The Cahn-Hilliard-Navier-Stokes system is based on a well-known diffuse interface model and describes the evolution of an incompressible isothermal mixture of binary fluids. A nonlocal variant consists of the Navier-Stokes equations suitably coupled with a nonlocal Cahn-Hilliard equation. The authors, jointly with P. Colli, have already proven the existence of a global weak solution to a nonlocal Cahn-Hilliard-Navier-Stokes system subject to no-slip and no-flux boundary conditions. Uniqueness is still an open issue even in dimension two. However, in this case, the energy identity holds. This property is exploited here to define, following J. M. Ball's approach, a generalized semiflow which has a global attractor. Through a similar argument, we can also show the existence of a (connected) global attractor for the convective nonlocal Cahn-Hilliard equation with a given velocity field, even in dimension three. Finally, we demonstrate that any weak solution fulfilling the energy inequality also satisfies a dissipative estimate. This allows us to establish the existence of the trajectory attractor also in dimension three with a time dependent external force.

Global and Trajectory Attractors for a Nonlocal Cahn-Hilliard-Navier-Stokes System / S. Frigeri, M. Grasselli. - In: JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS. - ISSN 1040-7294. - 24:4(2012), pp. 827-856. [10.1007/s10884-012-9272-3]

Global and Trajectory Attractors for a Nonlocal Cahn-Hilliard-Navier-Stokes System

S. Frigeri;
2012

Abstract

The Cahn-Hilliard-Navier-Stokes system is based on a well-known diffuse interface model and describes the evolution of an incompressible isothermal mixture of binary fluids. A nonlocal variant consists of the Navier-Stokes equations suitably coupled with a nonlocal Cahn-Hilliard equation. The authors, jointly with P. Colli, have already proven the existence of a global weak solution to a nonlocal Cahn-Hilliard-Navier-Stokes system subject to no-slip and no-flux boundary conditions. Uniqueness is still an open issue even in dimension two. However, in this case, the energy identity holds. This property is exploited here to define, following J. M. Ball's approach, a generalized semiflow which has a global attractor. Through a similar argument, we can also show the existence of a (connected) global attractor for the convective nonlocal Cahn-Hilliard equation with a given velocity field, even in dimension three. Finally, we demonstrate that any weak solution fulfilling the energy inequality also satisfies a dissipative estimate. This allows us to establish the existence of the trajectory attractor also in dimension three with a time dependent external force.
Navier-Stokes equations; Nonlocal Cahn-Hilliard equations; Incompressible binary fluids; Global attractors; Trajectory attractors
Settore MAT/05 - Analisi Matematica
2012
Article (author)
File in questo prodotto:
File Dimensione Formato  
FG1.pdf

accesso aperto

Tipologia: Post-print, accepted manuscript ecc. (versione accettata dall'editore)
Dimensione 359.26 kB
Formato Adobe PDF
359.26 kB Adobe PDF Visualizza/Apri
Frigeri-Grasselli2012_Article_GlobalAndTrajectoryAttractorsF.pdf

accesso riservato

Tipologia: Publisher's version/PDF
Dimensione 384.22 kB
Formato Adobe PDF
384.22 kB Adobe PDF   Visualizza/Apri   Richiedi una copia
Pubblicazioni consigliate

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/724680
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 55
  • ???jsp.display-item.citation.isi??? 51
social impact