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15
Diffusion in PoroElastic Media
 Jour. Math. Anal. Appl
, 1998
"... . Existence, uniqueness and regularity theory is developed for a general initialboundaryvalue problem for a system of partial differential equations which describes the Biot consolidation model in poroelasticity as well as a coupled quasistatic problem in thermoelasticity. Additional effects of se ..."
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. Existence, uniqueness and regularity theory is developed for a general initialboundaryvalue problem for a system of partial differential equations which describes the Biot consolidation model in poroelasticity as well as a coupled quasistatic problem in thermoelasticity. Additional effects of secondary consolidation and pore fluid exposure on the boundary are included. This quasistatic system is resolved as an application of the theory of linear degenerate evolution equations in Hilbert space, and this leads to a precise description of the dynamics of the system. 1. Introduction We shall consider a system modeling diffusion in an elastic medium in the case for which the inertia effects are negligible. This quasistatic assumption arises naturally in the classical Biot model of consolidation for a linearly elastic and porous solid which is saturated by a slightly compressible viscous fluid. The fluid pressure is denoted by p(x; t) and the displacement of the structure by u(x; t). ...
Singlephase Flow in Composite Poroelastic Media
 Math. Methods Appl. Sci
, 2002
"... . The mathematical formulation and analysis of the BarenblattBiot model of elastic deformation and laminar flow in a heterogeneous porous medium is discussed. This describes consolidation processes in a fluidsaturated doublediffusion model of fractured rock. The model includes various degenerate ..."
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Cited by 2 (2 self)
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. The mathematical formulation and analysis of the BarenblattBiot model of elastic deformation and laminar flow in a heterogeneous porous medium is discussed. This describes consolidation processes in a fluidsaturated doublediffusion model of fractured rock. The model includes various degenerate cases, such as incompressible constituents or totally fissured components, and it is extended to include boundary conditions arising from partially exposed pores. The quasistatic initialboundary problem is shown to have a unique weak solution, and this solution is strong when the data are smoother. 1. Introduction Any model of fluid flow through a deformable solid matrix must account for the coupling between the mechanical behavior of the matrix and the fluid dynamics. For example, compression of the medium leads to increased pore pressure, if the compression is fast relative to the fluid flow rate. Conversely, an increase in pore pressure induces a dilation of the matrix in response to t...
Uniform Stabilization of the Higher Dimensional System of Thermoelasticity with a Nonlinear Boundary Feedback
 Quarterly Appl. Math
"... Using multipler techniques and Lyapunov methods, we derive explicit decay rates for the energy in the higherdimensional system of thermoelasticity with a nonlinear velocity feedback on part of the boundary of a thermoelastic body, which is clamped along the rest of its boundary. Key Words: Thermoe ..."
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Using multipler techniques and Lyapunov methods, we derive explicit decay rates for the energy in the higherdimensional system of thermoelasticity with a nonlinear velocity feedback on part of the boundary of a thermoelastic body, which is clamped along the rest of its boundary. Key Words: Thermoelasticity; Uniform boundary stabilization; Lyapunov methods. AMS subject classification: 35B35, 73B30. To appear in Quarterly of Applied Mathematics # This work was supported in part by the Killam Postdoctoral Fellowship and done in part while the author was with UC San Diego. + Supported by grant PB960663 of the DGES (Spain). 1 Introduction Let# be a bounded domain in R n with smooth boundary # = ## of class C 2 , and consider a ndimensional linear, homogeneous, isotropic, and thermoelastic body occupying# in its nondeformed state. For a material point with configuration x = (x 1 , , x n ) at time t, let u(x, t) = (u 1 (x, t), , u n (x, t)) and #(x, t) denote the disp...
The Exponential Stabilization Of The HigherDimensional Linear System Of Thermoviscoelasticity
, 1998
"... .  Using multipler techniques and Lyapunov methods, we prove that the energy in the higherdimensional linear thermoviscoelasticity decays to zero exponentially by introducing a velocity feedback on part of the boundary of a thermoviscoelastic body, which is clamped along the rest of its boundary, ..."
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.  Using multipler techniques and Lyapunov methods, we prove that the energy in the higherdimensional linear thermoviscoelasticity decays to zero exponentially by introducing a velocity feedback on part of the boundary of a thermoviscoelastic body, which is clamped along the rest of its boundary, to increase the loss of energy. Key words: Thermoviscoelasticity, exponential stabilization, boundary velocity feedback, Lyapunov functional. AMS Subject Classification: 35B35, 45K05, 73B30, 73F15. 1. Introduction In this paper, we shall be concerned with the problem of exponential stabilization of the linear thermoviscoelastic model # # # # # # # # # # # # # u ##  #u  (# + )#divu +g # #u + (# + )g # #divu + ### = 0 in# (0, #), # # ## + #div u # = 0 in# (0, #), u = 0, # = 0 on # (0, #), u(x, 0) = u 0 (x), u # (x, 0) = u 1 (x), #(x, 0) = # 0 (x) in# , u(x, 0)  u(x, s) = w 0 (x, s) in# (0, #), (1.1) where the sign " # " denotes the conv...
Asymptotic behaviour in ndimensional Thermoelasticity
, 1997
"... We study the thermoelastic system and we prove that the divergence of the displacement vector field and the thermal difference decay exponentially as time goes to infinity. Moreover we show that the decay can not holds in general. AMS classification code: 35B40, 73B30, 35L70 Keywords and phrases: Th ..."
