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ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

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Upper-semicontinuity of the Global Attractors for a Class of Nonlocal Cahn-hilliard Equations

Author(s):

Joseph L. Shomberg

Assoc.Prof.
Department of Mathematics and Computer Science
Providence College
Providence, RI 02918, USA

jshomber@providence.edu

Abstract:

The aim of this work is to examine the upper-semicontinuity properties of the family of global attractors admitted by a non-isothermal viscous relaxation of some nonlocal Cahn-Hilliard equations. We prove that the family of global attractors is upper-semicontinuous as the perturbation parameters vanish. Additionally, under suitable assumptions, we prove that the family of global attractors satisfies a further upper-semicontinuity type estimate whereby the difference between trajectories of the relaxation problem and the limit isothermal non-viscous problem is explicitly controlled, in the topology of the relaxation problem, in terms of the relaxation parameters.

Keywords

References:

  1. Robert A. Adams and John J. F. Fournier. Sobolev Spaces. Pure and Applied Mathematics - Volume 140. Academic Press / Elsevier Science, Oxford, second edition, 2003
  2. Fuensanta Andreu-Vaillo, José M. Mazón, Julio D. Rossi, and J. Julián Toledo-Melero. Nonlocal diffusion problems, volume 165 of Mathematical Surveys and Monographs. American Mathematical Society, Real Sociedad Matemática Española, 2010
  3. A. V. Babin and M. I. Vishik. Attractors of Evolution Equations. North-Holland, Amsterdam, 1992
  4. J. M. Ball. "Global attractors for damped semilinear wave equations". Discrete Contin. Dyn. Syst., 10(2):31—52, 2004
  5. Pierluigi Colli, Sergio Frigeri, and Maurizio Grasselli. "Global existence of weak solutions to a nonlocal Cahn—Hilliard—Navier—Stokes system." J. Math. Anal. Appl., 386(1):428—444, 2012
  6. Monica Conti and Gianluca Mola. "3-D viscous Cahn—Hilliard equation with memory." Math. Models Methods Appl. Sci. , 32(11):1370-1395, 2008
  7. A. Eden, C. Foias, B. Nicolaenko, and R. Temam. Exponential Attractors for Dissipative Evolution Equations. Research in Applied Mathematics. John Wiley and Sons Inc., 1995
  8. Sergio Frigeri and Maurizio Grasselli. "Global and trajectory attractors for a nonlocal Cahn—Hilliard—Navier—Stokes system." Dynam. Differential Equations, 24(4):827—856, 2012
  9. Sergio Frigeri, Maurizio Grasselli, and Elisabetta Rocca. "A diffuse interface model for two-phase incompressible flows with non-local interactions and non-constant mobility." Nonlinearity, 28(5):1257—1293, 2015
  10. Ciprian G. Gal. "Robust family of exponential attractors for a conserved Cahn—Hilliard model with singularly perturbed boundary conditions." Commun. Pure Appl. Anal. , 7(4):819—836, 2008
  11. Ciprian G. Gal. "Well-posedness and long time behavior of the non-isothermal viscous Cahn—Hilliard equation with dynamic boundary conditions." Dyn. Partial Differ. Equ. , 5(1):39—67, 2008
  12. Ciprian G. Gal and Maurizio Grasselli. "The non-isothermal Allen—Cahn equation with dynamic boundary conditions." Discrete Contin. Dyn. Syst. , 22(4):1009—1040, 2008
  13. Ciprian G. Gal, Maurizio Grasselli, and Alain Miranville. "Robust exponential attractors for singularly perturbed phase-field equations with dynamic boundary conditions." NoDEA Nonlinear Differential Equations Appl. , 15(4-5):535—556, 2008
  14. Ciprian G. Gal and Murizio Grasselli. "Longtime behavior of nonlocal Cahn—Hilliard equations. Discrete Contin. Dyn. Syst. , 34(1):145—179, 2014
  15. Ciprian G. Gal and Alain Miranville. "Robust exponential attractors and convergence to equilibria for non-isothermal Cahn—Hilliard equations with dynamic boundary conditions." Discrete Contin. Dyn. Syst. Ser. S, 2(1):113—147, 2009
