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24
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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Cited by 11 (7 self)
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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). ...
MicroStructure Models Of Porous Media
 Birkhauser Verlag Basel
, 1995
"... Introduction Every attempt to exactly model laminar flow through highly inhomogeneous media, e.g., fissured or layered media, leads to very singular problems of partial differential equations with rapidly oscillating coefficients. Various methods of averaging will yield corresponding types of doubl ..."
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Cited by 7 (4 self)
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Introduction Every attempt to exactly model laminar flow through highly inhomogeneous media, e.g., fissured or layered media, leads to very singular problems of partial differential equations with rapidly oscillating coefficients. Various methods of averaging will yield corresponding types of double porosity models, and we shall describe some of these. As a first approximation to flow in a region G which consists of such a composite of two finely interspersed materials, one can consider averaged solutions, one for each material and both defined at every point x 2 G. This leads to a pair of partial differential equations, one identified with each of the two components, and a coupling term that describes the flow across the interface between these components. The values at each point x of the two dependent variables in this system (the solutions) have been obtained by averaging in the
Analysis of a class of degenerate reactiondiffusion systems and the bidomain model of cardiac tissue
 Netw. Heterog. Media
"... Abstract. We prove wellposedness (existence and uniqueness) results for a class of degenerate reactiondiffusion systems. A prototype system belonging to this class is provided by the bidomain model, which is frequently used to study and simulate electrophysiological waves in cardiac tissue. The ex ..."
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Cited by 6 (2 self)
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Abstract. We prove wellposedness (existence and uniqueness) results for a class of degenerate reactiondiffusion systems. A prototype system belonging to this class is provided by the bidomain model, which is frequently used to study and simulate electrophysiological waves in cardiac tissue. The existence result, which constitutes the main thrust of this paper, is proved by means of a nondegenerate approximation system, the FaedoGalerkin method, and the compactness method.
Asymptotic Behavior Of Solutions To Some Pseudoparabolic Equations
"... . The aim of this paper is to investigate the behavior as t !1 of solutions to the Cauchy problem u t \Gamma \Deltau t \Gamma \Deltau \Gamma (b; ru) = r \Delta F (u); u(x; 0) = u 0 (x), where ? 0 is a fixed constant, t 0, x 2 IR n . First, we prove that if u is the solution to the linearized e ..."
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Cited by 2 (0 self)
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. The aim of this paper is to investigate the behavior as t !1 of solutions to the Cauchy problem u t \Gamma \Deltau t \Gamma \Deltau \Gamma (b; ru) = r \Delta F (u); u(x; 0) = u 0 (x), where ? 0 is a fixed constant, t 0, x 2 IR n . First, we prove that if u is the solution to the linearized equation, i.e with r \Delta F (u) j 0, then u decays like a solution for the analogous problem to the heat equation. Moreover, the long time behavior of u is described by the heat kernel. Next, analogous results are established for the nonlinear equation with some assumptions imposed on F , p, and the initial condition u 0 . Mathematical Methods in the Applies Sciences, 20 (1997), 271289 1. Introduction Various physical phenomena lead to a study of mixed boundary value problems or the Cauchy problem for the partial differential equation u t \Gamma j\Deltau t \Gamma \Deltau = f(x; u; ru); (1.1) where u = u(x; t), x 2\Omega ae IR n , t 0, j and are nonnegative constants, \Delta denote...
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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Cited by 2 (0 self)
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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.
Electromagnetic processes in doublynonlinear composites
 Communications
"... Electromagnetic processes in inhomogeneous conductors are here described by coupling the Maxwell equations with nonlinear constitutive relations of the form B = B � H�x � and J = J�E � H�x�, neglecting hysteresis and displacement currents. The latter equality may also account for the Hall effect. A ..."
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Cited by 1 (1 self)
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Electromagnetic processes in inhomogeneous conductors are here described by coupling the Maxwell equations with nonlinear constitutive relations of the form B = B � H�x � and J = J�E � H�x�, neglecting hysteresis and displacement currents. The latter equality may also account for the Hall effect. A doublynonlinear parabolichyperbolic equation is formulated, and existence of a solution is proved via approximation by timediscretization, derivation of a priori estimates, and passage to the limit via compensated compactness and compactness by strict convexity. It is then assumed that the medium is a composite that exhibits periodic oscillations in space. Convergence to a corresponding homogenized twoscale problem is proved as the oscillation period vanishes, via Nguetseng’s notion of twoscale convergence. Finally this twoscale formulation is proved to be equivalent to a coarsescale problem.
