Blow-up and global asymptotics of the limit unstable Cahn--Hilliard equation

Evans, Jonathan D.; Галактионов, В. А.; Williams, J. F.

doi:10.1137/s0036141004440289

Cited by 40 publications

(89 citation statements)

References 27 publications

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“…It is well known that this equation manifests self-similar blow-up [57,58,10,13,35] in which |u(x, t)| becomes infinite at a point in finite time; here we show that it has multiple self-similar blow-up solutions and analyze their stability. Analysis of finite-time blow-up and rupture in many higher order PDEs starts along similar lines [35,11,76,77,79,34,22,25].…”

Section: Self-similar Finite-time Blow-up In a Higher-order Pdementioning

confidence: 68%

“…Analysis of finite-time blow-up and rupture in many higher order PDEs starts along similar lines [35,11,76,77,79,34,22,25].…”

Section: Self-similar Finite-time Blow-up In a Higher-order Pdementioning

confidence: 99%

“…Bernoff & Bertozzi [10] showed that these solutions must have zero mass, Ū dη = 0, to satisfy conservation of mass for u(x, t). Evans et al [35] give a matched asymptotic construction of the self-similar solutions for a more general version of (4.3) in the limit of a large number of extrema. Their computational approach is comparable to [73,75,76] and is described in terms of a one-parameter shooting search as a function of the far-field parameter c 0 .…”

Section: Similarity Variablesmentioning

confidence: 99%

“…Separating variables asŨ (η, s) = φ(η)e λs yields the eigenvalue problem, λφ = Lφ, (4.10a) subject to the boundary conditions WKB analysis of the far-field [10,35] shows that suppressing exponentially growing modes yields four boundary conditions (two each at η → ±∞) and produces a countable discrete spectrum. Numerical calculation of the eigenvalues for the first few self-similar solutions are shown in Fig.…”

Section: Linear Stabilitymentioning

confidence: 99%

“…Blow-up for m > 0 follows from the earlier work of Levine [53] on a broad set of formally parabolic equations. This connection is described in the review by Evans et al [35], who also describe extensively the existence of countable sets of similarity solutions for the Cahn-Hilliard equation and its generalizations to higherdimensional axisymmetric problems. Our numerical results on this problem follow the work of Bernoff & Bertozzi [10] closely, using the ideas described in [73].…”

Section: Introductionmentioning

confidence: 97%

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Stability and dynamics of self-similarity in evolution equations

Bernoff

Witelski

2009

J Eng Math

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Abstract. We discuss a methodology for studying the linear stability of self-similar solutions. We will illustrate these fundamental ideas on three prototype problems: a simple ODE with finite-time blowup, a second-order semi-linear heat equation with infinite-time spreading solutions, and the fourth-order Sivashinsky equation with finite-time self-similar blow-up. These examples are used to show that self-similar dynamics can be studied using many of the ideas arising in the study of dynamical systems. In particular, we will discuss the use of dimensional analysis to derive scaling invariant similarity variables, and the role of symmetries in the context of stability of self-similar dynamics. The spectrum of the linear stability problem determines the rate at which the solution will approach a self-similar profile. For blow-up solutions we will demonstrate that the symmetries give rise to positive eigenvalues associated with the symmetries, and show how this stability analysis can identify a unique stable (and observable) attracting solution from a countable infinity of similarity solutions.

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Section: Self-similar Finite-time Blow-up In a Higher-order Pdementioning

confidence: 68%

“…Analysis of finite-time blow-up and rupture in many higher order PDEs starts along similar lines [35,11,76,77,79,34,22,25].…”

Section: Self-similar Finite-time Blow-up In a Higher-order Pdementioning

confidence: 99%

Section: Similarity Variablesmentioning

confidence: 99%

Section: Linear Stabilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

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Stability and dynamics of self-similarity in evolution equations

Bernoff

Witelski

2009

J Eng Math

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Infinite‐energy solutions for the Cahn–Hilliard equation in cylindrical domains

Eden

Kalantarov

Zelik

2013

Math Methods in App Sciences

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Communicated by P. ColliWe give a detailed study of the infinite-energy solutions of the Cahn-Hilliard equation in the 3D cylindrical domains in uniformly local phase space. In particular, we establish the well-posedness and dissipativity for the case of regular potentials of arbitrary polynomial growth as well as for the case of sufficiently strong singular potentials. For these cases, we prove the further regularity of solutions and the existence of a global attractor. For the cases where we have failed to prove the uniqueness (e.g., for the logarithmic potentials), we establish the existence of the trajectory attractor and study its properties.

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Analysis for a new phase‐field model with periodic boundary conditions

Zhao

Mai

Cheng

2020

Math Methods in App Sciences

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This article is concerned with an initial‐boundary value problem (IBVP) for a new phase‐field model describing the evolution of structural phase transition in elastically deformable solid materials. The model consists of an elliptic‐parabolic system in which the displacement field and the order parameter both satisfy periodic boundary conditions. We prove the existence of global solutions to this IBVP by applying the method of continuation of local solutions and perform numerical simulations to investigate the microstructure evolution of MnNi alloys by using this new phase‐field model.

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Blow-up and global asymptotics of the limit unstable Cahn--Hilliard equation

Cited by 40 publications

References 27 publications

Stability and dynamics of self-similarity in evolution equations

Stability and dynamics of self-similarity in evolution equations

Infinite‐energy solutions for the Cahn–Hilliard equation in cylindrical domains

Analysis for a new phase‐field model with periodic boundary conditions

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