We consider a nonlinear system which consists of the incompressible Navier-Stokes equations coupled with a convective nonlocal Cahn-Hilliard equation. This is a diffuse interface model which describes the motion of an incompressible isothermal mixture of two (partially) immiscible fluids having the same density. We assume that both the viscosity and mobility functions depend smoothly on the order parameter. Moreover, we assume that the mobility degenerates at the pure phases and that the potential is singular (e.g. of logarithmic type). This system is endowed with a no-slip boundary condition for the (average) velocity and a homogeneous Neumann boundary condition for the chemical potential. Thus the total mass is conserved. In the two-dimensional case, this problem was already analyzed in some joint papers of the first three authors. However, in the present general case, only the existence of a global weak solution, the (conditional) weak-strong uniqueness and the existence of the global attractor were proven. Here we are able to establish the existence of a (unique) strong solution through an approximation procedure based on time discretization. As a consequence, we can prove suitable uniform estimates which allow us to show some smoothness of the global attractor. Finally, we discuss the existence of strong solutions for the convective nonlocal Cahn-Hilliard equation, with a given velocity field, in the three dimensional case as well.

Two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems with variable viscosity, degenerate mobility and singular potential / S. Frigeri, C.G. Gal, M. Grasselli, J. Sprekels. - In: NONLINEARITY. - ISSN 0951-7715. - 32:2(2019), pp. 678-727. [10.1088/1361-6544/aaedd0]

Two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems with variable viscosity, degenerate mobility and singular potential

M. Grasselli;
2019

Abstract

We consider a nonlinear system which consists of the incompressible Navier-Stokes equations coupled with a convective nonlocal Cahn-Hilliard equation. This is a diffuse interface model which describes the motion of an incompressible isothermal mixture of two (partially) immiscible fluids having the same density. We assume that both the viscosity and mobility functions depend smoothly on the order parameter. Moreover, we assume that the mobility degenerates at the pure phases and that the potential is singular (e.g. of logarithmic type). This system is endowed with a no-slip boundary condition for the (average) velocity and a homogeneous Neumann boundary condition for the chemical potential. Thus the total mass is conserved. In the two-dimensional case, this problem was already analyzed in some joint papers of the first three authors. However, in the present general case, only the existence of a global weak solution, the (conditional) weak-strong uniqueness and the existence of the global attractor were proven. Here we are able to establish the existence of a (unique) strong solution through an approximation procedure based on time discretization. As a consequence, we can prove suitable uniform estimates which allow us to show some smoothness of the global attractor. Finally, we discuss the existence of strong solutions for the convective nonlocal Cahn-Hilliard equation, with a given velocity field, in the three dimensional case as well.
global attractors; Holder continuity; incompressible binary fluids; Navier-Stokes equations; nonlocal Cahn-Hilliard equations; regularization; strong solutions; time discretization schemes
Settore MAT/05 - Analisi Matematica
Article (author)
File in questo prodotto:
File Dimensione Formato  
FGGS_LAST.pdf

accesso aperto

Tipologia: Post-print, accepted manuscript ecc. (versione accettata dall'editore)
Dimensione 530.88 kB
Formato Adobe PDF
530.88 kB Adobe PDF Visualizza/Apri
Frigeri_2019_Nonlinearity_32_678.pdf

accesso riservato

Tipologia: Publisher's version/PDF
Dimensione 3.74 MB
Formato Adobe PDF
3.74 MB Adobe PDF   Visualizza/Apri   Richiedi una copia
Pubblicazioni consigliate

Caricamento 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/2434/724129
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 17
  • ???jsp.display-item.citation.isi??? 16
social impact