EXISTENCE AND UNIQUENESS OF A GLOBAL-IN-TIME SOLUTION TO A PHASE SEGREGATION PROBLEM OF THE ALLEN–CAHN TYPE

2010 ◽  
Vol 20 (04) ◽  
pp. 519-541 ◽  
Author(s):  
PIERLUIGI COLLI ◽  
GIANNI GILARDI ◽  
PAOLO PODIO-GUIDUGLI ◽  
JÜRGEN SPREKELS
Keyword(s):  
Existence And Uniqueness ◽  
Order Parameter ◽  
Chemical Potential ◽  
Phase Segregation ◽  
The Other ◽  
Memory Term ◽  
Limit Set ◽  
Smooth Solutions ◽  
Cahn Equation ◽  
Parameter Field

We study a model of phase segregation of the Allen–Cahn type, consisting in a system of two differential equations, one partial and the other ordinary, respectively interpreted as balances of microforces and microenergy; the two unknowns are the order parameter entering the standard A–C equation and the chemical potential. We introduce a notion of maximal solution to the o.d.e., parametrized on the order-parameter field; and, by substitution in the p.d.e. of the so-obtained chemical potential field, we give the latter equation the form of an Allen–Cahn equation for the order parameter, with a memory term. Finally, we prove the existence and uniqueness of global-in-time smooth solutions to this modified A–C equation, and we give a description of the relative ω-limit set.

scholarly journals Cahn–Hilliard equation on the boundary with bulk condition of Allen–Cahn type

2018 ◽  
Vol 9 (1) ◽  
pp. 16-38
Author(s):  
Pierluigi Colli ◽  
Takeshi Fukao
Keyword(s):  
Continuous Dependence ◽  
Order Parameter ◽  
Chemical Potential ◽  
Well Posedness ◽  
Dynamic Boundary Conditions ◽  
Cahn Equation ◽  
Hilliard Equation ◽  
The Poisson Equation ◽  
Dynamic Boundary ◽  
Cahn Hilliard Equation

Abstract The well-posedness of a system of partial differential equations with dynamic boundary conditions is discussed. This system is a sort of transmission problem between the dynamics in the bulk Ω and on the boundary Γ. The Poisson equation for the chemical potential and the Allen–Cahn equation for the order parameter in the bulk Ω are considered as auxiliary conditions for solving the Cahn–Hilliard equation on the boundary Γ. Recently, the well-posedness of this equation with a dynamic boundary condition, both of Cahn–Hilliard type, was discussed. Based on this result, the existence of the solution and its continuous dependence on the data are proved.

scholarly journals BSDEs with polynomial growth generators

2000 ◽  
Vol 13 (3) ◽  
pp. 207-238 ◽  
Author(s):  
Philippe Briand ◽  
René Carmona
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Existence And Uniqueness ◽  
Polynomial Growth ◽  
Probabilistic Representation ◽  
Existence And Uniqueness Results ◽  
State Variable ◽  
Cahn Equation ◽  
Uniqueness Results ◽  
Terminal Time

In this paper, we give existence and uniqueness results for backward stochastic differential equations when the generator has a polynomial growth in the state variable. We deal with the case of a fixed terminal time, as well as the case of random terminal time. The need for this type of extension of the classical existence and uniqueness results comes from the desire to provide a probabilistic representation of the solutions of semilinear partial differential equations in the spirit of a nonlinear Feynman-Kac formula. Indeed, in many applications of interest, the nonlinearity is polynomial, e.g, the Allen-Cahn equation or the standard nonlinear heat and Schrödinger equations.

scholarly journals A Fractional SAIDR Model in the Frame of Atangana–Baleanu Derivative

2021 ◽  
Vol 5 (2) ◽  
pp. 32
Author(s):  
Esmehan Uçar ◽  
Sümeyra Uçar ◽  
Fırat Evirgen ◽  
Necati Özdemir
Keyword(s):  
Existence And Uniqueness ◽  
Laplace Transformation ◽  
Cell Phones ◽  
Mobile Network ◽  
The Other ◽  
Banach Fixed Point Theorem ◽  
Propagation Models ◽  
Computer Worms ◽  
Computer Viruses ◽  
The Stability

It is possible to produce mobile phone worms, which are computer viruses with the ability to command the running of cell phones by taking advantage of their flaws, to be transmitted from one device to the other with increasing numbers. In our day, one of the services to gain currency for circulating these malignant worms is SMS. The distinctions of computers from mobile devices render the existing propagation models of computer worms unable to start operating instantaneously in the mobile network, and this is particularly valid for the SMS framework. The susceptible–affected–infectious–suspended–recovered model with a classical derivative (abbreviated as SAIDR) was coined by Xiao et al., (2017) in order to correctly estimate the spread of worms by means of SMS. This study is the first to implement an Atangana–Baleanu (AB) derivative in association with the fractional SAIDR model, depending upon the SAIDR model. The existence and uniqueness of the drinking model solutions together with the stability analysis are shown through the Banach fixed point theorem. The special solution of the model is investigated using the Laplace transformation and then we present a set of numeric graphics by varying the fractional-order θ with the intention of showing the effectiveness of the fractional derivative.