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We study the thermoelastic system and we prove that the divergence of the displacement vector field and the thermal difference decay exponentially as time goes to infinity. Moreover we show that the decay can not holds in general. AMS classification code: 35B40, 73B30, 35L70 Keywords and phrases: Thermoelasticity, Uniform rate of decay, exponential decay, initial boundary value problems 1 Introduction In one dimensional thermoelasticity, thanks to the works [4], [5], [6], [7], [8], [10], [13], [15], [16], [18], [19], it is well known that the energy associated to the solution of the thermoelastic system decays exponentially as time goes to infinity. Whereas for ndimensional materials the situation is more complicate and there are only a few results concerning asymptotic behaviour. In general is not true that the total energy decays to zero as was showed in [2]. For example, for materials that occupy the whole IR 3 , Dassios and Grillakis [3] showed that the heat difference and the ...
Uniform Rates of Decay in Nonlinear Viscoelasticity for Polynomial Decaying Kernels
, 1996
"... We prove a global existence theorem for the nonlinear viscoelastic equation for small data in H 2 (]0; L[) and that the solution decays algebraically to zero when the kernel decays also algebraically as t goes to infinity. That is, if the kernel g satisfies g 0 (t) \Gammac 0 g(t) 1+ 1 p ; ..."
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We prove a global existence theorem for the nonlinear viscoelastic equation for small data in H 2 (]0; L[) and that the solution decays algebraically to zero when the kernel decays also algebraically as t goes to infinity. That is, if the kernel g satisfies g 0 (t) \Gammac 0 g(t) 1+ 1 p ; and g; g 1+ 1 p 2 L 1 (IR) with p ? 2; then the firts and second order energy decay as 1 (1+t) p . AMS classification code: 35B40, 35L05, 35L70 Keywords and phrases: viscoelasticity, exponential and polynomial decay, global solution, initial boundary value problems 1 Introduction In nonlinear system of hyperbolic type, characteristic speeds are not constant. So weak waves are amplified and smooth solution may blow up in finite time due to the formations of shock waves. While, if there exists a damping mechanism, then it should be expected that the dissipation prevails and waves do not break when the initial data is small. This is the case for nonlinear models in thermoelasticity and...
Diffusion in Deforming Porous Media
"... We report on some recent progress in the mathematical theory of nonlinear fluid transport and poromechanics, specifically, the design, analysis and application of mathematical models for the flow of fluids driven by the coupled pressure and stress distributions within a deforming heterogeneous p ..."
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We report on some recent progress in the mathematical theory of nonlinear fluid transport and poromechanics, specifically, the design, analysis and application of mathematical models for the flow of fluids driven by the coupled pressure and stress distributions within a deforming heterogeneous porous structure. The goal of this work is to develop a set of mathematical models of coupled flow and deformation processes as a basis for fundamental research on the theoretical and numerical modeling and simulation of flow in deforming heterogeneous porous media.
Theoretical And Numerical Analysis On A ThermoElastic System With Discontinuities
 J. Comput. Appl. Math
, 1998
"... . A second order accurate numerical scheme is proposed for a thermoelastic system which models a bar made of two distinct materials. The physical parameters involved may be discontinuous across the joint of the two materials, where there might be also singular heat and/or force sources. The solutio ..."
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. A second order accurate numerical scheme is proposed for a thermoelastic system which models a bar made of two distinct materials. The physical parameters involved may be discontinuous across the joint of the two materials, where there might be also singular heat and/or force sources. The solution components, the temperature and the displacement, may change rapidly across the joint. By transforming the system into a different one, timemarching schemes can be used for the new system which is wellposed. The immersed interface method is employed to deal with the discontinuities of the coefficients and the singular sources. The proposed numerical method can fit both explicit and implicit formulation. For the implicit version, a stable and fast PredictionCorrection scheme is also developed. Convergence analysis shows that our method is second order accurate at all grid points in spite of the discontinuities across the interface. Numerical experiments are performed to support the theor...
Stability in Inhomogeneous and Anisotropic Thermoelasticity
 Ital. A
, 1997
"... In this paper we will prove that the first order energy in inhomogeneous and anisotropic thermoelasticity decays exponentially, when exists a dissipative mechanims in the boundary. 1 Introduction By now it is well known, thanks to the work of Dafermos [2] that the temperature gradient and the speci ..."
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In this paper we will prove that the first order energy in inhomogeneous and anisotropic thermoelasticity decays exponentially, when exists a dissipative mechanims in the boundary. 1 Introduction By now it is well known, thanks to the work of Dafermos [2] that the temperature gradient and the specific entropy always decay to zero and that in general the displascement also converge to zero in linear anisotropic inhomogeneous thermoelasticity. The exception is provided by a small class of combinations operators and boundary conditions for which the displacement converges to an undamped oscillation. For several application it is importat to know how fast the solution goes to zero, if there exist a rate of decay. Unfortunately in thermoelasticity, even in the best case of the convergence to zero occurs, it can be arbitrarily slow. So it is not possible to get uniformly rate of decay in this general situation. This means that the dissipation given by the thermal effect is not strong enough...
Existence and Exponential Decay in Nonlinear Thermoelasticity
"... We consider the nonlinear thermoelastic equations and we prove the global existence of smooth solution for small data (u 0 ; u 1 ) in H (0; L). That is third order derivative of u 0 and the second order derivative of u 1 can be large. As a consequence of our method we get the exponential decay o ..."
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We consider the nonlinear thermoelastic equations and we prove the global existence of smooth solution for small data (u 0 ; u 1 ) in H (0; L). That is third order derivative of u 0 and the second order derivative of u 1 can be large. As a consequence of our method we get the exponential decay of the solution. AMS classification code: 35K22, 73B30 Keywords and phrases: thermoelasticity, thermoelastic rod, exponential decay, global solution, initial boundary value problems 1