Parabolic PDE With Hysteresis
, 1996
"... . Our objective here is to describe some results on the wellposedness of initialboundaryvalue problems for a nonlinear parabolic partial differential equation with memory effects and with general boundary conditions. The system consists of a general porous medium equation together with a family o ..."
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. Our objective here is to describe some results on the wellposedness of initialboundaryvalue problems for a nonlinear parabolic partial differential equation with memory effects and with general boundary conditions. The system consists of a general porous medium equation together with a family of ordinary differential equations subject to constraints. Such a system includes a very general class of hysteresis functionals known as Preisach models of hysteresis. 1. Introduction We begin by describing a system consisting of a parabolic partial differential equation and an ordinary differential equation which are coupled naturally by terms which depend on the difference of the unknowns. This system takes the form @ @t a(u(x; t)) \Gamma \Deltau(x; t) \Gamma c(v(x; t) \Gamma u(x; t)) 3 f(x; t) ; x 2\Omega ; t 2 (0; T ] ; (1.1.a) @ @t b(v(x; t)) + c(v(x; t) \Gamma u(x; t)) 3 g(x; t) ; (1.1.b) \Gamma @ @ u(s; t) 2 d(u(s; t)) ; s 2 @\Omega ; (1.1.c) in which u = u(x; t) and v = v(x; ...
Vibration of a Shape Memory Alloy Wire
, 2000
"... Introduction. We shall prove localintime existence of the unique solution of an initialboundaryvalue problem for the nonlinear system v tt \Gamma \Gamma 1 v xt + T (`; u x ; u xt )v x \Gamma Rv xxx \Delta x = f 1 (x; t) (1.1.a) u tt \Gamma \Gamma / " (`; u x ) + 2 u xt ) x = f 2 ..."
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Introduction. We shall prove localintime existence of the unique solution of an initialboundaryvalue problem for the nonlinear system v tt \Gamma \Gamma 1 v xt + T (`; u x ; u xt )v x \Gamma Rv xxx \Delta x = f 1 (x; t) (1.1.a) u tt \Gamma \Gamma / " (`; u x ) + 2 u xt ) x = f 2 (x; t) (1.1.b) \Gamma ` \Gamma / ` (`; u x ) \Delta t \Gamma k(ff` xt + ` x ) x = 1 v 2 xt<F
A Note on Wave Equation in Einstein & de Sitter Spacetime
, 908
"... We consider the wave propagating in the the Einstein & de Sitter spacetime. The covariant d’Alambert’s operator in the Einstein & de Sitter spacetime belongs to the family of the nonFuchsian partial differential operators. We introduce the initial value problem for this equation and give th ..."
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We consider the wave propagating in the the Einstein & de Sitter spacetime. The covariant d’Alambert’s operator in the Einstein & de Sitter spacetime belongs to the family of the nonFuchsian partial differential operators. We introduce the initial value problem for this equation and give the explicit representation formulas for the solutions. We also show the L p − L q estimates for solutions.
VOIGTREGULARIZATION OF THE THREEDIMENSIONAL INVISCID RESISTIVE MAGNETOHYDRODYNAMIC EQUATIONS
, 2011
"... Abstract. We prove existence, uniqueness, and higherorder global regularity of strong solutions to a particular Voigtregularization of the threedimensional inviscid resistive Magnetohydrodynamic (MHD) equations. Specifically, the coupling of a resistive magnetic field to the EulerVoigt model is ..."
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Abstract. We prove existence, uniqueness, and higherorder global regularity of strong solutions to a particular Voigtregularization of the threedimensional inviscid resistive Magnetohydrodynamic (MHD) equations. Specifically, the coupling of a resistive magnetic field to the EulerVoigt model is introduced to form an inviscid regularization of the inviscid resistive MHD system. The results hold in both the whole space R3 and in the context of periodic boundary conditions. Weak solutions for this regularized model are also considered, and proven to exist globally in time, but the question of uniqueness for weak solutions is still open. Since the main purpose of this line of research is to introduce a reliable and stable inviscid numerical regularization of the underlying model we, in particular, show that the solutions of the Voigt regularized system converge, as the regularization parameter α → 0, to strong solutions of the original inviscid resistive MHD, on the corresponding time interval of existence of the latter. Moreover, we also establish a new criterion for blowup of solutions to the original MHD system inspired by this Voigt regularization.