×

Focusing of an elongated hole in porous medium flow. (English) Zbl 0989.35082

The authors study solutions to the porous medium equation \(\partial_t u=\Delta (u^m)\), whose initial distributions are positive in the exterior of a compact 2D region and zero outside. They assume that the initial interface is elongated and possesses reflectional symmetry with respect to the \(x\)- and \(y\)-axes. A numerical scheme is implemented, that adopts the numerical grid around the interface so as to maintain a high resolution as the interface shrinks to a point. For \(t\) tending to the focusing time \(T\), the interface becomes oval-like with lengths of the major and minor axes \(O(\sqrt{T-t})\), respectively. The aspect ratio is \(O(1/ \sqrt{T-t})\). By scaling and formal asymptotic arguments they derive an approximate solution which is valid for all \(m\). This approximation indicates that the numerically observed power law behavior for the major and minor axes is universal for all \(m>1\).

MSC:

35K65 Degenerate parabolic equations
37C50 Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics
35B40 Asymptotic behavior of solutions to PDEs
35A35 Theoretical approximation in context of PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Angenent, S. B.; Aronson, D. G., Intermediate asymptotics for convergent viscous gravity currents, Phys. Fluids, 7, 223 (1995) · Zbl 0832.76020
[2] Angenent, S. B.; Aronson, D. G., The focusing problem for the radially symmetric porous medium equation, Commun. PDE, 20, 1217 (1996) · Zbl 0830.35062
[3] S.B. Angenent, D.G. Aronson, Self-similar non-radial hole filling for the porous medium equation, Preprint, 1999.; S.B. Angenent, D.G. Aronson, Self-similar non-radial hole filling for the porous medium equation, Preprint, 1999. · Zbl 0973.35115
[4] Angenent, S. B.; Velazquez, J. J.L., Degenerate neckpinches in mean curvature flow, J. Reine Angew. Math., 482, 15-66 (1997) · Zbl 0866.58055
[5] D.G. Aronson, The porous medium equation, in: A. Fasano, M. Primicerio (Eds.), Some Problems in Nonlinear Diffusion, Lecture Notes in Mathematics, Vol. 1224, Springer, Berlin, 1986.; D.G. Aronson, The porous medium equation, in: A. Fasano, M. Primicerio (Eds.), Some Problems in Nonlinear Diffusion, Lecture Notes in Mathematics, Vol. 1224, Springer, Berlin, 1986. · Zbl 0626.76097
[6] Aronson, D. G.; Graveleau, J., A self-similar solution to the focusing problem for the porous medium equation, Eur. J. Appl. Math., 4, 65 (1993) · Zbl 0780.35079
[7] Aronson, D. G.; Gill, O.; Vasquez, J. L., Limit behaviour of focusing solutions to nonlinear diffusions, Commun. PDE, 23, 307-332 (1998) · Zbl 0895.35055
[8] Barenblatt, G. I., On some unsteady motions of fluids and gases in a porous medium, Prikl. Mat. Mekh., 16, 67 (1952) · Zbl 0049.41902
[9] G.I. Barenblatt, Scaling, Self-similarity and Intermediate Asymptotics, Cambridge University Press, Cambridge, 1996.; G.I. Barenblatt, Scaling, Self-similarity and Intermediate Asymptotics, Cambridge University Press, Cambridge, 1996.
[10] Berger, M.; Kohn, R., A rescaling algorithm for the numerical calculation of blowing-up solutions, Commun. Pure Appl. Math., 41, 841-863 (1988) · Zbl 0652.65070
[11] Betelu, S. I.; Aronson, D. G.; Angenent, S. B., Renormalization study of two-dimensional convergent solutions of the porous medium equation, Physica D, 138, 344-359 (2000) · Zbl 0958.76088
[12] Diez, J.; Gratton, R.; Gratton, J., Self-similar solution of the second kind for a convergent viscous gravity current, Phys. Fluids A, 4, 1148 (1992)
[13] Diez, J.; Thomas, L. P.; Betelu, S.; Gratton, R.; Marino, B.; Gratton, J.; Aronson, D. G.; Angenent, S. B., Noncircular converging flows in viscous gravity currents, Phys. Rev. E, 58, 6182-6187 (1998)
[14] E.J. Doedel, Private communication.; E.J. Doedel, Private communication.
[15] L.Y. Chen, N. Goldenfeld, Numerical renormalization-group calculations for similarity solutions and travelling waves, Phys. Rev. E 51 (1995) 5577.; L.Y. Chen, N. Goldenfeld, Numerical renormalization-group calculations for similarity solutions and travelling waves, Phys. Rev. E 51 (1995) 5577.
[16] N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, Addison-Wesley, Boston, 1992, p. 326.; N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, Addison-Wesley, Boston, 1992, p. 326. · Zbl 0825.76872
[17] Gratton, J.; Minotti, F., Self-similar viscous gravity currents: phase-plane formalism, J. Fluid Mech., 210, 155 (1990) · Zbl 0686.76024
[18] J. Hulshof, Private communication.; J. Hulshof, Private communication.
[19] Huppert, H. E., The propagation of two-dimensional viscous gravity currents over a rigid horizontal surface, J. Fluid Mech., 121, 43 (1982)
[20] Lions, P. L.; Souganidis, P. E.; Vazquez, J. L., The relation between the porous medium equation and the Eikonal equation in several space dimensions, Rev. Mat. Iberoamericana, 3, 275-310 (1987) · Zbl 0697.35012
[21] J.D. Murray, Mathematical Biology, Second Corrected Edition, Springer, Berlin, 1993, p. 238.; J.D. Murray, Mathematical Biology, Second Corrected Edition, Springer, Berlin, 1993, p. 238.
[22] J.A. Sethian, Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision and Material Science, Cambridge University Press, Cambridge, 1996.; J.A. Sethian, Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision and Material Science, Cambridge University Press, Cambridge, 1996. · Zbl 0859.76004
[23] Shu, C.-W.; Osher, S., Efficient implementation of essentially nonoscillatory shock capturing schemes, II, J. Comput. Phys., 83, 32-78 (1988) · Zbl 0674.65061
[24] Sussman, M.; Smereka, P.; Osher, S., A level-set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys., 114, 146-159 (1994) · Zbl 0808.76077
[25] Osher, S.; Shu, C.-W., High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations, SIAM J. Numer. Anal., 28, 907-922 (1991) · Zbl 0736.65066
[26] Osher, S.; Sethian, J., Fronts propagating with curvature dependent speed: algorithms based in Hamilton-Jacobi formulation, J. Comput. Phys., 79, 12-49 (1988) · Zbl 0659.65132
[27] W. Press, S. Teukolsky, W. Vatterling, B. Flannery, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge, 1992.; W. Press, S. Teukolsky, W. Vatterling, B. Flannery, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge, 1992. · Zbl 0845.65001
[28] C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. Advanced numerical approximation of nonlinear hyperbolic equations (Cetraro, 1997), Lecture Notes in Mathematics, Vol. 1697, Springer, Berlin, 1998, pp. 325-432.; C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. Advanced numerical approximation of nonlinear hyperbolic equations (Cetraro, 1997), Lecture Notes in Mathematics, Vol. 1697, Springer, Berlin, 1998, pp. 325-432.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.