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Transactions of the American Mathematical Society

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

 

 

Behaviour of the velocity of one-dimensional flows in porous media


Author: Juan Luis Vázquez
Journal: Trans. Amer. Math. Soc. 286 (1984), 787-802
MSC: Primary 35B40; Secondary 35K55, 35L65, 76S05
MathSciNet review: 760987
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Abstract: The one-dimensional flow of gas of density $ u$ through a porous medium obeys the equation $ {u_t} = {({u^m})_{xx}}$, where $ m > 1,x \in {\mathbf{R}}$ and $ t > 0$. We prove that the local velocity of the gas, given by $ \upsilon = - m{u^{m - 2}}{u_x}$, not only is bounded for $ t \geqslant \tau > 0$ but approaches an $ N$-wave profile as $ t \to \infty $. $ N$-waves are the typical asymptotic profiles for some first-order conservation laws, a class of nonlinear hyperbolic equations. The case $ m \leqslant 1$ is also studied: there are solutions with unbounded velocity while others have bounded velocity.


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DOI: https://doi.org/10.1090/S0002-9947-1984-0760987-X
Keywords: Velocity of propagation, $ N$-waves, flows in porous media, asymptotic behaviour, locally bounded variation
Article copyright: © Copyright 1984 American Mathematical Society