330 research outputs found
Global weak solutions to a sixth order Cahn--Hilliard type equation
In this paper we study a sixth order Cahn-Hilliard type equation that arises as a model for the faceting of a growing surface. We show global in time existence of weak solutions and uniform in time a priori estimates in the H^3 norm. These bounds enable us to show the uniqueness of weak solutions
Global solvability and blow up for the convective Cahn-Hilliard equations with concave potentials
We study initial boundary value problems for the convective Cahn-Hilliard
equation \Dt u +\px^4u +u\px u+\px^2(|u|^pu)=0. It is well-known that without
the convective term, the solutions of this equation may blow up in finite time
for any . In contrast to that, we show that the presence of the convective
term u\px u in the Cahn-Hilliard equation prevents blow up at least for
. We also show that the blowing up solutions still exist if is
large enough (). The related equations like
Kolmogorov-Sivashinsky-Spiegel equation, sixth order convective Cahn-Hilliard
equation, are also considered
Long-time Dynamics and Optimal Control of a Diffuse Interface Model for Tumor Growth
We investigate the long-time dynamics and optimal control problem of a
diffuse interface model that describes the growth of a tumor in presence of a
nutrient and surrounded by host tissues. The state system consists of a
Cahn-Hilliard type equation for the tumor cell fraction and a
reaction-diffusion equation for the nutrient. The possible medication that
serves to eliminate tumor cells is in terms of drugs and is introduced into the
system through the nutrient. In this setting, the control variable acts as an
external source in the nutrient equation. First, we consider the problem of
`long-time treatment' under a suitable given source and prove the convergence
of any global solution to a single equilibrium as . Then we
consider the `finite-time treatment' of a tumor, which corresponds to an
optimal control problem. Here we also allow the objective cost functional to
depend on a free time variable, which represents the unknown treatment time to
be optimized. We prove the existence of an optimal control and obtain first
order necessary optimality conditions for both the drug concentration and the
treatment time. One of the main aim of the control problem is to realize in the
best possible way a desired final distribution of the tumor cells, which is
expressed by the target function . By establishing the Lyapunov
stability of certain equilibria of the state system (without external source),
we see that can be taken as a stable configuration, so that the
tumor will not grow again once the finite-time treatment is completed
An Energetic Variational Approach for the Cahn--Hilliard Equation with Dynamic Boundary Condition: Model Derivation and Mathematical Analysis
The Cahn--Hilliard equation is a fundamental model that describes phase
separation processes of binary mixtures. In recent years, several types of
dynamic boundary conditions have been proposed in order to account for possible
short-range interactions of the material with the solid wall. Our first aim in
this paper is to propose a new class of dynamic boundary conditions for the
Cahn--Hilliard equation in a rather general setting. The derivation is based on
an energetic variational approach that combines the least action principle and
Onsager's principle of maximum energy dissipation. One feature of our model is
that it naturally fulfills three important physical constraints such as
conservation of mass, dissipation of energy and force balance relations. Next,
we provide a comprehensive analysis of the resulting system of partial
differential equations. Under suitable assumptions, we prove the existence and
uniqueness of global weak/strong solutions to the initial boundary value
problem with or without surface diffusion. Furthermore, we establish the
uniqueness of asymptotic limit as and characterize the stability
of local energy minimizers for the system.Comment: to appear in Arch. Rational Mech. Ana
Global existence and uniqueness for a singular/degenerate Cahn-Hilliard system with viscosity
Existence and uniqueness are investigated for a nonlinear diffusion problem
of phase-field type, consisting of a parabolic system of two partial
differential equations, complemented by Neumann homogeneous boundary conditions
and initial conditions. This system aims to model two-species phase segregation
on an atomic lattice; in the balance equations of microforces and microenergy,
the two unknowns are the order parameter and the chemical potential. A simpler
version of the same system has recently been discussed in arXiv:1103.4585v1. In
this paper, a fairly more general phase-field equation is coupled with a
genuinely nonlinear diffusion equation. The existence of a global-in-time
solution is proved with the help of suitable a priori estimates. In the case of
a constant atom mobility, a new and rather unusual uniqueness proof is given,
based on a suitable combination of variables.Comment: Key words: phase-field model, nonlinear laws, existence of solutions,
new uniqueness proo
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