Very singular diffusion equations: second and fourth order problems

Giga, Mi-Ho; Giga, Yoshikazu

doi:10.1007/s13160-010-0020-y

Cited by 53 publications

(73 citation statements)

References 86 publications

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“…The motivation to study this problem is twofold: (a) image analysis, see [1,5,27,34]; (b) crystal growth problems, see [4,25,32,36]. There are different physically relevant models, where a similar to ours surface energy appears, but the corresponding evolutionary problem is not necessarily set up, see for example, [11].…”

Section: Introductionmentioning

confidence: 99%

“…Equation (1.1) may be interpreted as a steepest descent of the total variation, that is, we can write (1.1) as a gradient flow u t ∈ −∂ E(u) for a functional E. This is why we can apply the abstract nonlinear semigroup theory of Komura, see [10,15], to obtain existence of solutions. This has been performed by [25,20,27] and also by [2,3,7,12,14,33]. However, the generality of this tool does not permit to study fine points of solutions to (1.1).…”

Section: Introductionmentioning

confidence: 99%

“…This is why we call the above behavior of solutions the sudden directional diffusion. However, even more dramatic effects of singular diffusion can be seen in the fourth order problems, see [21].…”

Section: Introductionmentioning

confidence: 99%

“…We leave aside the issue that in general this is a measure, not a function. This problem is resolved differently by the authors applying the semigroup approach, [5,12,20,25] etc. We treat u x as a Clarke differential (see (2.1) and the text below this formula).…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, there are two sorts of results available up to now to deal with (1.1): (1) the method based on the abstract semigroup theory, see for example, [5,20,25] and [12]. It is very general and elegant, it enables us to study the facet motion, but it does not capture all relevant information.…”

Section: Introductionmentioning

confidence: 99%

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Almost classical solutions to the total variation flow

2012

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Abstract. The paper examines the one-dimensional total variation flow equation with Dirichlet boundary conditions. Thanks to a new concept of "almost classical" solutions we are able to determine evolution of facets -flat regions of solutions. A key element of our approach is the natural regularity determined by the nonlinear elliptic operator, for which x 2 is an example of an irregular function. Such a point of view allows us to construct solutions. We apply this idea to numerical simulations for typical initial data. Due to the nature of Dirichlet data, any monotone function is an equilibrium. We prove that each solution reaches such a steady state in finite time.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Almost classical solutions to the total variation flow

2012

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Energy‐dissipation in a coupled system of Allen–Cahn‐type equation and Kobayashi–Warren–Carter‐type model of grain boundary motion

Watanabe

Shirakawa

2020

Math Methods in App Sciences

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In this paper, we consider a system of initial-boundary value problems for parabolic equations, as a generalized version of the "-model" of grain boundary motion, proposed by Kobayashi (2001). The system is a coupled system of an Allen-Cahn-type equation with a given temperature source and a phase-field model of grain boundary motion, known as "Kobayashi-Warren-Carter-type model." The focus of the study is on a special kind of solution, called energy-dissipative solution, which is to reproduce the energy-dissipation of the governing energy in time. Under suitable assumptions, two Main Theorems, concerned with the existence of energy-dissipative solution and and the large-time behavior, will be demonstrated as the results of this paper.

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A C⁰ interior penalty discontinuous Galerkin method for fourth‐order total variation flow. I: Derivation of the method and numerical results

Bhandari

Hoppe

Kumar

2019

Numerical Methods Partial

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We consider the numerical solution of a fourth-order total variation flow problem representing surface relaxation below the roughening temperature. Based on a regularization and scaling of the nonlinear fourth-order parabolic equation, we perform an implicit discretization in time and a C 0 Interior Penalty Discontinuous Galerkin (C 0 IPDG) discretization in space. The C 0 IPDG approximation can be derived from a mixed formulation involving numerical flux functions where an appropriate choice of the flux functions allows to eliminate the discrete dual variable. The fully discrete problem can be interpreted as a parameter dependent nonlinear system with the discrete time as a parameter. It is solved by a predictor corrector continuation strategy featuring an adaptive choice of the time step sizes. A documentation of numerical results is provided illustrating the performance of the C 0 IPDG method and the predictor corrector continuation strategy. The existence and uniqueness of a solution of the C 0 IPDG method will be shown in the second part of this paper. KEYWORDS C 0 interior penalty discontinuous Galerkin method, fourth-order total variation flow, surface relaxation

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Very singular diffusion equations: second and fourth order problems

Cited by 53 publications

References 86 publications

Almost classical solutions to the total variation flow

Almost classical solutions to the total variation flow

Energy‐dissipation in a coupled system of Allen–Cahn‐type equation and Kobayashi–Warren–Carter‐type model of grain boundary motion

A C⁰ interior penalty discontinuous Galerkin method for fourth‐order total variation flow. I: Derivation of the method and numerical results

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Very singular diffusion equations: second and fourth order problems

Cited by 53 publications

References 86 publications

Almost classical solutions to the total variation flow

Almost classical solutions to the total variation flow

Energy‐dissipation in a coupled system of Allen–Cahn‐type equation and Kobayashi–Warren–Carter‐type model of grain boundary motion

A C0 interior penalty discontinuous Galerkin method for fourth‐order total variation flow. I: Derivation of the method and numerical results

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A C⁰ interior penalty discontinuous Galerkin method for fourth‐order total variation flow. I: Derivation of the method and numerical results