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Well-posedness and global attractors for a non-isothermal viscous relaxationof nonlocal Cahn-Hilliard equations

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

Special Issues: Nonlinear Evolution PDEs, Interfaces and Applications

We investigate a non-isothermal viscous relaxation of some nonlocal Cahn-Hilliard equations. This perturbation problem generates a family of solution operators exhibiting dissipation and conservation. The solution operators admit a family of compact global attractors that are bounded in a more regular phase-space
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1. Robert A. Adams and John J. F. Fournier, Sobolev spaces, second ed., Pure and Applied Mathematics - Volume 140, Academic Press / Elsevier Science, Oxford, 2003.

2. Fuensanta Andreu-Vaillo, Jos´e M. Maz´on, Julio D. Rossi, and J. Juli´an Toledo-Melero, Nonlocal diffusion problems, Mathematical Surveys and Monographs, vol. 165, American Mathematical Society, Real Sociedad Matem´atica Espa˜nola, 2010.

3. A. V. Babin and M. I. Vishik, Attractors of evolution equations, North-Holland, Amsterdam, 1992.

4. Viorel Barbu, Nonlinear semigroups and differential equations in Banach spaces, Noordhoff International Publishing, Bucharest, 1976.

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 (2012), no. 1, 428–444.

6. Pierluigi Colli, Pavel Krejˇc´ı, Elisabetta Rocca, and J¨urgen Sprekels, Nonlinear evolution inclusions arising from phase change models, Czechoslovak Math. J. 57 (2007), no. 132, 1067–1098.

7. Sergio Frigeri and Maurizio Grasselli, Global and trajectory attractors for a nonlocal Cahn–Hilliard–Navier–Stokes system, J. Dynam. Differential Equations 24 (2012), no. 4, 827–856.

8. Sergio Frigeri, Maurizio Grasselli, and Elisabetta Rocca, A diffuse interface model for two-phase incompressible flows with nonlocal interactions and nonconstant mobility, arXiv:1303.6446v2 (2015).

9. Ciprian G. Gal and Maurizio Grasselli, The non-isothermal Allen–Cahn equation with dynamic boundary conditions, Discrete Contin. Dyn. Syst. 22 (2008), no. 4, 1009–1040.

10. 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 (2008), no. 4–5, 535–556.

11. Ciprian G. Gal and Murizio Grasselli, Longtime behavior of nonlocal Cahn–Hilliard equations, Discrete Contin. Dyn. Syst. 34 (2014), no. 1, 145–179.

12. 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 (2009), no. 3, 1738–1766.

13. Giambattista Giacomin and Joel L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. i. macroscopic limits, J. Statist. Phys. 87 (1997), no. 1–2, 37–61.

14. Maurizio Grasselli, Finite-dimensional global attractor for a nonlocal phase-field system, Istituto Lombardo (Rend. Scienze) Mathematica 146 (2012), 113–132.

15. Maurizio Grasselli, Hana Petzeltov´a, and Giulio Schimperna, Asymptotic behavior of a nonisothermal viscous cahn-hilliard equation with inertial term, J. Differential Equations 239 (2007), no. 1, 38–60.

16. L. Herrera and D. Pav´on, Hyperbolic theories of dissipation: Why and when do we need them?, Phys. A 307 (2002), 121–130.

17. D. D. Joseph and Luigi Preziosi, Heat waves, Rev. Modern Phys. 61 (1989), no. 1, 41–73.

18. D. D. Joseph and Luigi Preziosi, Addendum to the paper: “heat waves”, Rev. Modern Phys. 62 (1990), no. 2, 375–391.

19. Pavel Krejˇc´ı and J¨urgen Sprekels, Nonlocal phase-field models for non-isothermal phase transitions and hysteresis, Adv. Math. Sci. Appl. 14 (2004), no. 2, 593–612.

20. 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.

21. Alain Miranville and Sergey Zelik, Robust exponential attractors for singularly perturbed phasefield type equations, Electron. J. Differential Equations 2002 (2002), no. 63, 1–28.

22. Francesco Della Porta and Maurizio Grasselli, Convective nonlocal cahn-hilliard equations with reaction terms, Discrete Contin. Dyn. Syst. Ser. B 20 (2015), no. 5, 1529–1553.

23. Michael Renardy and Robert C. Rogers, An introduction to partial differential equations, second ed., Texts in Applied Mathematics - Volume 13, Springer-Verlag, New York, 2004.

24. James C. Robinson, Inertial manifolds for the Kuramoto–Sivashinsky equation, Physics Letters (1994), no. 184, 190–193.

25. Murizio Grasselli Giulio Schimperna, Nonlocal phase-field systems with general potentials, Discrete Contin. Dyn. Syst. 33 (2013), no. 11–12, 5089–5106.

26. Hiroki Tanabe, Equations of evolution, Pitman, London, 1979.

27. Roger Temam, Infinite-dimensional dynamical systems in mechanics and physics, Applied Mathematical Sciences - Volume 68, Springer-Verlag, New York, 1988.

28. Roger Temam, Navier-Stokes equations - theory and numerical analysis, reprint ed., AMS Chelsea Publishing, Providence, 2001.

29. Songmu Zheng, Nonlinear evolution equations, Monographs and Surveys in Pure and Applied Mathematics - Volume 133, Chapman & Hall/CRC, Boca Raton, 2004.

30. Songmu Zheng and Albert Milani, Global attractors for singular perturbations of the Cahn– Hilliard equations, J. Differential Equations 209 (2005), no. 1, 101–139.

Copyright Info: © 2016, Joseph L. Shomberg, licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

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