The geometry of the phase diffusion equation

The Cross-Newell phase diffusion equation, t(\(k) over right arrow\)Theta(T ) = -del . (B(\(k) over right arrow\) . (k) over right arrow), (k) over rig ht arrow = del Theta, and its regularization describes natural patterns and defects far from onset in large aspect ratio systems with rotational symme try. In this paper we construct explicit solutions of the unregularized equ ation and suggest candidates for its weak solutions. We confirm these ideas by examining a fourth-order regularized equation in the limit of infinite aspect ratio. The stationary solutions of this equation include the minimiz ers of a free energy, and we show these minimizers are remarkably well-appr oximated by a second-order "self-dual" equation. Moreover, the self-dual solutions give upper bounds for the free energy whi ch imply the existence of weak limits for the asymptotic minimizers. In cer tain cases, some recent results of Jin and Kohn [28] combined with these up per bounds enable us to demonstrate that the energy of the asymptotic minim izers converges to that of the self-dual solutions in a viscosity limit.