On a nonlocal Cahn-Hilliard/Navier-Stokes system with degenerate mobility and singular potential for incompressible fluids with different densities

We consider a diffuse interface model describing flow and phase separation of a binary isothermal mixture of (partially) immiscible viscous incompressible Newtonian fluids having different densities. The model is the nonlocal version of the one derived by Abels, Garcke and Grün and consists in a inhomogeneous Navier-Stokes type system coupled with a convective nonlocal Cahn-Hilliard equation. This model was already analyzed in a paper by the same author, for the case of singular potential and non-degenerate mobility. Here, we address the physically more relevant situation of degenerate mobility and we prove existence of global weak solutions satisfying an energy inequality. The proof relies on a regularization technique based on a careful approximation of the singular potential. Existence and regularity of the pressure field is also discussed. Moreover, in two dimensions and for slightly more regular solutions, we establish the validity of the energy identity. We point out that in none of the existing contributions dealing with the original (local) Abels, Garcke Grün model, an energy identity in two dimensions is derived (only existence of weak solutions has been proven so far).     

一类非局部Cahn-Hilliard方程弱解的存在唯一性

研究一类对流非局部Cahn-Hilliard方程的Neumann问题.通过一致Schauder估计和Leray-Schauder不动点定理,得到了该问题经典解的存在唯一性.进而,利用弱收敛方法得到了该问题弱解的存在唯一性.     

Global existence of weak solutions to a diffuse interface model for magnetic fluids

This article is devoted to the derivation and analysis of a system of partial differential equations modeling a diffuse interface flow of two Newtonian incompressible magnetic fluids. The system consists of the incompressible Navier–Stokes equations coupled with an evolutionary equation for the magnetization vector and the Cahn–Hilliard equations. We show global in time existence of weak solutions to the system using the time discretization method.     

Numerical solutions to a BBM‐Burgers model with a nonlocal viscous term

In this paper, we numerically investigate the BBM‐Burgers equation with a nonlocal viscous term (1) where is the Riemann‐Liouville half derivative. In particular, we implement different numerical schemes to approximate the solution and its asymptotical behavior. Also, we compare our numerical results with those given in 2013, 2014 for similar models.     

Learning patient-specific parameters for a diffuse interface glioblastoma model from neuroimaging data

Parameters in mathematical models for glioblastoma multiforme (GBM) tumour growth are highly patient specific. Here, we aim to estimate parameters in a Cahn–Hilliard type diffuse interface model in an optimised way using model order reduction (MOR) based on proper orthogonal decomposition (POD). Based on snapshots derived from finite element simulations for the full-order model (FOM), we use POD for dimension reduction and solve the parameter estimation for the reduced-order model (ROM). Neuroimaging data are used to define the highly inhomogeneous diffusion tensors as well as to define a target functional in a patient-specific manner. The ROM heavily relies on the discrete empirical interpolation method, which has to be appropriately adapted in order to deal with the highly nonlinear and degenerate parabolic partial differential equations. A feature of the approach is that we iterate between full order solvers with new parameters to compute a POD basis function and sensitivity-based parameter estimation for the ROM problems. The algorithm is applied using neuroimaging data for two clinical test cases, and we can demonstrate that the reduced-order approach drastically decreases the computational effort.     

Mathematical modeling of the three-dimensional evolution of the interface between fluids of different viscosities and densities in an inhomogeneous ground

The three-dimensional evolution of the interface between fluids of different viscosities and densities in the case of a piston displacement is considered. The problem is reduced to a system of integral and differential equations, which are solved numerically by the method of discrete singularities. The practical convergence of the numerical scheme is analyzed, and the numerical results are compared with an analytical solution.     

The asymptotic limits of Riemann solutions for the isentropic drift-flux model of compressible two-phase flows

The formation of vacuum state and delta shock wave are observed and studied in the limits of Riemann solutions for the one-dimensional isentropic drift-flux model of compressible two-phase flows by letting the pressure in the mixture momentum equation tend to zero. It is shown that the Riemann solution containing two rarefaction waves and one contact discontinuity turns out to be the solution containing two contact discontinuities with the vacuum state between them in the limiting situation. By comparison, it is also proved rigorously in the sense of distributions that the Riemann solution containing two shock waves and one contact discontinuity converges to a delta shock wave solution under this vanishing pressure limit.     

Existence of weak solutions for a non-classical sharp interface model for a two-phase flow of viscous,incompressible fluids

We introduce a new sharp interface model for the flow of two immiscible, viscous, incompressible fluids. In contrast to classical models for two-phase flows we prescribe an evolution law for the interfaces that takes diffusional effects into account. This leads to a coupled system of Navier–Stokes and Mullins–Sekerka type parts that coincides with the asymptotic limit of a diffuse interface model. We prove the long-time existence of weak solutions, which is an open problem for the classical two-phase model. We show that the phase interfaces have in almost all points a generalized mean curvature.     

Analysis of finite element approximations of a phase field model for two-phase fluids

This paper studies a phase field model for the mixture of two immiscible and incompressible fluids. The model is described by a nonlinear parabolic system consisting of the nonstationary Stokes equations coupled with the Allen-Cahn equation through an extra phase induced stress term in the Stokes equations and a fluid induced transport term in the Allen-Cahn equation. Both semi-discrete and fully discrete finite element methods are developed for approximating the parabolic system. It is shown that the proposed numerical methods satisfy a discrete energy law which mimics the basic energy law for the phase field model. Error estimates are derived for the semi-discrete method, and the convergence to the phase field model and to its sharp interface limiting model are established for the fully discrete finite element method by making use of the discrete energy law. Numerical experiments are also presented to validate the theory and to show the effectiveness of the combined phase field and finite element approach.

