American Institute of Mathematical Sciences

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

 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.
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:

show all references

References:
 [1] Markus Dick, Martin Gugat, Günter Leugering. A strict $H^1$-Lyapunov function and feedback stabilization for the isothermal Euler equations with friction. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 225-244. doi: 10.3934/naco.2011.1.225 [2] Hua Qiu, Shaomei Fang. A BKM's criterion of smooth solution to the incompressible viscoelastic flow. Communications on Pure & Applied Analysis, 2014, 13 (2) : 823-833. doi: 10.3934/cpaa.2014.13.823 [3] Peter Bella, Arianna Giunti. Green's function for elliptic systems: Moment bounds. Networks & Heterogeneous Media, 2018, 13 (1) : 155-176. doi: 10.3934/nhm.2018007 [4] Zvi Artstein. Invariance principle in the singular perturbations limit. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3653-3666. doi: 10.3934/dcdsb.2018309 [5] Lars Grüne, Peter E. Kloeden, Stefan Siegmund, Fabian R. Wirth. Lyapunov's second method for nonautonomous differential equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 375-403. doi: 10.3934/dcds.2007.18.375 [6] Louis Tebou. Stabilization of some coupled hyperbolic/parabolic equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1601-1620. doi: 10.3934/dcdsb.2010.14.1601 [7] Xiaoyu Fu. Stabilization of hyperbolic equations with mixed boundary conditions. Mathematical Control & Related Fields, 2015, 5 (4) : 761-780. doi: 10.3934/mcrf.2015.5.761 [8] Martin Gugat, Günter Leugering, Ke Wang. Neumann boundary feedback stabilization for a nonlinear wave equation: A strict $H^2$-lyapunov function. Mathematical Control & Related Fields, 2017, 7 (3) : 419-448. doi: 10.3934/mcrf.2017015 [9] Adriano Da Silva, Christoph Kawan. Invariance entropy of hyperbolic control sets. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 97-136. doi: 10.3934/dcds.2016.36.97 [10] Doyoon Kim, Seungjin Ryu. The weak maximum principle for second-order elliptic and parabolic conormal derivative problems. Communications on Pure & Applied Analysis, 2020, 19 (1) : 493-510. doi: 10.3934/cpaa.2020024 [11] Wanwan Wang, Hongxia Zhang, Huyuan Chen. Remarks on weak solutions of fractional elliptic equations. Communications on Pure & Applied Analysis, 2016, 15 (2) : 335-340. doi: 10.3934/cpaa.2016.15.335 [12] Xia Huang. Stable weak solutions of weighted nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 293-305. doi: 10.3934/cpaa.2014.13.293 [13] Iasson Karafyllis, Lars Grüne. Feedback stabilization methods for the numerical solution of ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 283-317. doi: 10.3934/dcdsb.2011.16.283 [14] Maria Francesca Betta, Rosaria Di Nardo, Anna Mercaldo, Adamaria Perrotta. Gradient estimates and comparison principle for some nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2015, 14 (3) : 897-922. doi: 10.3934/cpaa.2015.14.897 [15] Cleverson R. da Luz, Gustavo Alberto Perla Menzala. Uniform stabilization of anisotropic Maxwell's equations with boundary dissipation. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 547-558. doi: 10.3934/dcdss.2009.2.547 [16] Diane Denny. A unique positive solution to a system of semilinear elliptic equations. Conference Publications, 2013, 2013 (special) : 193-195. doi: 10.3934/proc.2013.2013.193 [17] Tobias Breiten, Karl Kunisch. Feedback stabilization of the three-dimensional Navier-Stokes equations using generalized Lyapunov equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (7) : 4197-4229. doi: 10.3934/dcds.2020178 [18] Sergio Grillo, Jerrold E. Marsden, Sujit Nair. Lyapunov constraints and global asymptotic stabilization. Journal of Geometric Mechanics, 2011, 3 (2) : 145-196. doi: 10.3934/jgm.2011.3.145 [19] Chérif Amrouche, María Ángeles Rodríguez-Bellido. On the very weak solution for the Oseen and Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 159-183. doi: 10.3934/dcdss.2010.3.159 [20] Tong Li, Anthony Suen. Existence of intermediate weak solution to the equations of multi-dimensional chemotaxis systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 861-875. doi: 10.3934/dcds.2016.36.861

Impact Factor: