Stability of global and exponential attractors for a three‐dimensional conserved phase‐field system with memory |
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Authors: | Gianluca Mola |
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Affiliation: | 1. Dipartimento di Matematica ‘F. Brioschi’, Politecnico di Milano, Via Bonardi 9, I‐20133 Milano, Italy;2. Department of Applied Physics, Graduate School of Engineering, Osaka University, Suita, Osaka 565‐0871, Japan |
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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 χ. The latter equation contains a nonmonotone nonlinearity ? and the viscosity effects are taken into account by a term ?αΔ?tχ, for some α?0. Rescaling the kernel k with a relaxation time ε>0, we formulate a Cauchy–Neumann problem depending on ε and α. Assuming a suitable decay of k, we prove the existence of a family of exponential attractors {?α,ε} for our problem, whose basin of attraction can be extended to the whole phase–space in the viscous case (i.e. when α>0). Moreover, we prove that the symmetric Hausdorff distance of ?α,ε from a proper lifting of ?α,0 tends to 0 in an explicitly controlled way, for any fixed α?0. In addition, the upper semicontinuity of the family of global attractors {??α,ε} as ε→0 is achieved for any fixed α>0. Copyright © 2009 John Wiley & Sons, Ltd. |
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Keywords: | conserved phase‐field models memory effects global attractor exponential attractors dynamical systems |
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