Robust exponential attractors for a phase-field system with memory |
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Authors: | Maurizio Grasselli Vittorino Pata |
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Institution: | (1) Dipartimento di Matematica “F.Brioschi”, Politecnico di Milano, Via Bonardi 9, I-20133 Milano, Italy |
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Abstract: | H.G. Rotstein et al. proposed a nonconserved phase-field system characterized by the presence of memory terms both in the heat conduction and
in the order parameter dynamics. These hereditary effects are represented by time convolution integrals whose relaxation kernels
k and h are nonnegative, smooth and decreasing. Rescaling k and h properly, we obtain a system of coupled partial integrodifferential equations depending on two relaxation times ɛ and σ.
When ɛ and σ tend to 0, the formal limiting system is the well-known nonconserved phase-field model proposed by G. Caginalp.
Assuming the exponential decay of the relaxation kernels, the rescaled system, endowed with homogeneous Neumann boundary conditions,
generates a dissipative strongly continuous semigroup Sɛ, σ(t) on a suitable phase space, which accounts for the past histories of the temperature as well as of the order parameter. Our
main result consists in proving the existence of a family of exponential attractors
for Sɛ, σ(t), with ɛ, σ ∈ 0, 1], whose symmetric Hausdorff distance from
tends to 0 in an explicitly controlled way. |
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Keywords: | 35B41 37L25 37L30 45K05 80A22 |
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