American Institute of Mathematical Sciences

Advanced
May  2003, 9(3): 705-726. doi: 10.3934/dcds.2003.9.705

Well-posedness results for phase field systems with memory effects in the order parameter dynamics

S. Gatti 1, and  Elena Sartori 2,

1. 

Dipartimento di Matematica, Università di Ferrara, Via Machiavelli 35, I-44100 Ferrara, Italy
2. 

Dipartimento di Matematica - Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 - Milano, Italy

Received  November 2001 Revised  October 2002 Published  February 2003

We study two models arising in phase transition dynamics. The state of the system is described by the pair $(\theta,\chi)$, where $\theta$ is the (relative) temperature and $\chi$ is the order parameter or phase field. The main difference between the two models relies on whether global constraints on $\chi$ are imposed or not: accordingly, the resulting models will be called conserved or nonconserved. Memory effects influencing both the heat flux and the dynamics of $\chi$ have been considered in a number of recent papers. Here we assume the Fourier law for the heat flux in the energy balance equation, while we consider memory effects in the order parameter dynamics. We analyze the well-posedness of corresponding Cauchy-Neumann problems for both conserved and nonconserved models. Various results are derived according to properties of the memory kernel involved.
Keywords: integrodifferential equations, existence and uniqueness., Phase field systems.
Mathematics Subject Classification: 45K05, 80A2.
Citation: S. Gatti, Elena Sartori. Well-posedness results for phase field systems with memory effects in the order parameter dynamics. Discrete & Continuous Dynamical Systems, 2003, 9 (3) : 705-726. doi: 10.3934/dcds.2003.9.705
[1]

Vicent Caselles. An existence and uniqueness result for flux limited diffusion equations. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1151-1195. doi: 10.3934/dcds.2011.31.1151
[2]

Ariane Piovezan Entringer, José Luiz Boldrini. A phase field $\alpha$-Navier-Stokes vesicle-fluid interaction model: Existence and uniqueness of solutions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 397-422. doi: 10.3934/dcdsb.2015.20.397
[3]

Emil Minchev. Existence and uniqueness of solutions of a system of nonlinear PDE for phase transitions with vector order parameter. Conference Publications, 2005, 2005 (Special) : 652-661. doi: 10.3934/proc.2005.2005.652
[4]

Luiz F. O. Faria. Existence and uniqueness of positive solutions for singular biharmonic elliptic systems. Conference Publications, 2015, 2015 (special) : 400-408. doi: 10.3934/proc.2015.0400
[5]

Alberto Bressan, Truyen Nguyen. Non-existence and non-uniqueness for multidimensional sticky particle systems. Kinetic & Related Models, 2014, 7 (2) : 205-218. doi: 10.3934/krm.2014.7.205
[6]

Hiroshi Watanabe. Existence and uniqueness of entropy solutions to strongly degenerate parabolic equations with discontinuous coefficients. Conference Publications, 2013, 2013 (special) : 781-790. doi: 10.3934/proc.2013.2013.781
[7]

Peiying Chen. Existence and uniqueness of weak solutions for a class of nonlinear parabolic equations. Electronic Research Announcements, 2017, 24: 38-52. doi: 10.3934/era.2017.24.005
[8]

Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310
[9]

Aaron Hoffman, Benjamin Kennedy. Existence and uniqueness of traveling waves in a class of unidirectional lattice differential equations. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 137-167. doi: 10.3934/dcds.2011.30.137
[10]

T. Tachim Medjo. Existence and uniqueness of strong periodic solutions of the primitive equations of the ocean. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1491-1508. doi: 10.3934/dcds.2010.26.1491
[11]

Tonny Paul, A. Anguraj. Existence and uniqueness of nonlinear impulsive integro-differential equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1191-1198. doi: 10.3934/dcdsb.2006.6.1191
[12]

Matteo Bonforte, Yannick Sire, Juan Luis Vázquez. Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5725-5767. doi: 10.3934/dcds.2015.35.5725
[13]

Dominique Blanchard, Olivier Guibé, Hicham Redwane. Existence and uniqueness of a solution for a class of parabolic equations with two unbounded nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (1) : 197-217. doi: 10.3934/cpaa.2016.15.197
[14]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 4927-4962. doi: 10.3934/dcdsb.2020320
[15]

Craig Cowan. Uniqueness of solutions for elliptic systems and fourth order equations involving a parameter. Communications on Pure & Applied Analysis, 2016, 15 (2) : 519-533. doi: 10.3934/cpaa.2016.15.519
[16]

Tian Ma, Shouhong Wang. Cahn-Hilliard equations and phase transition dynamics for binary systems. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 741-784. doi: 10.3934/dcdsb.2009.11.741
[17]

Daoyi Xu, Weisong Zhou. Existence-uniqueness and exponential estimate of pathwise solutions of retarded stochastic evolution systems with time smooth diffusion coefficients. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 2161-2180. doi: 10.3934/dcds.2017093
[18]

Alexey Cheskidov, Songsong Lu. The existence and the structure of uniform global attractors for nonautonomous Reaction-Diffusion systems without uniqueness. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 55-66. doi: 10.3934/dcdss.2009.2.55
[19]

Anne Mund, Christina Kuttler, Judith Pérez-Velázquez. Existence and uniqueness of solutions to a family of semi-linear parabolic systems using coupled upper-lower solutions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5695-5707. doi: 10.3934/dcdsb.2019102
[20]

F. R. Guarguaglini, R. Natalini. Global existence and uniqueness of solutions for multidimensional weakly parabolic systems arising in chemistry and biology. Communications on Pure & Applied Analysis, 2007, 6 (1) : 287-309. doi: 10.3934/cpaa.2007.6.287

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (44)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences