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Register a new userDiffuse Interface models for incompressible binary fluids and the mass-conserving Allen-Cahn approximation
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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.
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We introduce a fractional variant of the Cahn-Hilliard equation settled in a bounded domain $Omega$ of $R^N$ and complemented with homogeneous Dirichlet boundary conditions of solid type (i.e., imposed in the entire complement of $Omega$). After setting a proper functional framework, we prove existence and uniqueness of weak solutions to the related initial-boundary value problem. Then, we investigate some significant singular limits obtained as the order of either of the fractional Laplacians appearing in the equation is let tend to 0. In particular, we can rigorously prove that the fractional Allen-Cahn, fractional porous medium, and fractional fast-diffusion equations can be obtained in the limit. Finally, in the last part of the paper, we discuss existence and qualitative properties of stationary solutions of our problem and of its singular limits.
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.
This paper is concerned with a fully nonlinear variant of the Allen-Cahn equation with strong irreversibility, where each solution is constrained to be non-decreasing in time. Main purposes of the paper are to prove the well-posedness, smoothing effect and comparison principle, to provide an equivalent reformulation of the equation as a parabolic obstacle problem and to reveal long-time behaviors of solutions. More precisely, by deriving emph{partial} energy-dissipation estimates, a global attractor is constructed in a metric setting, and it is also proved that each solution $u(x,t)$ converges to a solution of an elliptic obstacle problem as $t to +infty$.
This article is mainly devoted to the asymptotic analysis of a fractional version of the (elliptic) Allen-Cahn equation in a bounded domain $Omegasubsetmathbb{R}^n$, with or without a source term in the right hand side of the equation (commonly called chemical potential). Compare to the usual Allen-Cahn equation, the Laplace operator is here replaced by the fractional Laplacian $(-Delta)^s$ with $sin(0,1/2)$, as defined in Fourier space. In the singular limit $varepsilonto 0$, we show that arbitrary solutions with uniformly bounded energy converge both in the energetic and geometric sense to surfaces of prescribed nonlocal mean curvature in $Omega$ whenever the chemical potential remains bounded in suitable Sobolev spaces. With no chemical potential, the notion of surface of prescribed nonlocal mean curvature reduces to the stationary version of the nonlocal minimal surfaces introduced by L.A. Caffarelli, J.M. Roquejoffre, and O. Savin. Under the same Sobolev regularity assumption on the chemical potential, we also prove that surfaces of prescribed nonlocal mean curvature have a Minkowski codimension equal to one, and that the associated sets have a locally finite fractional $2s^prime$-perimeter in $Omega$ for every $s^primein(0,1/2)$.
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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