     

Global existence of classical solution for a viscous liquid-gas two-phase model with mass-dependent viscosity and vacuum

In this work, we obtain the global existence and uniqueness of classical solutions to a viscous liquid-gas two-phase model with mass-dependent viscosity and vacuum in one dimension, where the initial vacuum is allowed. We get the upper and lower bounds of gas and liquid masses n and m by the continuity methods which we use to study the compressible Navier-Stokes equations.     

Existence and blow-up of solutions for a porous medium equation with nonlocal boundary condition

In this article, a porous medium equation with nonlocal boundary condition and a localized source is studied. The results of the existence of global solutions or blow-up of solutions are given. The blow-up rate estimates are also obtained under some conditions.     

Extinction and decay estimates of solutions for a polytropic filtration equation with the nonlocal source and interior absorption

This paper deals with the extinction properties of solutions for the homogeneous Dirichlet boundary value problem with the nonlocal source and interior absorption where m,λ,k,q > 0, 0 < m(p ? 1) < 1, r ≤ 1, and . By using L^p‐integral norm estimate method, we obtain the sufficient conditions of extinction solutions. Moreover, we also give the precise decay estimates of the extinction solutions. Copyright © 2012 John Wiley & Sons, Ltd.     

Uniqueness and existence of solutions for a singular system with nonlocal operator via perturbation method

In this work, we investigate the existence and the uniqueness of solutions for the nonlocal elliptic system involving a singular nonlinearity as follows: $$ \left\{\begin{array}{ll} (-\Delta_p)^su = a(x)|u|^{q-2}u +\frac{1-\alpha}{2-\alpha-\beta} c(x)|u|^{-\alpha}|v|^{1-\beta}, \quad \text{in }\Omega,\ (-\Delta_p)^s v= b(x)|v|^{q-2}v +\frac{1-\beta}{2-\alpha-\beta} c(x)|u|^{1-\alpha}|v|^{-\beta}, \quad \text{in }\Omega,\ u=v = 0 ,\;\;\mbox{ in }\,\mathbb{R}^N\setminus\Omega, \end{array} \right. $$ where $\Omega $ is a bounded domain in $\mathbb{R}^{n}$ with smooth boundary, $0<\alpha <1,$ $0<\beta <1,$ $2-\alpha -\beta 相似文献   

Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits

We propose and analyze a fully discrete finite element scheme for the phase field model describing the solidification process in materials science. The primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical method, in particular, by focusing on the dependence of the error bounds on the parameter , known as the measure of the interface thickness. Optimal order error bounds are shown for the fully discrete scheme under some reasonable constraints on the mesh size and the time step size . In particular, it is shown that all error bounds depend on only in some lower polynomial order for small . The cruxes of the analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of Chen, and to establish a discrete counterpart of it for a linearized phase field operator to handle the nonlinear effect. Finally, as a nontrivial byproduct, the error estimates are used to establish convergence of the solution of the fully discrete scheme to solutions of the sharp interface limits of the phase field model under different scaling in its coefficients. The sharp interface limits include the classical Stefan problem, the generalized Stefan problems with surface tension and surface kinetics, the motion by mean curvature flow, and the Hele-Shaw model.

     

An existence and uniqueness principle for a nonlinear version of the Lebowitz‐Rubinow model with infinite maximum cycle length

The present article deals with existence and uniqueness results for a nonlinear evolution initial‐boundary value problem, which originates in an age‐structured cell population model introduced by Lebowitz and Rubinow (1974) describing the growth of a cell population. Cells of this population are distinguished by age a and cycle length l. In our framework, daughter and mother cells are related by a general reproduction rule that covers all known biological ones. In this paper, the cycle length l is allowed to be infinite. This hypothesis introduces some mathematical difficulties. We consider both local and nonlocal boundary conditions.     

Decay of global solutions, stability and blowup for a reaction-diffusion problem with free boundary

We consider a one-phase Stefan problem for the heat equation with a nonlinear reaction term. We first exhibit an energy condition, involving the initial data, under which the solution blows up in finite time in norm. We next prove that all global solutions are bounded and decay uniformly to 0, and that either: (i) the free boundary converges to a finite limit and the solution decays at an exponential rate, or (ii) the free boundary grows up to infinity and the decay rate is at most polynomial. Finally, we show that small data solutions behave like (i).

     

Existence of multiple and sign-changing solutions for a second-order nonlinear functional difference equation with periodic coefficients

In the present paper, we apply the method of invariant sets of descending flow to establish a series of criteria to ensure that a second-order nonlinear functional difference equation with periodic boundary conditions possesses at least one trivial solution and three nontrivial solutions. These nontrivial solutions consist of sign-changing solutions, positive solutions and negative solutions. Moreover, as an application of our theoretical results, an example is elaborated. Our results generalize and improve some existing ones.     

Existence and asymptotic behavior of solutions for a mathematical ecology model with herd behavior

In this work, we consider a prey-predator model with herd behavior under Neumann boundary conditions. For the system without diffusion, we establish a sufficient condition to guarantee the local asymptotic stability of all nontrivial equilibria and prove the existence of limit cycle of our proposed model. For the system with diffusion, we consider the long time behavior of the model including global attractor and local stability, and the Hopf and steady-state bifurcation analysis from the unique homogeneous positive steady state are carried out in detail. Furthermore, some numerical simulations to illustrate the theoretical analysis are performed to expand our theoretical results.     

带混合边界条件的Holling-Ⅱ型捕食模型正稳态解的存在性

研究了一个带混合边界条件的Holling-Ⅱ型捕食模型．通过构造一个微分紧算子K,借助锥内不动点指数理论、拓扑度理论、椭圆方程估计理论和极值原理证得了算子K存在正的不动点,此结论表明捕食模型存在正解,即物种共存状态．