Reaction-diffusion equations, interfaces and geometric flows

The subject of this post is a reaction-diffusion equation of the form \partial_t u=\Delta u-\frac{1}{\epsilon^2} f'(u) for a real-valued function $u$ with t\in {\bf R}^+ and x\in {\bf R}^n. Here \epsilon is a small parameter. This equation is often known as the Allen-Cahn equation and the phenomena discussed in the following were described in a 1979 paper of Allen and Cahn in Acta Metallurgica (which I have not seen). My primary source is a paper by Xinfu Chen (J. Diff. Eq. 96, 116) which includes both a nice introduction to the subject and rigorous proofs of some key theorems. The function f in the equation is a smooth function with certainly qualitative properties: two minima u_1^- and u_2^- separated by a maximum u^+.

In physical terms there are two opposing effects at work in this system. There is reaction which drives the evolution to resemble that of solutions of the corresponding ODE (limit \epsilon\to\infty) and diffusion which causes the solution to spread out in space. Generic solutions of the ODE tend to one of the two minima of the potential at late times. If the diffusion is set to zero this leads to two spatial regions where the solution has almost constant values u_1^- and u_2^- respectively. They are separated by an narrow interface where the spatial derivatives of u are large and the the values of u are close to u^+. These ideas are captured mathematically by proving asympotic expansions in the parameter \epsilon for the solutions on time intervals whose lengths depend on \epsilon in a suitable way.

The following phenomena arise. Here I assume that the space dimension n is at least two. The case n=1 is special. The first statement is that on a suitable time interval, which is short when \epsilon is small, an interface forms. It can be described by a hypersurface. Without going into details it can be thought of as the level hypersurface u=u^+. The second describes how the interface moves in space on a longer timescale. If the function f has unequal values at the minima then the interface moves in the direction of its normal towards the region corresponding to the smaller minimum with a velocity which is proportional to the difference of the two values. On the timescale on which this motion takes place the interface stands still in the case that the two values are equal. It can, however, be shown that in the case of equal minima the interface moves in a definite way on an even longer timescale. The interface moves in the direction of its normal with a velocity which is proportional to its mean curvature at the given point. In other words, the hypersurface is a solution of mean curvature flow, one of the best-known geometric flows.

To finish I will state a question which I have no answer to, not even vaguely. Are there any useful analogues of these results for the hyperbolic equation \partial_t^2 u+\partial_t u=\Delta u-\frac{1}{\epsilon^2} f'(u)? Here there is a potential interaction between reaction, damping and dispersion.

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2 Responses to “Reaction-diffusion equations, interfaces and geometric flows”

  1. hydrobates Says:

    Addendum: It was pointed out to me by Roger Bieli that an equation similar to the one I wrote at the end of this post has been considered in a very recent preprint by Bellettini, Novaga and Orlandi (arXiv: 0811.3741). One difference is that I included a term with the first time derivative of the unknown. I thought that this type of dissipation could help to promote or stabilize the formation of a front. In view of the preprint just mentioned this may be unnecessary. The work of Bellettini et. al. also includes the case of a vector-valued unknown. I found out that the equation which I called ‘Allen-Cahn equation’ is also known under the name ‘parabolic Ginzburg-Landau equation’.

  2. hydrobates Says:

    There is a new paper out by Robert Jerrard on the hyperbolic equation discussed above without extra dissipation (, thanks to Roger Bieli for pointing this out to me). This paper is not just a technical advance but seems to do a good job of explaining the wider context of the result. Among many other things the author indicates a relation to the work of David Stuart on deriving the equations of motion of material bodies in general relativity from the Einstein equations.

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