Regularity and convergence results for a phase-field model with memory

Bonfanti, Giovanna; Luterotti, Fabio

doi:10.1002/(sici)1099-1476(199808)21:12<1085::aid-mma986>3.0.co;2-6

Cited by 13 publications

(10 citation statements)

References 12 publications

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“…Moreover, the unique solution (u, v) continuously depends on the convolution kernel h, i.e., for every (u 2 …”

Section: The Main Abstract Resultsmentioning

confidence: 99%

“…The phase-field problems that motivate the present paper, have been derived in [1,2,11] to which we refer the reader for more details. A classical phase-field model with memory consists of two integrodifferential equations governing the dynamics of the solid-liquid phase transitions represented by the temperature u and the phase-field v. The function v may stand for the local proportion of one of the two phases in a threedimensional body Ω.…”

Section: Some Modelsmentioning

confidence: 99%

“…Such models consist of a system of equations describing the evolution of the temperature u and the phase-field v, where v may stand for the local proportion of one of the two phases in a three-dimensional body Ω. Among the many important problems which have been proposed and solved in this area, we mention the fixed and free boundary problems, [3][4][5][6][7][8][9] and the recent papers [1,2,14], in which phase-field models with memory have been considered. All the above mentioned papers and many others deal with direct problems in which the only unknowns are u and v.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Quasilinear systems with memory kernels

Colombo

2005

Journal of Mathematical Analysis and Applications

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We consider a quasilinear integrodifferential system in non-normal form. Such a system is a generalization of a phase-field model with memory and includes, as a particular case, the system describing the combustion of a material with memory. In this paper, we study both the direct and the inverse problems. Our fundamental tools are: the theory of analytic semigroups, optimal regularity results and fixed point arguments.  2005 Elsevier Inc. All rights reserved.Keywords: Quasilinear integrodifferential system in non-normal form; Phase-field model with memory;Combustion of a material with memory; Theory of analytic semigroups; Optimal regularity results; Fixed point arguments; Identification problem

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“…Moreover, the unique solution (u, v) continuously depends on the convolution kernel h, i.e., for every (u 2 …”

Section: The Main Abstract Resultsmentioning

confidence: 99%

Section: Some Modelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Quasilinear systems with memory kernels

Colombo

2005

Journal of Mathematical Analysis and Applications

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“…Many important problems in this area have been proposed and solved. Among them, without claiming completeness, we mention References [1][2][3][4][5][6][7][8][9][10] and, for phase-ÿeld models with memory, References [11][12][13][14][15] and the literature therein.…”

Section: Introductionmentioning

confidence: 99%

Identification of two memory kernels and the time dependence of the heat source for a parabolic conserved phase‐field model

Colombo

Guidetti

Vespri

2005

Math Methods in App Sciences

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SUMMARYIn this paper we consider a system of two integro-di erential evolution equations coming from a conservative phase-ÿeld model in which the principal part of the elliptic operators, involved in the two evolution equations, have di erent orders. The inverse problem consists in ÿnding the evolution of: the temperature, the phase-ÿeld, the two memory kernels and the time dependence of the heat source when we suppose to know additional measurements of the temperature on some part of the body . Our results are set within the framework of H older continuous function spaces with values in a Banach space X . We prove that the inverse problem admits a local in time solution, but we are also able to prove a global in time uniqueness result. Our setting, when we choose for example X = C( ), allows us to make additional measurements of the temperature on the boundary of the body .

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“…In particular, existence and uniqueness when A is a quadratic nonlinearity as well as a detailed characterization of the w-limit set have been shown. In this framework, it is also worth quoting [5,6], which are devoted to studying a phase-field model with memory based on a constitutive law for the heat flux, proposed by Coleman and Gurtin, where q depends both on the present value of the temperature gradient and on its past history. Regarding the longtime behavior and existence of a maximal attractor for other phase transition models without memory effects, the reader is referred, e.g., to [28,32] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Well-posedness and longtime behavior of the phase-field model with memory in a history space setting

Giorgi¹,

Grasselli²,

Pata³

2001

Quart. Appl. Math.

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Abstract.A thermodynamically consistent phase-field model with memory, based on the linearized version of the Gurtin-Pipkin heat conduction law, is considered. The formulation of an initial and boundary value problem for the phase-field evolution system is framed in a history space setting. Namely, the summed past history of the temperature is regarded itself as a variable along with the temperature and the phase-field. Wellposedness results are discussed, as well as longtime behavior of solutions. Under suitable conditions, the existence of an absorbing set can be achieved.

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Regularity and convergence results for a phase-field model with memory

Cited by 13 publications

References 12 publications

Quasilinear systems with memory kernels

Quasilinear systems with memory kernels

Identification of two memory kernels and the time dependence of the heat source for a parabolic conserved phase‐field model

Well-posedness and longtime behavior of the phase-field model with memory in a history space setting

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