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Exponential Attractors (exponential + attractor)

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Mathematics and Statistics100%

Selected Abstracts

Exponential attractor for a planar shear-thinning flow

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 17 2007
Dalibor Pra
Abstract We study the dynamics of an incompressible, homogeneous fluid of a power-law type, with the stress tensor T = ,(1 + µ|Dv|)p,2Dv, where Dv is a symmetric velocity gradient. We consider the two-dimensional problem with periodic boundary conditions and p , (1, 2). Under these assumptions, we estimate the fractal dimension of the exponential attractor, using the so-called method of ,,-trajectories. Copyright © 2007 John Wiley & Sons, Ltd. [source]

Stability of global and exponential attractors for a three-dimensional conserved phase-field system with memory

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 18 2009
Gianluca Mola
Abstract We consider a conserved phase-field system on a tri-dimensional bounded domain. The heat conduction is characterized by memory effects depending on the past history of the (relative) temperature ,, which is represented through a convolution integral whose relaxation kernel k is a summable and decreasing function. Therefore, the system consists of a linear integrodifferential equation for ,, which is coupled with a viscous Cahn,Hilliard type equation governing the order parameter ,. The latter equation contains a nonmonotone nonlinearity , and the viscosity effects are taken into account by a term ,,,,t,, for some ,,0. Rescaling the kernel k with a relaxation time ,>0, we formulate a Cauchy,Neumann problem depending on , and ,. Assuming a suitable decay of k, we prove the existence of a family of exponential attractors {,,,,} for our problem, whose basin of attraction can be extended to the whole phase,space in the viscous case (i.e. when ,>0). Moreover, we prove that the symmetric Hausdorff distance of ,,,, from a proper lifting of ,,,0 tends to 0 in an explicitly controlled way, for any fixed ,,0. In addition, the upper semicontinuity of the family of global attractors {,,,,,} as ,,0 is achieved for any fixed ,>0. Copyright © 2009 John Wiley & Sons, Ltd. [source]

Finite-dimensional attractors and exponential attractors for degenerate doubly nonlinear equations

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 13 2009
M. Efendiev
Abstract We consider the following doubly nonlinear parabolic equation in a bounded domain ,,,3: where the nonlinearity f is allowed to have a degeneracy with respect to ,tu of the form ,tu|,tu|p at some points x,,. Under some natural assumptions on the nonlinearities f and g, we prove the existence and uniqueness of a solution of that problem and establish the finite-dimensionality of global and exponential attractors of the semigroup associated with this equation in the appropriate phase space. Copyright © 2009 John Wiley & Sons, Ltd. [source]

3-D viscous Cahn,Hilliard equation with memory

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2009
Monica Conti
Abstract We deal with the memory relaxation of the viscous Cahn,Hilliard equation in 3-D, covering the well-known hyperbolic version of the model. We study the long-term dynamic of the system in dependence of the scaling parameter of the memory kernel , and of the viscosity coefficient ,. In particular we construct a family of exponential attractors, which is robust as both , and , go to zero, provided that , is linearly controlled by ,. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Global and exponential attractors for 3-D wave equations with displacement dependent damping

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2006
Vittorino Pata
Abstract A weakly damped wave equation in the three-dimensional (3-D) space with a damping coefficient depending on the displacement is studied. This equation is shown to generate a dissipative semigroup in the energy phase space, which possesses finite-dimensional global and exponential attractors in a slightly weaker topology. Copyright © 2006 John Wiley & Sons, Ltd. [source]

Well-posedness and long time behavior of a parabolic-hyperbolic phase-field system with singular potentials

MATHEMATISCHE NACHRICHTEN, Issue 13-14 2007
Maurizio Grasselli
Abstract In this article, we study the long time behavior of a parabolic-hyperbolic system arising from the theory of phase transitions. This system consists of a parabolic equation governing the (relative) temperature which is nonlinearly coupled with a weakly damped semilinear hyperbolic equation ruling the evolution of the order parameter. The latter is a singular perturbation through an inertial term of the parabolic Allen,Cahn equation and it is characterized by the presence of a singular potential, e.g., of logarithmic type, instead of the classical double-well potential. We first prove the existence and uniqueness of strong solutions when the inertial coefficient , is small enough. Then, we construct a robust family of exponential attractors (as , goes to 0). (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]