A perturbation of the Cahn-Hilliard equation with logarithmic nonlinearity

Article Mathematics, Applied

A Cahn-Hilliard Model Based on Microconcentrations

Samuel Forest et al.

Summary: The aim of this paper is to study a Cahn-Hilliard type system based on microconcentrations. We prove the existence and uniqueness of solutions to this system and show the convergence of the solutions to the original Cahn-Hilliard equation as a small parameter approaches zero, within finite time intervals.

MEDITERRANEAN JOURNAL OF MATHEMATICS (2023)

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Article Mathematics, Applied

THE SEPARATION PROPERTY FOR 2D CAHN-HILLIARD EQUATIONS: LOCAL, NONLOCAL AND FRACTIONAL ENERGY CASES

Ciprian G. Gal et al.

Summary: We study the separation property for Cahn-Hilliard type equations with constant mobility and singular potentials in two dimensions. The solutions with initial finite energy stay away from the pure phases +/- 1 after a certain time. Our new proofs provide direct entropy estimates using Sobolev inequality in the local case and De Giorgi estimates in the nonlocal case. The methods can be applied to a more general class of singular potentials and have technical advantages for analyzing physical systems that couple Cahn-Hilliard equations with other equations.

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS (2023)

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Article Mathematics

Global well-posedness of a Navier-Stokes-Cahn-Hilliard system with chemotaxis and singular potential in 2D

Jingning He et al.

Summary: This study focuses on a diffuse interface model for incompressible two-phase flows with chemotaxis effects, considering mechanisms such as active transport and nonlocal interactions of Oono's type. The research proves the existence and uniqueness of global strong solutions in a smooth bounded domain, as well as the continuous dependence of the strong solution on initial data and source terms in energy norms. Additionally, the study demonstrates the propagation of regularity for global weak solutions.

JOURNAL OF DIFFERENTIAL EQUATIONS (2021)

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Article Mathematics, Applied

An Allen-Cahn equation based on an unconstrained order parameter and its Cahn-Hilliard limit

Alain Miranville

Summary: This paper investigates an Allen-Cahn model based on a microforce balance and an unconstrained order parameter. The study demonstrates the existence, uniqueness, and regularity of solutions, and shows that solutions converge to those of the original Cahn-Hilliard equation on finite time intervals as a small parameter approaches zero.

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS (2021)

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Article Mechanics

Coupled diffusion and phase transition: Phase fields, constraints, and the Cahn-Hilliard equation

Fernando P. Duda et al.

Summary: The theory developed provides a systematic treatment and characterization of reactions needed to maintain the internal constraint given by the coincidence of mass fraction and phase field. It also introduces boundary conditions for situations where the interface is structureless. Additionally, it offers a new interpretation of the Cahn-Hilliard equation as a limiting variant of an Allen-Cahn type diffusion system.

MECCANICA (2021)

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Article Mathematics

The Cahn-Hilliard equation with a nonlinear source term

Alain Miranville

Summary: The paper aims to prove the existence of solutions to the Cahn-Hilliard equation with a general nonlinear source term. By considering logarithmic nonlinear terms instead of regular nonlinear terms, sufficient conditions on the source term are provided to ensure the existence of a global in time weak solution.

JOURNAL OF DIFFERENTIAL EQUATIONS (2021)

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Article Mathematics