Research on SUE DTA Model with Constant Demand Based on Probit Model

2011 ◽  
Vol 374-377 ◽  
pp. 2605-2609
Author(s):  
Lei Shi ◽  
Li Gao
Keyword(s):  
Logit Model ◽  
Existence And Uniqueness ◽  
Probit Model ◽  
The Other ◽  
A Algorithm ◽  
Model Based

Logit model is among the most important model in SUE DTA study. A lot of work have been done based on Logit model. As the other very important SUE DTA model, Probit model has not been the focus of many researcher. This paper presents a SUE model based on Probit model, which aims at building up the Probit model with constant demand. The existence and uniqueness of the model is presented, Finally, a algorithm is given.

scholarly journals Nonstationary Superconductivity: Quantum Dissipation and Time-Dependent Ginzburg-Landau Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-10
Author(s):  
Anatoly A. Barybin
Keyword(s):  
Order Parameter ◽  
Continuity Equation ◽  
Chemical Potential ◽  
Landau Equation ◽  
Time Dependent ◽  
Quantum Dissipation ◽  
Ginzburg Landau ◽  
Superfluid Component ◽  
Ginzburg Landau Equation ◽  
Conservation Property

Transport equations of the macroscopic superfluid dynamics are revised on the basis of a combination of the conventional (stationary) Ginzburg-Landau equation and Schrödinger's equation for the macroscopic wave function (often called the order parameter) by using the well-known Madelung-Feynman approach to representation of the quantum-mechanical equations in hydrodynamic form. Such an approach has given (a) three different contributions to the resulting chemical potential for the superfluid component, (b) a general hydrodynamic equation of superfluid motion, (c) the continuity equation for superfluid flow with a relaxation term involving the phenomenological parameters and , (d) a new version of the time-dependent Ginzburg-Landau equation for the modulus of the order parameter which takes into account dissipation effects and reflects the charge conservation property for the superfluid component. The conventional Ginzburg-Landau equation also follows from our continuity equation as a particular case of stationarity. All the results obtained are mutually consistent within the scope of the chosen phenomenological description and, being model-neutral, applicable to both the low- and high- superconductors.

scholarly journals Free vs. locally free Kleinian groups

2019 ◽  
Vol 2019 (746) ◽  
pp. 149-170
Author(s):  
Pekka Pankka ◽  
Juan Souto
Keyword(s):  
Hausdorff Dimension ◽  
Kleinian Group ◽  
Cantor Set ◽  
Cantor Sets ◽  
Kleinian Groups ◽  
The Other ◽  
Limit Set ◽  
Limit Sets ◽  
Other Hand

Abstract We prove that Kleinian groups whose limit sets are Cantor sets of Hausdorff dimension < 1 are free. On the other hand we construct for any ε > 0 an example of a non-free purely hyperbolic Kleinian group whose limit set is a Cantor set of Hausdorff dimension < 1 + ε.

scholarly journals Equilibrium measures for the Hénon map at the first bifurcation: uniqueness and geometric/statistical properties

2014 ◽  
Vol 36 (1) ◽  
pp. 215-255 ◽  
Author(s):  
SAMUEL SENTI ◽  
HIROKI TAKAHASI
Keyword(s):  
Existence And Uniqueness ◽  
Thermodynamic Formalism ◽  
Invariant Probability Measure ◽  
Statistical Properties ◽  
Limit Set ◽  
Bifurcation Parameter ◽  
Large Interval ◽  
Equilibrium Measures ◽  
Uniform Hyperbolicity ◽  
Unstable Direction

For strongly dissipative Hénon maps at the first bifurcation parameter where the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set, we establish a thermodynamic formalism, i.e. we prove the existence and uniqueness of an invariant probability measure that minimizes the free energy associated with a non-continuous geometric potential$-t\log J^{u}$, where$t\in \mathbb{R}$is in a certain large interval and$J^{u}$denotes the Jacobian in the unstable direction. We obtain geometric and statistical properties of these measures.

A nonlinear system for irreversible phase changes

1990 ◽  
Vol 1 (1) ◽  
pp. 91-100 ◽  
Author(s):  
Dominique Blanchard ◽  
Hamid Ghidouche
Keyword(s):  
Nonlinear System ◽  
Existence And Uniqueness ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Phase Changes ◽  
The Other ◽  
System Modelling ◽  
Mathematical Study ◽  
Uniqueness Of The Solution ◽  
Change Problem

This paper is concerned with the mathematical study of a nonlinear system modelling an irreversible phase change problem. Uniqueness of the solution is proved using the accretivity of the system in (L1)2. Expressing one of the two unknowns as an explicit functional of the other reduces the system to a single nonlinear evolution equation and ultimately leads to an existence theorem.In this paper the existence and uniqueness of the solution of a nonlinear system modelling some irreversible phase changes is established.

Sign in / Sign up

Export Citation Format

Share Document