American Institute of Mathematical Sciences

Advanced
2015, 2015(special): 66-74. doi: 10.3934/proc.2015.0066

Stabilization of a hyperbolic/elliptic system modelling the viscoelastic-gravitational deformation in a multilayered Earth

Alicia Arjona 1, and  Jesús Ildefonso Díaz 2,

1. 

European Center for Geodynamics and Seismology, Rue Josy Welter, 19. L-7256, Walferdange, Grand-Duchy of Luxembourg
2. 

Instituto de Matemática Interdisciplinar and Dpto. Mat. Aplicada. (UCM), Facultad de Matemáticas, Plaza de las Ciencias, 3. 28040, Madrid, Spain

Received  September 2014 Revised  December 2014 Published  November 2015

In the last 30 years several mathematical studies have been devoted to the viscoelastic-gravitational coupling in stationary and transient regimes either for static case or for hyperbolic case. However, to the best of our knowledge there is a lack of mathematical study of the stabilization as $t$ goes to infinity of a viscoelastic-gravitational models crustal deformations of multilayered Earth. Here we prove that, under some additional conditions on the data, the difference of the viscoelastic and elastic solutions converges to zero, as $t$ goes to infinity, in a suitable functional space. The proof of that uses a reformulation of the hyperbolic/elliptic system in terms of a nonlocal hyperbolic system.
Keywords: Stabilization, LaSalle's Invariance principle, weak solution, viscoelastic-gravitational, hyperbolic/elliptic equations, Lyapunov function..
Mathematics Subject Classification: 34H15, 37B25, 35Q74, 35Q8.
Citation: Alicia Arjona, Jesús Ildefonso Díaz. Stabilization of a hyperbolic/elliptic system modelling the viscoelastic-gravitational deformation in a multilayered Earth. Conference Publications, 2015, 2015 (special) : 66-74. doi: 10.3934/proc.2015.0066
References:
[1]

A. Arjona, J.I. Díaz, J. Fernández and J.B. Rundle, On the Mathematical Analysis of an Elastic-gravitational Layered Earth Model for Magmatic Intrusion: The Stationary Case,, Pure Appl. Geophys., 165 (2008), 1465.   Google Scholar

[2]

A. Arjona and J.I. Díaz, On the mathematical analysis of a viscoealstic-gravitational layered earth model for magmatic intrusion: The dynamic Case,, Submitted., ().   Google Scholar

[3]

T. Cazenave and A. Haraux, Introduction aux Problèmes d'évolution Semi-Linéaires. Ellipses ,, Paris, (1990).   Google Scholar

[4]

J.I. Díaz and F. de Thelin, On a nonlinear parabolic problems arising in some models related to turbulence flows,, SIAM Journal of Mathematical Analysis, 25 (1994), 1085.   Google Scholar

[5]

J. Fernández, J.M. Carrasco, J.B. Rundle and V. Araña, Geodetic methods for detecting volcanic unrest: a theoretical approach,, Bulletin of Volcanology, 60 (1999), 534.   Google Scholar

[6]

J. Fernández, M. Charco, K.F. Tiampo, G. Jentzsch and J.B. Rundle, Joint interpretation of displacement and gravity data in volcanic areas. A test example: Long Valley Caldera, California,, J.Volcanology and Geothermal Research, 28 (2001), 1063.   Google Scholar

[7]

J. Fernández and J.B. Rundle, Postseismic visoelastic-gravitational half space computations: Problems and solutions,, Geophysical Research Letters, 31 (2004).   Google Scholar

[8]

A. Folch, J. Fernández, J.B. Rundle and J. Martí, Ground deformation in a viscoelastic medium composed of a layer overlying a half space: A comparison between point and extended sources,, Geophys.J.Int., 140 (2000), 37.   Google Scholar

[9]

A.E.H. Love, Some problems in Geodynamics,, Cambridge University Press, (1911).   Google Scholar

[10]

J.B. Rundle, Static elastic-gravitational deformation of a layared half space by point couple sources,, J. Geophys.Res., 85 (1980), 5355.   Google Scholar

[11]

J.B. Rundle, Numerical Evaluation of static elastic-gravitational deformation of a layared half space by point couple sources,, Rep., (1980), 80.   Google Scholar

[12]

J.B. Rundle, Deformation, gravity and potential changes due to volcanic loading of the crust,, J. Geophys.R, 87 (1982), 729.   Google Scholar

[13]

J.B. Rundle, Viscoeslastic-Gravitational Deformation by a Rectangular Thrust Fault in a Layered Earth,, J. Geophys.Res., 87 (): 7787.   Google Scholar

show all references

References:
[1]

A. Arjona, J.I. Díaz, J. Fernández and J.B. Rundle, On the Mathematical Analysis of an Elastic-gravitational Layered Earth Model for Magmatic Intrusion: The Stationary Case,, Pure Appl. Geophys., 165 (2008), 1465.   Google Scholar

[2]

A. Arjona and J.I. Díaz, On the mathematical analysis of a viscoealstic-gravitational layered earth model for magmatic intrusion: The dynamic Case,, Submitted., ().   Google Scholar

[3]

T. Cazenave and A. Haraux, Introduction aux Problèmes d'évolution Semi-Linéaires. Ellipses ,, Paris, (1990).   Google Scholar

[4]

J.I. Díaz and F. de Thelin, On a nonlinear parabolic problems arising in some models related to turbulence flows,, SIAM Journal of Mathematical Analysis, 25 (1994), 1085.   Google Scholar

[5]

J. Fernández, J.M. Carrasco, J.B. Rundle and V. Araña, Geodetic methods for detecting volcanic unrest: a theoretical approach,, Bulletin of Volcanology, 60 (1999), 534.   Google Scholar

[6]

J. Fernández, M. Charco, K.F. Tiampo, G. Jentzsch and J.B. Rundle, Joint interpretation of displacement and gravity data in volcanic areas. A test example: Long Valley Caldera, California,, J.Volcanology and Geothermal Research, 28 (2001), 1063.   Google Scholar

[7]

J. Fernández and J.B. Rundle, Postseismic visoelastic-gravitational half space computations: Problems and solutions,, Geophysical Research Letters, 31 (2004).   Google Scholar

[8]

A. Folch, J. Fernández, J.B. Rundle and J. Martí, Ground deformation in a viscoelastic medium composed of a layer overlying a half space: A comparison between point and extended sources,, Geophys.J.Int., 140 (2000), 37.   Google Scholar

[9]

A.E.H. Love, Some problems in Geodynamics,, Cambridge University Press, (1911).   Google Scholar

[10]

J.B. Rundle, Static elastic-gravitational deformation of a layared half space by point couple sources,, J. Geophys.Res., 85 (1980), 5355.   Google Scholar

[11]

J.B. Rundle, Numerical Evaluation of static elastic-gravitational deformation of a layared half space by point couple sources,, Rep., (1980), 80.   Google Scholar

[12]

J.B. Rundle, Deformation, gravity and potential changes due to volcanic loading of the crust,, J. Geophys.R, 87 (1982), 729.   Google Scholar

[13]

J.B. Rundle, Viscoeslastic-Gravitational Deformation by a Rectangular Thrust Fault in a Layered Earth,, J. Geophys.Res., 87 (): 7787.   Google Scholar

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