Xinfu Chen

Convergence of the Cahn-Hilliard equation to the Hele-Shaw model

We prove that level surfaces of solutions to the Cahn-Hilliard equation tend to solutions of the Hele-Shaw problem under the assumption that classical solutions of the latter exist. The method is based on a new matched asymptotic expansion for solutions, a spectral analysis for linearizd operators, and an estimate for the difference between the true solutions and certain approximate ones.

Generation and propagation of interfaces for reaction-diffusion equations

Abstract This paper is concerned with the asymptotic behavior as e ↖ 0 of the solution ue of the reaction-diffusion equation in R N × R +: u t − δ +( 1 ∈ 2 )= 0 where φ is the derivative of a bistable potential. We show that if the initial data u(·, 0) has values in both domains of attraction of the potential, then an interface will develop in a short time O(e 2 ¦ ln e¦) . We also show that if the wells of the potential are of equal depth, then this interface will propagate with normal velocity equal to its mean curvature. Our result is valid as long as the interface remains smooth. If the initial interface is compact and N ⩾ 2 then the interface will disappear in a finite time (but not if N = 1). In case the depths of the wells are not equal then in order to obtain get “reasonable” results, we must work on the scaled time s = t e (slower time scale). In this scale we show that the interface moves with a constant speed proportional to the difference of the depths of the two wells, along the normal, towards the domain of the deeper well; this result is valid for all s ϵ (0, ∞) and does not actually depend on the regularity of the interface. We also extend the above results to the homogeneous second initial-boundary value problem. In case the depths of the wells are equal and the initial interface is orthogonal to the boundary, we prove that the interface moves with normal velocity equal to its mean curvature provided that there is a family of hypersurfaces which moves according to their mean curvature and intersect the boundary orthogonally.

Convergence of the phase field model to its sharp interface limits

We consider the distinguished limits of the phase eld equations and prove that the corresponding free boundary problem is attained in each case. These include the classical Stefan model, the surface tension model (with or without kinetics), the surface tension model with zero specic heat, the two phase Hele{Shaw, or quasi-static, model. The Hele{Shaw model is also a limit of the Cahn{Hilliard equation, which is itself a limit of the phase eld equations. Also included in the distinguished limits is the motion by mean curvature model that is a limit of the Allen{Cahn equation, which can in turn be attained from the phase eld equations.

Traveling Waves of Bistable Dynamics on a Lattice

We prove the existence of stationary or traveling waves ina lattice dynamical system arising in the theory of binary phase transitions. The system allows infinite-range couplings with positive and ...

A free boundary problem arising in a model of wound healing

In this paper we consider a system of two semilinear parabolic reaction-diffusion equations with a free boundary, which arises in a model of corneal epithelial wound healing. We prove that the init...

The Hele-Shaw problem and area-preserving curve-shortening motions

We prove existence, locally in time, of a solution of the following Hele-Shaw problem: Given a simply connected curve contained in a smooth bounded domainΩ, find the motion of the curve such that its normal velocity equals the jump of the normal derivatives of a function which is harmonic in the complement of the curve inΩ and whose boundary value on the curve equals its curvature. We show that this motion is a curve-shortening motion which does not change the area of the region enclosed by the curve. In caseΩ is the whole plane ℛ2, we also show that if the initial curve is close to an equilibrium curve, i.e., to a circle, then there exists a global solution and the global solution tends to some circle exponentially fast as time tends to infinity.

Periodic traveling waves and locating oscillating patterns in multidimensional domains

We establish the existence and robustness of layered, time-periodic solutions to a reaction-diffusion equation in a bounded domain in Rn, when the diffusion coefficient is sufficiently small and the reaction term is periodic in time and bistable in the state variable. Our results suggest that these patterned, oscillatory solutions are stable and locally unique. The location of the internal layers is characterized through a periodic traveling wave problem for a related one-dimensional reaction-diffusion equation. This one-dimensional problem is of independent interest and for this we establish the existence and uniqueness of a heteroclinic solution which, in constant-velocity moving coodinates, is periodic in time. Furthermore, we prove that the manifold of translates of this solution is globally exponentially asymptotically stable.

Local existence and uniqueness of solutions of the Stefan problem with surface tension and kinetic undercooling

Abstract We consider the Stefan Problem with surface tension and kinetic undercooling effects, that is, with the temperature u satisfying the condition u = − σκ − βV on the interface Γ(σ, β = const. >0), (∗) where κ and V are the mean curvature and the normal velocity of Γ, respectively. We establish short time existence and uniqueness of classical solutions for the resulting free-boundary problem. The key observation is that, when translated to local coordinates, equation (∗) is a quasi-linear parabolic equation on a manifold (without boundary). To prove existence and uniqueness we look for the solution u as a fixed point of a contracting map R . We start by solving the equation of motion (∗) for a given left-hand side u . This provides us with an interface Γ which we use to solve the parabolic problem in the bulk (with the conservation of energy condition across Γ), thereby obtaining a new temperature function ū= R ( u ). Finally, it is the regularizing character of the operator R that allows us to show that R is a contraction on a small time interval.

A mathematical analysis of the optimal exercise boundary for american put options

We study a free boundary problem arising from American put options. In particular we prove existence and uniqueness for this problem, and we derive and rigorously prove high order asymptotic expansions for the early exercise boundary near expiry. We provide four approximations for the boundary: one is explicit and is valid near expiry (weeks); two others are implicit involving inverse functions and are accurate for longer time to expiry (months); the fourth is an ODE initial value problem which is very accurate for all times to expiry, is extremely stable, and hence can be solved instantaneously on any computer. We further provide an ode iterative scheme which can reach its numerical fixed point in five iterations for all time to expiry. We also provide a large time (equivalent to regular expiration times but large interest rate and/or volatility) behavior of the exercise boundary. To demonstrate the accuracy of our approximations, we present the results of a numerical simulation.

A nonlinear elliptic equation arising from gauge field theory and cosmology

We study radially symmetric solutions of a nonlinear elliptic partial differential equation in R2 with critical Sobolev growth, i. e. the nonlinearity is of exponential type. This problem arises from a wide variety of important areas in theoretical physics including superconductivity and cosmology. Our results lead to many interesting implications for the physical problems considered. For example, for the self-dual Chern–Simons theory, we are able to conclude that the electric charge, magnetic flux, or energy of a non-topological N-vortex solution may assume any prescribed value above an explicit lower bound. For the Einstein-matter-gauge equations, we find a necessary and sufficient condition for the existence of a self-dual cosmic string solution. Such a condition imposes an obstruction for the winding number of a string in terms of the universal gravitational constant.