Stability of Transition Front Solutions in Multidimensional Cahn–Hilliard Systems

Howard, Peter

doi:10.1007/s00332-016-9295-8

Cited by 3 publications

(1 citation statement)

References 29 publications

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“…The analysis by Howard makes use of pointwise estimates on the Green's function for the linearized operator in order to locate the shifts of the planar wave, through the local tracking method. This analysis has been extended by the same author to the non-linear case [10,11] as well as to systems [12]. Let us mention that a similar (in spirit) result to ours has been obtained by Carlen and Orlandi in [1].…”

Section: Introductionsupporting

confidence: 77%

A gradient flow approach to relaxation rates for the multi-dimensional Cahn–Hilliard equation

2018

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The aim of this paper is to study relaxation rates for the Cahn-Hilliard equation in dimension larger than one. We follow the approach of Otto and Westdickenberg based on the gradient flow structure of the equation and establish differential and algebraic relationships between the energy, the dissipation, and the squaredḢ −1 distance to a kink. This leads to a scale separation of the dynamics into two different stages: a first fast phase of the order t − 1 2 where one sees convergence to some kink, followed by a slow relaxation phase with rate t − 1 4 where convergence to the centered kink is observed.

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Section: Introductionsupporting

confidence: 77%

A gradient flow approach to relaxation rates for the multi-dimensional Cahn–Hilliard equation

2018

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Conditional stability in a backward Cahn–Hilliard equation via a Carleman estimate

Shang

2020

Journal of Inverse and Ill-Posed Problems

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We consider a Cahn–Hilliard equation in a bounded domain Ω in {\mathbb{R}^{n}} over a time interval {(0,T)} and discuss the backward problem in time of determining intermediate data {u(x,\theta)}, {\theta\in(0,T)}, {x\in\Omega} from the measurement of the final data {u(x,T)}, {x\in\Omega}. Under suitable a priori boundness assumptions on the solutions {u(x,t)}, we prove a conditional stability estimate for the semilinear Cahn–Hilliard equation\lVert u(\,\cdot\,,\theta)\rVert_{L^{2}(\Omega)}\leq C\lVert u(\,\cdot\,,T)% \rVert_{H^{2}(\Omega)}^{\kappa_{0}},and a conditional stability estimate for the linear Cahn–Hilliard equation\lVert u(\,\cdot\,,\theta)\rVert_{H^{\beta}(\Omega)}\leq C\lVert u(\,\cdot\,,T% )\rVert_{H^{2}(\Omega)}^{\kappa_{1}},where {\theta\in(0,T)}, {\beta\in(0,4)} and {\kappa_{0},\kappa_{1}\in(0,1)}. The proof is based on a Carleman estimate with the weight function {\mathrm{e}^{2s\mathrm{e}^{\lambda t}}} with large parameters {s,\lambda\in\mathbb{R}^{+}}.

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Traveling fronts bifurcating from stable layers in the presence of conservation laws

Pogan

Scheel²

2017

Discrete &Amp; Continuous Dynamical Systems - A

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Stability of Transition Front Solutions in Multidimensional Cahn–Hilliard Systems

Cited by 3 publications

References 29 publications

A gradient flow approach to relaxation rates for the multi-dimensional Cahn–Hilliard equation

A gradient flow approach to relaxation rates for the multi-dimensional Cahn–Hilliard equation

Conditional stability in a backward Cahn–Hilliard equation via a Carleman estimate

Traveling fronts bifurcating from stable layers in the presence of conservation laws

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