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Register a new userA diffuse interface model for two-phase incompressible flows with nonlocal interactions and nonconstant mobility
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We consider a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids with matched constant densities. This model consists of the Navier-Stokes system coupled with a convective nonlocal Cahn-Hilliard equation with non-constant mobility. We first prove the existence of a global weak solution in the case of non-degenerate mobilities and regular potentials of polynomial growth. Then we extend the result to degenerate mobilities and singular (e.g. logarithmic) potentials. In the latter case we also establish the existence of the global attractor in dimension two. Using a similar technique, we show that there is a global attractor for the convective nonlocal Cahn-Hilliard equation with degenerate mobility and singular potential in dimension three.
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We study a diffuse interface model describing the motion of two viscous fluids driven by the surface tension in a Hele-Shaw cell. The full system consists of the Cahn-Hilliard equation coupled with the Darcys law. We address the physically relevant case in which the two fluids have different viscosities (unmatched viscosities case) and the free energy density is the logarithmic Helmholtz potential. In dimension two we prove the uniqueness of weak solutions under a regularity criterion, and the existence and uniqueness of global strong solutions. In dimension three we show the existence and uniqueness of strong solutions, which are local in time for large data or global in time for appropriate small data. These results extend the analysis obtained in the matched viscosities case by Giorgini, Grasselli and Wu (Ann. Inst. H. Poincar{e} Anal. Non Lin{e}aire 35 (2018), 318-360). Furthermore, we prove the uniqueness of weak solutions in dimension two by taking the well-known polynomial approximation of the logarithmic potential.
In this paper, we propose and analyze a diffuse interface model for inductionless magnetohydrodynamic fluids. The model couples a convective Cahn-Hilliard equation for the evolution of the interface, the Navier-Stokes system for fluid flow and the possion quation for electrostatics. The model is derived from Onsagers variational principle and conservation laws systematically. We perform formally matched asymptotic expansions and develop several sharp interface models in the limit when the interfacial thickness tends to zero. It is shown that the sharp interface limit of the models are the standard incompressible inductionless magnetohydrodynamic equations coupled with several different interface conditions for different choice of the mobilities. Numerical results verify the convergence of the diffuse interface model with different mobilitiess.
We consider a general family of regularized models for incompressible two-phase flows based on the Allen-Cahn formulation in n-dimensional compact Riemannian manifolds for n=2,3. The system we consider consists of a regularized family of Navier-Stokes equations (including the Navier-Stokes-{alpha}-like model, the Leray-{alpha} model, the Modified Leray-{alpha} model, the Simplified Bardina model, the Navier-Stokes-Voight model and the Navier-Stokes model) for the fluid velocity suitably coupled with a convective Allen-Cahn equation for the (phase) order parameter. We give a unified analysis of the entire three-parameter family of two-phase models using only abstract mapping properties of the principal dissipation and smoothing operators, and then use assumptions about the specific form of the parametrizations, leading to specific models, only when necessary to obtain the sharpest results. We establish existence, stability and regularity results, and some results for singular perturbations, which as special cases include the inviscid limit of viscous models and the {alpha}->0 limit in {alpha}-models. Then, we also show the existence of a global attractor and exponential attractor for our general model, and then establish precise conditions under which each trajectory converges to a single equilibrium by means of a LS inequality. We also derive new results on the existence of global and exponential attractors for the regularized family of Navier-Stokes equations and magnetohydrodynamics models which improve and complement the results of Holst et. al. [J. Nonlinear Science 20, 2010, 523-567]. Finally, our analysis is applied to certain regularized Ericksen-Leslie (RSEL) models for the hydrodynamics of liquid crystals in n-dimensional compact Riemannian manifolds.
This paper is devoted to the mathematical analysis of some Diffuse Interface systems which model the motion of a two-phase incompressible fluid mixture in presence of capillarity effects in a bounded smooth domain. First, we consider a two-fluids parabolic-hyperbolic model that accounts for unmatched densities and viscosities without diffusive dynamics at the interface. We prove the existence and uniqueness of local solutions. Next, we introduce dissipative mixing effects by means of the mass-conserving Allen-Cahn approximation. In particular, we consider the resulting nonhomogeneous Navier- Stokes-Allen-Cahn and Euler-Allen-Cahn systems with the physically relevant Flory-Huggins potential. We study the existence and uniqueness of global weak and strong solutions and their separation property. In our analysis we combine energy and entropy estimates, a novel end-point estimate of the product of two functions, and a logarithmic type Gronwall argument.
We consider a diffuse interface model of tumor growth proposed by A.~Hawkins-Daruud et al. This model consists of the Cahn-Hilliard equation for the tumor cell fraction $varphi$ nonlinearly coupled with a reaction-diffusion equation for $psi$, which represents the nutrient-rich extracellular water volume fraction. The coupling is expressed through a suitable proliferation function $p(varphi)$ multiplied by the differences of the chemical potentials for $varphi$ and $psi$. The system is equipped with no-flux boundary conditions which entails the conservation of the total mass, that is, the spatial average of $varphi+psi$. Here we prove the existence of a weak solution to the associated Cauchy problem, provided that the potential $F$ and $p$ satisfy sufficiently general conditions. Then we show that the weak solution is unique and continuously depends on the initial data, provided that $p$ satisfies slightly stronger growth restrictions. Also, we demonstrate the existence of a strong solution and that any weak solution regularizes in finite time. Finally, we prove the existence of the global attractor in a phase space characterized by an a priori bounded energy.
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