A nonlinear partial differential equation for the volume preserving mean curvature flow
Dimitra Antonopoulou Georgia Karali
Networks & Heterogeneous Media 2013, 8(1): 9-22 doi: 10.3934/nhm.2013.8.9
We analyze the evolution of multi-dimensional normal graphs over the unit sphere under volume preserving mean curvature flow and derive a non-linear partial differential equation in polar coordinates. Furthermore, we construct finite difference numerical schemes and present numerical results for the evolution of non-convex closed plane curves under this flow, to observe that they become convex very fast.
keywords: Nonlinear parabolic equations geometric evolution equations normal graphs volume preserving mean curvature flow numerics.
Existence of solution for a generalized stochastic Cahn-Hilliard equation on convex domains
Dimitra Antonopoulou Georgia Karali
Discrete & Continuous Dynamical Systems - B 2011, 16(1): 31-55 doi: 10.3934/dcdsb.2011.16.31
We consider a generalized Stochastic Cahn-Hilliard equation with multiplicative white noise posed on bounded convex domains in $R^d$, $d=1,2,3$, with piece-wise smooth boundary, and introduce an additive time dependent white noise term in the chemical potential. Since the Green's function of the problem is induced by a convolution semigroup, we present the equation in a weak stochastic integral formulation and prove existence of solution when $d\leq 2$ for general domains, and for $d=3$ for domains with minimum eigenfunction growth, without making use of any explicit expression of the spectrum and the eigenfunctions. The analysis is based on stochastic integral calculus, Galerkin approximations and the asymptotic spectral properties of the Neumann Laplacian operator. Existence is also derived for some non-convex cases when the boundary is smooth.
keywords: convex domains space-time white noise convolution semigroup Galerkin approximations. Stochastic Cahn-Hilliard Neumann Laplacian eigenfunction basis Green's function
On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation
Georgia Karali Yuko Nagase
Discrete & Continuous Dynamical Systems - S 2014, 7(1): 127-137 doi: 10.3934/dcdss.2014.7.127
In this manuscript, we consider a Cahn-Hilliard/Allen-Cahn equation is introduced in [17]. We give an existence of the solution, slightly improved from [18]. We also present a stochastic version of this equation in [3].
keywords: Galerkin method space-time white noise existence of solution stochastic partial differential equation. Cahn-Hilliard/Allen-Cahn equation
Asymptotics for a generalized Cahn-Hilliard equation with forcing terms
Dimitra Antonopoulou Georgia Karali Georgios T. Kossioris
Discrete & Continuous Dynamical Systems - A 2011, 30(4): 1037-1054 doi: 10.3934/dcds.2011.30.1037
Motivated by the physical theory of Critical Dynamics the Cahn-Hilliard equation on a bounded space domain is considered and forcing terms of general type are introduced. For such a rescaled equation the limiting inter-face problem is studied and the following are derived: (i) asymptotic results indicating that the forcing terms may slow down the equilibrium locally or globally, (ii) the sharp interface limit problem in the multidimensional case demonstrating a local influence in phase transitions of terms that stem from the chemical potential, while free energy independent terms act on the rest of the domain, (iii) a limiting non-homogeneous linear diffusion equation for the one-dimensional problem in the case of deterministic forcing term that follows the white noise scaling.
keywords: Generalized Cahn-Hilliard equation Hele-Shaw problem. asymptotics
Global-in-time behavior of the solution to a Gierer-Meinhardt system
Georgia Karali Takashi Suzuki Yoshio Yamada
Discrete & Continuous Dynamical Systems - A 2013, 33(7): 2885-2900 doi: 10.3934/dcds.2013.33.2885
Gierer-Meinhardt system is a mathematical model to describe biological pattern formation due to activator and inhibitor. Turing pattern is expected in the presence of local self-enhancement and long-range inhibition. The long-time behavior of the solution, however, has not yet been clarified mathematically. In this paper, we study the case when its ODE part takes periodic-in-time solutions, that is, $\tau=\frac{s+1}{p-1}$. Under some additional assumptions on parameters, we show that the solution exists global-in-time and absorbed into one of these ODE orbits. Thus spatial patterns eventually disappear if those parameters are in a region without local self-enhancement or long-range inhibition.
keywords: Hamilton structure Turing pattern asymptotic behavior of the solution. Reaction-diffusion equation Gierer-Meinhardt system

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