No Access

Global well-posedness and attractors for the hyperbolic Cahn–Hilliard–Oono equation in the whole space

Anton Savostianov

Département de Mathématiques, Université de Cergy-Pontoise, Cergy-Pontoise Cedex, F-95000, France

E-mail Address: anton.savostianov@u-cergy.fr

Search for more papers by this author

and

Sergey Zelik

Department of Mathematics, University of Surrey, Guildford, GU2 7XH, UK

E-mail Address: s.zelik@surrey.ac.uk

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S0218202516500329Cited by:8 (Source: Crossref)

Abstract

We prove the global well-posedness of the so-called hyperbolic relaxation of the Cahn–Hilliard–Oono equation in the whole space $ℝ^{3}$ with the nonlinearity of the sub-quintic growth rate. Moreover, the dissipativity and the existence of a smooth global attractor in the naturally defined energy space is also verified. The result is crucially based on the Strichartz estimates for the linear Schrödinger equation in $ℝ^{3}$ .

Communicated by F. Brezzi

Keywords:

AMSC: 35B40, 35B45

References

1. F. Abergel, Existence and finite-dimensionality of the global attractor for evolution equations on unbounded domains, J. Differential Equations 83 (1990) 85–108. Crossref, Web of Science, Google Scholar
2. A. Babin and M. Vishik, Attractors of partial differential equations in an unbounded domain, Proc. Royal Soc. Edinburgh 116A (1990) 221–243. Crossref, Web of Science, Google Scholar
3. A. Babin and M. Vishik, Attractors of Evolutionary Equations (North-Holland, 1992). Google Scholar
4. H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations (Springer, 2011). Crossref, Google Scholar
5. F. Bai, C. Elliott, A. Gardiner, A. Spence and A. Stuart, The viscous Cahn–Hilliard equation. Part I: Computations, Nonlinearity 8 (1995) 131–160. Crossref, Web of Science, Google Scholar
6. J. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst. 10 (2004) 31–52. Crossref, Web of Science, Google Scholar
7. M. Blair, H. Smith and C. Sogge, Strichartz estimates and the nonlinear Schrödinger equation on manifolds with boundary, Math. Ann. 354 (2012) 1397–1430. Crossref, Web of Science, Google Scholar
8. A. Bonfoh, Finite-dimensional attractor for the viscous Cahn–Hilliard equation in an unbounded domain, Quart. Appl. Math. 64 (2006) 94–104. Crossref, Web of Science, Google Scholar
9. N. Burq, G. Lebeau and F. Planchon, Global existence for energy critical waves in 3D domains, J. Amer. Math. Soc. 21 (2008) 831–845. Crossref, Web of Science, Google Scholar
10. N. Burq and F. Planchon, Global existence for energy critical waves in 3D domains: Neumann boundary conditions, Amer. J. Math. 131 (2009) 1715–1742. Crossref, Web of Science, Google Scholar
11. L. Caffarelli and N. Muler, An $L^{\infty}$ bound for solutions of the Cahn–Hilliard equation, Arch. Ration. Mech. Anal. 133 (1995) 129–144. Crossref, Web of Science, Google Scholar
12. J. Cahn and J. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys. 28 (1958) 258–267. Crossref, Web of Science, Google Scholar
13. L. Cherfils, A. Miranville and S. Zelik, The Cahn–Hilliard equation with logarithmic potentials, Milan J. Math. 79 (2011) 561–596. Crossref, Web of Science, Google Scholar
14. V. Chepyzhov and M. Vishik, Evolution equations and their trajectory attractors, J. Math. Pures Appl. 76 (1997) 913–964. Crossref, Web of Science, Google Scholar
15. I. Chueshov and I. Lasiecka, Von Karman Evolution Equations: Well-posedness and Long-time Dynamics (Springer, 2010). Crossref, Google Scholar
