The geometry of the phase diffusion equation
The authors consider the Cross-Newell phase diffusion equation \(\tau(|\vec k|) \Theta_T= -\nabla\cdot (B(|\vec k|) \cdot\vec k)\), \(\vec k=\nabla \Theta\) and its regularization, which describes natural patterns and defects far from onsel in large aspect ratio systems with rotational symmetry. They construct explicit solutions of the nonregularized equation and examine its weak solutions. This idea is confirmed by studying a fourth-order regularized equation in the limit of infinite aspect ratio. The stationary solutions of this equation include the minimizers of free energy, and it is shown that these minimizers can be remarkably well approximated by a second-order ``self-dual'' equation. Using the results of \textit{W. Jin} and \textit{R. V. Kohn} [J. Nonlinear Sci. 10, No. 3, 355-390, doi:10.1007/s003329910014 (2000; Zbl 0973.49009)], the authors demonstrate that the energy of asymptotic minimizers converges to self-dual solutions in a viscosity limit.
scientific article; zbMATH DE number 1439880