How long does it take for a gas to fill a porous container?
HTML articles powered by AMS MathViewer
- by Carmen Cortázar and Manuel Elgueta PDF
- Proc. Amer. Math. Soc. 122 (1994), 449-453 Request permission
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.References
- D. G. Aronson, Nonlinear diffusion problems, Free boundary problems: theory and applications, Vol. I, II (Montecatini, 1981) Res. Notes in Math., vol. 78, Pitman, Boston, MA, 1983, pp. 135–149. MR 714914
- D. G. Aronson and L. A. Caffarelli, The initial trace of a solution of the porous medium equation, Trans. Amer. Math. Soc. 280 (1983), no. 1, 351–366. MR 712265, DOI 10.1090/S0002-9947-1983-0712265-1
- Luis A. Caffarelli and Avner Friedman, Regularity of the free boundary of a gas flow in an $n$-dimensional porous medium, Indiana Univ. Math. J. 29 (1980), no. 3, 361–391. MR 570687, DOI 10.1512/iumj.1980.29.29027
- L. A. Caffarelli, J. L. Vázquez, and N. I. Wolanski, Lipschitz continuity of solutions and interfaces of the $N$-dimensional porous medium equation, Indiana Univ. Math. J. 36 (1987), no. 2, 373–401. MR 891781, DOI 10.1512/iumj.1987.36.36022
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 122 (1994), 449-453
- MSC: Primary 35K65; Secondary 76S05
- DOI: https://doi.org/10.1090/S0002-9939-1994-1232138-0
- MathSciNet review: 1232138