16. E. Cordero and D. Zucco, Strichartz estimates for the vibrating plate equation, J. Evol. Equations 11 (2011) 827–845. Crossref, Web of Science, Google Scholar
17. A. Debussche, A singular perturbation of the Cahn–Hilliard equation, Asympt. Anal. 4 (1991) 161–185. Google Scholar
18. T. Dlotko, M. Kania and C. Sun, Analysis of the viscous Cahn–Hilliard equation in $ℝ^{N}$ , J. Differential Equations 252 (2012) 2771–2791. Crossref, Web of Science, Google Scholar
19. A. Eden and V. Kalantarov, The convective Cahn–Hilliard equation, Appl. Math. Lett. 20 (2007) 455–461. Crossref, Web of Science, Google Scholar
20. A. Eden and V. Kalantarov, 3D convective Cahn–Hilliard equation, Commun. Pure Appl. Anal. 6 (2007) 1075–1086. Crossref, Web of Science, Google Scholar
21. A. Eden, V. Kalantarov and S. Zelik, Infinite-energy solutions for the Cahn–Hilliard equation in cylindrical domains, Math. Methods Appl. Sci. 37 (2014) 1884–1908. Crossref, Web of Science, Google Scholar
22. M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn–Hilliard system, Math. Nachr. 272 (2004) 11–31. Crossref, Web of Science, Google Scholar
23. C. Elliott, The Cahn–Hilliard model for the kinetics of phase separation, mathematical models for phase change problems, Int. Ser. Numer. Math. 88 (1989) 35–73. Google Scholar
24. C. Elliott and A. Stuart, Viscous Cahn–Hilliard equation. II. Analysis, J. Differential Equations 128 (1996) 387–414. Crossref, Web of Science, Google Scholar
25. P. Galenko and D. Jou, Diffuse-interface model for rapid phase transformations in nonequilibrium systems, Phys. Rev. E 71 (2005) 046125. Crossref, Web of Science, Google Scholar
26. P. Galenko and V. Lebedev, Analysis of the dispersion relation in spinodal decomposition of a binary system, Philos. Mag. Lett. 87 (2007) 821–827. Google Scholar
27. P. Galenko and V. Lebedev, Local nonequilibrium effect on spinodal decomposition in a binary system, Int. J. Thermodynam. 11 (2008) 21–28. Google Scholar
28. P. Galenko and V. Lebedev, Nonequilibrium effects in spinodal decomposition of a binary system, Phys. Lett. A 372 (2008) 985–989. Crossref, Web of Science, Google Scholar
29. S. Gatti, M. Grasselli, A. Miranville and V. Pata, On the hyperbolic relaxation of the one-dimensional Cahn–Hilliard equation, J. Math. Anal. Appl. 312 (2005) 230–247. Crossref, Web of Science, Google Scholar
30. 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 (2005) 165–198. Link, Web of Science, Google Scholar
31. S. Gatti, M. Grasselli, A. Miranville and V. Pata, Memory relaxation of the one-dimensional Cahn–Hilliard equation, dissipative phase transitions, Ser. Adv. Math. Appl. Sci. 71 (2006) 101–114. Link, Google Scholar
32. M. Grasselli, G. Schimperna, A. Segatti and S. Zelik, On the 3D Cahn–Hilliard equation with inertial term, J. Evol. Equations 9 (2009) 371–404. Crossref, Web of Science, Google Scholar
33. M. Grasselli, G. Schimperna and S. Zelik, On the 2D Cahn–Hilliard equation with inertial term, Comm. Partial Differential Equations 34 (2009) 137–170. Crossref, Web of Science, Google Scholar
34. M. Grasselli, G. Schimperna and S. Zelik, Trajectory and smooth attractors for Cahn–Hilliard equations with inertial term, Nonlinearity 23 (2010) 707–737. Crossref, Web of Science, Google Scholar
35. M. Grasselli, H. Petzeltova and G. Schimperna, Asymptotic behavior of a nonisothermal viscous Cahn–Hilliard equation with inertial term, J. Differential Equations 239 (2007) 38–60. Crossref, Web of Science, Google Scholar
36. V. Kalantarov, A. Savostianov and S. Zelik, Attractors for damped quintic wave equations in bounded domains, arXiv:1309.6272, to appear in Ann. I. H. Poincaré. Google Scholar
