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

Luca, Lucia De; Goldman, Michael; Strani, Marta

doi:10.1007/s00208-018-1765-x

Cited by 3 publications

(1 citation statement)

References 19 publications

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“…Recently an application of the relaxation framework from [19] to the multi-dimensional Cahn-Hilliard equation for initial data that are close in L ∞ to the planar profile was carried out in [9].…”

Section: Previous Resultsmentioning

confidence: 99%

Relaxation to a planar interface in the Mullins–Sekerka problem

Chugreeva

Otto

Westdickenberg

2019

Interfaces Free Bound.

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We analyze the convergence rates to a planar interface in the Mullins-Sekerka model by applying a relaxation method based on relationships among distance, energy, and dissipation. The relaxation method was developed by two of the authors in the context of the 1-d Cahn-Hilliard equation and the current work represents an extension to a higher dimensional problem in which the curvature of the interface plays an important role. The convergence rates obtained are optimal given the assumptions on the initial data.

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Section: Previous Resultsmentioning

confidence: 99%

Relaxation to a planar interface in the Mullins–Sekerka problem

Chugreeva

Otto

Westdickenberg

2019

Interfaces Free Bound.

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On the initial value problem for a class of nonlinear biharmonic equation with time-fractional derivative

Nguyễn¹,

Caraballo²,

Tuan

2021

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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In this study, we investigate the intial value problem (IVP) for a time-fractional fourth-order equation with nonlinear source terms. More specifically, we consider the time-fractional biharmonic with exponential nonlinearity and the time-fractional Cahn–Hilliard equation. By using the Fourier transform concept, the generalized formula for the mild solution as well as the smoothing effects of resolvent operators are proved. For the IVP associated with the first one, by using the Orlicz space with the function $\Xi (z)={\textrm {e}}^{|z|^{p}}-1$ and some embeddings between it and the usual Lebesgue spaces, we prove that the solution is a global-in-time solution or it shall blow up in a finite time if the initial value is regular. In the case of singular initial data, the local-in-time/global-in-time existence and uniqueness are derived. Also, the regularity of the mild solution is investigated. For the IVP associated with the second one, some modifications to the generalized formula are made to deal with the nonlinear term. We also establish some important estimates for the derivatives of resolvent operators, they are the basis for using the Picard sequence to prove the local-in-time existence of the solution.

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A note on the slow convergence of solutions to conservation laws with mean curvature diffusions

Strani

2019

Complex Variables and Elliptic Equations

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Since 1984, he has worked on boundary value problems in complex analysis and numerical methods to singular integral equations and has become the leading expert in this area in P. R. China. His research interests include boundary value problems in hyper-complex analysis and asymptotic analysis of orthogonal polynomials using Riemann-Hilbert technique. Professor Jinyuan Du has not only devoted his scientific activities to fundamental theoretical research, but in his early years also to research mathematical applications to agriculture.

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

Cited by 3 publications

References 19 publications

Relaxation to a planar interface in the Mullins–Sekerka problem

Relaxation to a planar interface in the Mullins–Sekerka problem

On the initial value problem for a class of nonlinear biharmonic equation with time-fractional derivative

A note on the slow convergence of solutions to conservation laws with mean curvature diffusions

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