Abstract
Shape of many volcanic edifices depend on different phenomena, such as parasitic cones, erosion or coral growth. A nonlinear model proposed in 1981 proves that the shape of volcanoes is determined by the hydraulic resistance to the flow of magma, along a line, through the porous edifice. This model was later extended to include the shape of aseismic and submarine ridges. In this paper we propose a modification of the above mentioned models in order to simulate the more realistic case of volcanoes growth assuming they have a limited base. We present the 3D extension and a generalization of the model. We formulate a new model including the case of a possible outpointing flow.
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References
Alt, H.W., Luckhaus, S.: Quasilinear elliptic-parabolic differential equations. Math. Z. 183, 311–341 (1983)
Angevine, C.L., Turcotte, D.L., Ockendon, J.R.: Geometrical form of aseismic ridges, volcanoes, and seamounts. J. Geophys. Res. Solid Earth 89(B13), 11287–11292 (1984)
Antonsev, S., Díaz, J.I., Shmarev, S.: Energy Methods for Free Boundary Problems. Birkäuser, Boston (2002)
Arjona, A., Díaz, J.I., Fernández, J.: Geometric form of volcanoes with a limited base, CD-Actas de XXI CEDYA (XI Congreso de Matemáticas Aplicada, Universidad de Castilla la Mancha. Ciudad Real (2009, ISBN 978-84-692-6473-7)
Benilan, Ph, Díaz, J.I.: Pointwise gradient estimates of solutions of onedimensional nonlinear parabolic problems. J. Evol. Equ. 3, 557–602 (2004)
Benilan, Ph., Wittbold, P.: On mild and weak solutions of elliptic-parabolic problems. Adv. Differ. Equ. 1, 919–1122 (1996)
Boccardo, L., Giacheti, D., Díaz, J.I., Murat, F.: Existence and regularity of renormalized solutions for some elliptic problems involving derivatives of nonlinear termes. J. Differ. Equ. 106, 215–237 (1993)
Bonafede, M., Cenni, A.: A porous flow model of magma migration within Mt. Etna: the influence of extended sources and permeability anisotrophy. J. Volcanol. Geotherm. Res. 81, 51–68 (1998)
Bouhsis, F.: Etude d’un problème parabolique par les semi-groupes non linéaires. Publications Mathematiques de Besançon-Analyse non linéaire 15, 133–141 (1995/97)
Brézis, H.: Analyses Fonctionelle: Théorie et Applications. Alianza, Marson-Pons, Madrid (1984)
Carrillo, J.: On the uniqueness of the solution of the evolution dam problem. Nonlinear Anal. 22, 573–607 (1994)
Carrillo, J.: Entropy solutions for nonlinear degenerate problems. Arch. Rational Mech. Anal. 147, 269–361 (1999)
Carrillo, J., Wittbold, P.: Uniqueness of renormalized solutions of degenerate elliptic parabolic problems. J. Differ. Equ. 156, 93–121 (1999)
Díaz, J.I., Kersner, R.: On a nonlinear degenerate parabolic equation in filtration or evaporation through a porous medium. J. Differ. Equ. 69(3), 368–403 (1987)
Díaz, J.I., Kersner, R.: On the behaviour and cases of nonexistence of the free boundary in a semibounded porous medium. J. Math. Anal. Appl. 132(1), 281–289 (1988)
Díaz, J.I., Shmarev, S.I.: On the behaviour of the interface in nonlinear processes with convection dominating diffusion via Lagrangian coordinates. Adv. Math. Sci. Appl. 1(1), 19–45 (1992)
Díaz, J.I.: Qualitative study of nonlinear parabolic equations: an introduction. Extracta Mathematicae 16(2), 303–341 (2001)
Díaz, J.I.: Estimates on the location of the free boundary for the obstacle and Stefan problems by means of some energy methods. Georgian Math. J. 15(3), 455–484 (2008)
Evans, L.C.: Nonlinear evolution equations in an arbitrary Banach space. Israel J. Math. 26, 1–42 (1977)
Gilding, B.H.: Improved theory for a nonlinear degenerate parabolic equation. Annali della Scuola Normale Superiore di Pisa Classe di Scienze Sér. 4 16(2), 165–224 (1989)
Gurtin, M.E.: An introduction to Continuum Mechanics. Academic Press Inc., London (1981)
Kačur, J.: Solution of nonlinear and degenerate convection-diffusion problems. Nonlinear Anal. 47, 123–134 (2001)
Klashnikov, A.S.: Some problems of qualitative theory of nonlinear degenerate second-order parabolic equations. Uspeekhi Mat. Nauk. 42, 135–176 (1987)
Lacey, A., Ockendon, J.R., Turcotte, D.L.: On the geometrical form of volcanoes. Earth Planet. Sci. Lett. 54, 139–143 (1981)
Marshak, S.: Earth. Portrait of a Planet. W.W. Norton & Company, New York (2005)
Milne, J.: On the form of volcanoes. Geol. Mag. 15, 337–345 (1878)
Milne, J.: Further notes on the form of volcanoes. Geol. Mag. 16, 506–514 (1879)
Otto, F.: L\(^{1}\)-contraction and uniqueness for quasilinear elliptic-parabolic equations. J. Differ. Equ. 131, 20–38 (1996)
Polubarinova-Kochina, P.Y.: Theory of Ground Water Movement. Princeton University Press, Princeton (1972)
Schmincke, H.U.: Volcanism. Springer-Verlag, Berlin Heidelberg (2004)
Turcotte, D.L., Shubert, G.: Geodynamics. Cambridge University Press, United Kingdom (2002)
Wadge, G., Francis, P.: A porous flow model for the geometrical form of volcanoes critical comments. Earth Planet. Sci. Lett. 57, 453–455 (1981)
Acknowledgments
The authors thank Professor J. Fernández for several useful conversations on this subject and to the two anonymous referees for their careful reading of the manuscript. The research of A. Arjona was supported by the National Research Fund of Luxembourg (AFR Grant 4832278). The research of J. I. Díaz was partially supported by the project ref. MTM2011-26119 of the DGISPI (Spain) and the Research Group MOMAT (Ref. 910480) supported by UCM. He has received also support from the ITN FIRST of the Seventh Framework Program of the European Community’s (Grant Agreement Number 238702).
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Arjona, A., Díaz, J.I. Geometrical evolution of volcanoes: a theoretical approach. RACSAM 109, 511–534 (2015). https://doi.org/10.1007/s13398-014-0198-y
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DOI: https://doi.org/10.1007/s13398-014-0198-y