37. C. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Commun. Pure Appl. Math. 46 (1993) 527–620. Crossref, Web of Science, Google Scholar
38. P.-G. Lemarié-Rieusset, Recent Developments in the Navier–Stokes Problem, Research Notes in Mathematics (Chapman & Hall/CRC, 2002). Crossref, Google Scholar
39. F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, 2nd edn. (Springer, 2015). Crossref, Google Scholar
40. A. Miranville, Asymptotic behavior of the Cahn–Hilliard–Oono equation, J. Appl. Anal. Comput. 1 (2011) 523–536. Web of Science, Google Scholar
41. A. Miranville and S. Zelik, Robust exponential attractors for Cahn–Hilliard type equations with singular potentials, Math. Methods Appl. Sci. 27 (2004) 545–582. Crossref, Web of Science, Google Scholar
42. A. Miranville and S. Zelik, Exponential attractors for the Cahn–Hilliard equation with dynamic boundary conditions, Math. Methods Appl. Sci. 28 (2005) 709–735. Crossref, Web of Science, Google Scholar
43. A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handbook of Differential Equations: Evolutionary Equations, Vol. 4 (North-Holland, 2008), pp. 103–200. Crossref, Google Scholar
44. A. Miranville and S. Zelik, Doubly nonlinear Cahn–Hilliard–Gurtin equations, Hokkaido Math. J. 38 (2009) 315–360. Crossref, Web of Science, Google Scholar
45. I. Moise, R. Rosa and X. Wang, Attractors for non-compact semigroups via energy equations, Nonlinearity 11 (1998) 1369–1393. Crossref, Web of Science, Google Scholar
46. A. Novick-Cohen, The Cahn–Hilliard equation: Mathematical and modeling perspectives, Adv. Math. Sci. Appl. 8 (1998) 965–985. Google Scholar
47. Y. Oono and S. Puri, Computionally efficient modeling of ordering of quenched phases, Phys. Rev. Lett. 58 (1987) 836–839. Crossref, Web of Science, Google Scholar
48. F. Planchon, On the cubic NLS on 3D compact domains, J. Inst. Math. Jussieu 13 (2014) 1–18. Crossref, Web of Science, Google Scholar
49. A. Robertson and W. Robertson, Topological Vector Spaces, reprint of the 2nd edn. Cambridge Tracts in Mathematics, Vol. 53 (Cambridge Univ. Press, 1980). Google Scholar
50. A. Savostianov and S. Zelik, Finite dimensionality of the attractor for the hyperbolic Cahn–Hilliard–Oono equation in $ℝ^{3}$ , Math. Methods Appl. Sci. 39 (2016) 1254–1267. Crossref, Web of Science, Google Scholar
51. A. Savostianov and S. Zelik, Exponential attractors for the hyperbolic Cahn–Hilliard–Oono equation in a whole space, in preparation. Google Scholar
52. J. Shatah and M. Struwe, Regularity results for nonlinear wave equations, Ann. Math. 138 (1993) 503–518. Crossref, Web of Science, Google Scholar
53. C. Sogge, Lectures on Nonlinear Wave Equations, 2nd edn. (International Press, 2008). Google Scholar
54. W. Strauss, Nonlinear Wave Equations, CBMS Regional Conference Series in Mathematics, Vol. 73 (Amer. Math. Soc., 1989). Google Scholar
55. T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Regional Conference Series in Mathematics (Amer. Math. Soc., 2006). Google Scholar
56. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edn., Applied Mathematical Sciences, Vol. 68 (Springer-Verlag, 1997). Crossref, Google Scholar
57. S. Zelik, Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity, Commun. Pure Appl. Math. 56 (2003) 584–637. Crossref, Web of Science, Google Scholar
58. S. Zelik and J. Pennant, Global well-posedness in uniformly local spaces for the Cahn–Hilliard equation in $ℝ^{3}$ , Commun. Pure Appl. Anal. 12 (2013) 461–480. Web of Science, Google Scholar