Abstract
A model about the dynamic of vesicle membranes in incompressible viscous fluids is introduced. The system consists of the Navier-Stokes equations with an extra stress depending on the membrane, coupled with a Cahn-Hilliard phase-field equation in 3D domains. This problem has a time dissipative energy which leads, in particular, to the existence of global in time weak solutions. By using some extra regular estimates, we prove that every weak solution is strong and unique for sufficiently large times. Moreover, the asymptotic behavior of these solutions is analyzed. We prove that the w-limit set is a subset of the set of equilibrium points. By using a Lojasiewic-Simon type inequality and a continuity result with respect to the initial values, we demonstrate the convergence of the whole trajectory to a single equilibrium.
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References
Campelo, F., Hernández-Machado, A.: Model for curvature-driven pearling instability in membranes. Phys. Rev. Lett. 99 (8), 1–4 (2007)
Climent-Ezquerra, B., Guillén-González, F.: Convergence to equilibrium for smectic-A liquid crystals in 3D domains without constraints for the viscosity. Nonlinear Anal. 102, 208–219 (2014)
Climent-Ezquerra, B., Guillén-González, F., Rodríguez-Bellido, M.A.: Stability for Nematic liquid crystals with stretching terms. Int. J. Bifurcations Chaos 20, 2937–2942 (2010)
Du, Q., Liu, C., Wang, X.: A phase field approach in the numerical study of the elastic bending energy for vesicle membranes. J. Comput. Phys. 198, 450–468 (2004)
Du, Q., Li, M., Liu, C.: Analysis of a phase field Navier-Stokes vesicle-fluid interaction model. Discret. Contin. Dyn. Syst. B 8 (3), 539–556 (2007)
Gal, C.G., Grasselli, M.: Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D. Ann. I. H. Poincaré (C) Non Linear Anal. 27, 401–436 (2010)
Helfrich, W.: Elastic properties of lipid bilayers-theory and possible experiments. Z. Naturforsch. C 28, 693–703 (1973)
Liu, C., Takahashi, T., Tucsnak, M.: Strong solutions for a phase-filed Navier-Stokes Vesicle-Fluid interaction model. J. Math. Fluid Mech. 14, 177–195 (2012)
Wu, H., Xu, X.: Strong solutions, global regularity and stability of a hydrodynamic system modeling vesicle and fluid interactions. SIAM J. Math. Anal. 45 (1), 181–214 (2013)
Acknowledgements
This research was partially supported by MINECO grant MTM2012-32325 with the participation of FEDER.
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Climent-Ezquerra, B., Guillén-González, F. (2016). Long-Time Behavior of a Cahn-Hilliard-Navier-Stokes Vesicle-Fluid Interaction Model. In: Ortegón Gallego, F., Redondo Neble, M., Rodríguez Galván, J. (eds) Trends in Differential Equations and Applications. SEMA SIMAI Springer Series, vol 8. Springer, Cham. https://doi.org/10.1007/978-3-319-32013-7_8
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DOI: https://doi.org/10.1007/978-3-319-32013-7_8
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