}$ and $u_{-}$ for the scalar viscous conservation law in two space dimensions. We assume that the initial data $u_{0}(x,y)$ tends to constant states $u_{\pm }$ as $x \rightarrow \pm \infty $, respectively. Then, the convergence rate to $r(\frac{x}{t})$ of the solution $u(t,x,y)$ is investigated without the smallness conditions of $|u_{+}-u_{-}|$ and the initial disturbance. The proof is given by elementary $L^{2}$-energy method.


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Additional Information

Masataka Nishikawa
Affiliation: Department of Mathematics, School of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169, Japan
Email: masataka@mn.waseda.ac.jp

Kenji Nishihara
Affiliation: School of Political Science and Economics, Waseda University Tokyo, 169-50, Japan
Email: kenji@mn.waseda.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-99-02290-4
Keywords: Nonlinear stable, viscous conservation law, planar rarefaction wave, $L^2$-energy method.
Received by editor(s): July 8, 1996
Received by editor(s) in revised form: October 14, 1997
Published electronically: September 20, 1999
Article copyright: © Copyright 1999 American Mathematical Society

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