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Transactions of the American Mathematical Society

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

 

 

Large-time behavior of solutions to a scalar conservation law in several space dimensions


Authors: Patricia Bauman and Daniel Phillips
Journal: Trans. Amer. Math. Soc. 298 (1986), 401-419
MSC: Primary 35L65; Secondary 35B40
DOI: https://doi.org/10.1090/S0002-9947-1986-0857450-6
MathSciNet review: 857450
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Abstract: We consider solutions of the Cauchy problem in $ {\mathbf{R}}_ + ^{n + 1}$ for the equation $ {u_t} + {\operatorname{div}_x}f(u) = 0$. The initial data is assumed to be a compact perturbation of a function of the form, $ \varphi (x) = a$ for $ \left\langle {x,\,\mu } \right\rangle > 0$, $ \varphi (x) = b$ for $ \left\langle {x,\,\mu } \right\rangle < 0$, where $ a$ and $ b$ are constants and $ \mu $ is a given unit vector. The Cauchy problem together with an entropy condition on $ u$ is known to be well posed. The solution with unperturbed initial data, $ \varphi (x)$, is a traveling shock, $ \varphi (x - \overrightarrow k t)$, provided that $ \varphi (x - \overrightarrow k t)$ satisfies the entropy condition (an inequality on $ a,\,b,\,\mu $, and $ f$). Assuming this type of condition on $ \varphi $, we study the large-time behavior of $ u$. In particular, we show that $ u$ converges to a traveling shock whose profile agrees with $ \left\langle {x,\,\mu } \right\rangle = 0$ outside of a compact set.


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DOI: https://doi.org/10.1090/S0002-9947-1986-0857450-6
Article copyright: © Copyright 1986 American Mathematical Society