We prove convergence of suitable subsequences of weak solutions of a diffuse interface model for the two-phase flow of incompressible fluids with different densities with a nonlocal Cahn-Hilliard equation to weak solutions of the corresponding system with a standard "local" Cahn-Hilliard equation. The analysis is done in the case of a sufficiently smooth bounded domain with no-slip boundary condition for the velocity and Neumann boundary conditions for the Cahn-Hilliard equation. The proof is based on the corresponding result in the case of a single Cahn-Hilliard equation and compactness arguments used in the proof of existence of weak solutions for the diffuse interface model.
On a thermodynamically consistent diffuse interface model for incompressible two-phase flows with unmatched densities: Energy equality and Lyapunov stability
Universität Regensburg（雷根斯堡大学）；Fudan University（复旦大学）
In this article we prove the global existence of weak solutions for a diffuse interface model in a bounded domain (both in 2D and 3D) involving incompressible magnetic fluids with unmatched densities. The model couples the incompressible Navier-Stokes equations, gradient flow of the magnetization vector and the Cahn-Hilliard dynamics describing the partial mixing of two fluids. The density of the mixture depends on an order parameter and the modelling, specifically the density dependence, is inspired from Abels, Garcke and Grün 2011.
We prove existence of weak solutions for a diffuse interface model for the flow of two viscous incompressible Newtonian fluids with different densities in a bounded domain in two and three space dimensions. In contrast to previous works, we study a model with a singular non-local free energy, which controls the $H^{α/2}$-norm of the volume fraction. We show existence of weak solutions for large times with the aid of an implicit time discretization.
We study an initial-boundary value problem for the incompressible Navier-Stokes-Cahn-Hilliard system with non-constant density proposed by Abels, Garcke and Grün in 2012. This model arises in the diffuse interface theory for binary mixtures of viscous incompressible fluids. This system is a generalization of the well-known model H in the case of fluids with unmatched densities. In three dimensions, we prove that any global weak solution (for which uniqueness is not known) exhibits a propagation of regularity in time and stabilizes towards an equilibrium state as $t \rightarrow \infty$. More pr…
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 Darcy's 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 un…
Global Quasi-strong Solutions to the Voigt Regularization of a Diffuse Interface Model for Incompressible Two-phase Flows with Bulk-surface Interaction
Faculty for Mathematics, University of Regensburg（雷根斯堡大学数学学院）；Institute for Analysis, Department of Mathematics, Karlsruhe Institute of Technology (KIT)（卡尔斯鲁厄理工学院数学分析研究所）；School of Mathematical Sciences, Fudan University（复旦大学数学科学学院）
We investigate the nonlocal version of the Abels-Garcke-Grün (AGG) system, which describes the motion of a mixture of two viscous incompressible fluids. This consists of the incompressible Navier-Stokes-Cahn-Hilliard system characterized by concentration-dependent density and viscosity, and an additional flux term due to interface diffusion. In particular, the Cahn-Hilliard dynamics of the concentration (phase-field) is governed by the aggregation/diffusion competition of the nonlocal Helmholtz free energy with singular (logarithmic) potential and constant mobility. We first prove the existenc…
Global Weak Solutions of a Thermodynamically Consistent Diffuse Interface Model for Nonhomogeneous Incompressible Two-phase Flows with a Soluble Surfactant
Well-posedness of Strong Solutions for a Thermodynamically Consistent Diffuse-interface Model for Incompressible Two-phase Flows with a Soluble Surfactant
Politecnico di Milano（米兰理工大学）；Fudan University（复旦大学）
Nonlocal Cahn-Hilliard-Navier-Stokes systems with singular potentials
在这里，我们考虑一个Cahn-Hilliard-Navier-Stokes系统，其特征在于非局部Cahn-Hilliard方程具有奇异（例如，对数）电势。该系统起源于两种不混溶流体的不可压缩等温混合物的扩散界面模型。我们已经分析了具有任意多项式增长的光滑势的情况。在这里，利用以前的结果，我们研究这个更具挑战性（和物理相关）的情况下。我们首先建立了一个整体弱解的存在性与无滑移和无通量边界条件。然后证明了二维广义半流的整体吸引子的存在性（在J.M.球）。我们记得，即使在2D中，唯一性仍然是一个悬而未决的问题。作为副产品，我们还获得了（对流）非局部Cahn-Hilliard方程的连通全局吸引子的存在性。最后，在3D的情况下，我们建立了一个轨迹吸引子的存在性（在V. V. Chepyzhov和M.I. Vishik）。
In this paper we consider the diffuse interface generalized antiferromagnetic model with local/nonlocal attractive/repulsive terms in competition studied in Daneri-Kerschbaum-Runa arXiv:1907.06419. The parameters of the model are denoted by $τ$ and $\varepsilon$: the parameter $τ$ represents the relative strength of the local term with respect to the nonlocal one, while the parameter $\varepsilon$ describes the transition scale in the Modica-Mortola type term. Restricting to a periodic box of size $L$, with $L$ multiple of the period of the minimal one-dimensional minimizers, in Daneri-Kerschb…
In this paper we study a diffuse interface generalized antiferromagnetic model. The functional describing the model contains a Modica-Mortola type local term and a nonlocal generalized antiferromagnetic term in competition. The competition between the two terms results in a frustrated system which is believed to lead to the emergence of a wide variety of patterns. The sharp interface limit of our model is considered in \cite{GR} and in \cite{DR}. In the discrete setting it has been previously studied in \cite{GLL, GLS, GS}. The model contains two parameters: $τ$ and $\varepsilon$. The paramete…
In this paper, we consider a two-dimensional diffuse interface model for the phase separation of an incompressible and isothermal binary fluid mixture with matched densities. This model consists of the Navier--Stokes equations, nonlinearly coupled with a convective nonlocal Cahn--Hilliard equation. The system rules the evolution of the volume-averaged velocity $\uvec$ of the mixture and the (relative) concentration difference $φ$ of the two phases. The aim of this work is to study an optimal control problem for such a system, the control being a time-dependent external force acting on the flui…
On the nonlocal Cahn-Hilliard equation with nonlocal dynamic boundary condition and singular potential: well-posedness, regularity and asymptotic limits
We propose here a motivation for a mixed local/nonlocal problem with a new type of Neumann condition. Our description is based on formal expansions and approximations. In a nutshell, a biological species is supposed to diffuse either by a random walk or by a jump process, according to prescribed probabilities. If the process makes an individual exit the niche, it must come to the niche right away, by selecting the return point according to the underlying stochastic process. More precisely, if the random particle exits the domain, it is forced to immediately reenter the domain, and the new poin…
A nonlocal Cahn-Hilliard model with a nonsmooth potential of double-well obstacle type that promotes sharp interfaces in the solution is presented. To capture long-range interactions between particles, a nonlocal Ginzburg-Landau energy functional is defined which recovers the classical (local) model for vanishing nonlocal interactions. In contrast to the local Cahn-Hilliard problem that always leads to diffuse interfaces, the proposed nonlocal model can lead to a strict separation into pure phases of the substance. Here, the lack of smoothness of the potential is essential to guarantee the afo…
We analyze a diffuse interface model that couples a viscous Cahn-Hilliard equation for the phase variable with a diffusion-reaction equation for the nutrient concentration. The system under consideration also takes into account some important mechanisms like chemotaxis, active transport as well as nonlocal interaction of Oono's type. When the spatial dimension is three, we prove the existence and uniqueness of global weak solutions to the model with singular potentials including the physically relevant logarithmic potential. Then we obtain some regularity properties of the weak solutions when …
