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How long does it take for a gas to fill a porous container?

Authors: Carmen Cortázar and Manuel Elgueta
Journal: Proc. Amer. Math. Soc. 122 (1994), 449-453
MSC: Primary 35K65; Secondary 76S05
MathSciNet review: 1232138
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Abstract: Let us consider the problem $ {u_t}(x,t) = \Delta {u^m}(x,t)$ for $ (x,t) \in D \times [0, + \infty ),u(x,0) = {u_0}(x)$ for $ x \in D$, and $ (\partial {u^m}/\partial n)(x,t) = h(x,t)$ for $ (x,t) \in \partial D \times [0, + \infty )$. Here we assume $ D \subset {R^N},m > 1,{u_0} \geq 0$, and $ h \geq 0$. It is well known that solutions to this problem have the property of finite speed propagation of the perturbations. By this we mean that if z is an interior point of D and exterior to the support of $ {u_0}$, then there exists a time $ T(z) > 0$ so that $ u(z,t) = 0$ for $ t < T(z)$ and $ u(z,t) > 0$ for $ t > T(z)$. In this note we give, in an elementary way, an upper bound for $ T(z)$ for the case of bounded convex domains and in the case of a half space.

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  • [A] D. G. Aronson, Nonlinear diffusion problems, Free Boundary Problems: Theory and Applications (A. Fasano and M. Primicerio, eds.), Pitman, New York, 1983, pp. 135-149. MR 714914 (85g:76023)
  • [AC] D. G. Aronson and L. A. Caffarelli, The initial trace of the solution of the porous medium equation, Trans. Amer. Math. Soc. 280 (1983), 351-366. MR 712265 (85c:35042)
  • [CF] L. A. Caffarelli and A. Friedman, Regularity of the free boundary of a gas flow in an n-dimensional porous medium, Indiana. Univ. Math. J. 29 (1980), 361-391. MR 570687 (82a:35096)
  • [CVW] L. A. Caffarelli, J. L. Vazquez, and N. I. Wolanski, Lipschitz continuity of solutions and interfaces of the N-dimensional porous medium equation, Indiana Univ. Math. J. 36 (1987), 373-401. MR 891781 (88k:35221)

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Keywords: Diffusion, porous media
Article copyright: © Copyright 1994 American Mathematical Society

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