  16. Ciprian G. Gal and Alain Miranville. "Uniform global attractors for non-isothermal viscous and non-viscous Cahn—Hilliard equations with dynamic boundary conditions." Nonlinear Anal. Real World Appl. , 10(3):1738—1766, 2009
  17. S. Gatti, M. Grasselli, A. Miranville, and V. Pata. "Hyperbolic relaxation of the viscous Cahn—Hilliard equation in 3D." Math. Models Methods Appl. Sci. , 15(2):165—198, 2005
  18. S. Gatti, M. Grasselli, A. Miranville, and V. Pata. "On the hyperbolic relaxation of the one-dimensional Cahn—Hilliard equation." Math. Anal. Appl. , 312:230—247, 2005
  19. S. Gatti, M. Grasselli, A. Miranville, and V. Pata. "A construction of a robust family of exponential attractors." Amer. Math. Soc. , 134(1):117—127, 2006
  20. S. Gatti, A. Miranville, V. Pata, and S. Zelik. "Continuous families of exponential attractors for singularly perturbed equations with memory." Proc. Roy. Soc. Edinburgh Sect. A, 140:329—366, 2010
  21. Giambattista Giacomin and Joel L. Lebowitz. "Phase segregation dynamics in particle systems with long range interactions I. Macroscopic limits." J. Statist. Phys. , 87(1-2):37—61, 1997
  22. Maurizio Grasselli. "Finite-dimensional global attractor for a nonlocal phase-field system." Lombardo (Rend. Scienze) Mathematica, 146:113—132, 2012
  23. Maurizio Grasselli, Hana Petzeltová, and Giulio Schimperna. "Asymptotic behavior of a nonisothermal viscous Cahn—Hilliard equation with inertial term." J. Differential Equations, 239(1):38—60, 2007
  24. Murizio Grasselli and Giulio Schimperna. "Nonlocal phase-field systems with general potentials." Discrete Contin. Dyn. Syst. , 33(11-12):5089—5106, 2013
  25. J. Hale and G. Raugel. "Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation." J. Differential Equations, 73(2):197—214, 1988
  26. Jack K. Hale. Asymptotic Behavior of Dissipative Systems. Mathematical Surveys and Monographs - No. 25. American Mathematical Society, Providence, 1988
  27. L. Herrera and D. Pavó n. "Hyperbolic theories of dissipation: Why and when do we need them?" Phys. A, 307:121—130, 2002
  28. D. D. Joseph and Luigi Preziosi. "Heat waves." Rev. Modern Phys. , 61(1):41—73, 1989
  29. D. D. Joseph and Luigi Preziosi. "Addendum to the paper: ‘Heat waves’." Rev. Modern Phys. , 62(2):375—391, 1990
  30. I. N. Kostin. "Rate of attraction to a non-hyperbolic attractor." Asymptotic Anal. , 16(3):203—222, 1998
  31. J. L. Lions. Quelques mé thodes de ré solution des problè mes aux limites non liné aires. Dunod, Paris, 1969
  32. Albert J. Milani and Norbert J. Koksch. An Introduction to Semiflows. Monographs and Surveys in Pure and Applied Mathematics - Volume 134. Chapman & Hall/CRC, Boca Raton, 2005
  33. Alain Miranville and Sergey Zelik. "Robust exponential attractors for singularly perturbed phase-field type equations." Electron. J. Differential Equations, 2002(63):1—28, 2002
  34. Francesco Della Porta and Maurizio Grasselli. "Convective nonlocal Cahn—Hilliard equations with reaction terms." Discrete Contin. Dyn. Syst. Ser. B, 20(5):1529—1553, 2015
  35. James C. Robinson. Infinite-Dimensional Dynamical Systems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2001
  36. Joseph L. Shomberg. "Well-posedness and global attractors for a non-isothermal viscous relaxation of nonlocal Cahn—Hilliard equations." AIMS Mathematics: Nonlinear Evolution PDEs, Interfaces and Applications, 1(2):102—136, 2016
  37. Roger Temam. Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Applied Mathematical Sciences - Volume 68. Springer-Verlag, New York, 1988
  38. Roger Temam. Navier—Stokes Equations - Theory and Numerical Analysis. AMS Chelsea Publishing, Providence, reprint edition, 2001
  39. Songmu Zheng. Nonlinear Evolution Equations. Monographs and Surveys in Pure and Applied Mathematics - Volume 133. Chapman & Hall/CRC, Boca Raton, 2004
  40. Songmu Zheng and Albert Milani. "Global attractors for singular perturbations of the Cahn—Hilliard equations." J. Differential Equations, 209(1):101—139, 2005

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