caa a22 4500 475805445 CHVBK 20180406123745.0 cr unu---uuuuu 170329e20000401xx s 000 0 eng 10.1007/s003329910010 doi (NATIONALLICENCE)springer-10.1007/s003329910010 The Geometry of the Phase Diffusion Equation [Elektronische Daten] [N. M. Ercolani, R. Indik, A. C. Newell, T. Passot] $ \tau(|{{\vec k}}|) \mbox{\bf $\Theta$}_T = -\nabla\cdot (B(|{{\vec k}}|)\cdot {{\vec k}}), \,\, {{\vec k}} = \nabla \mbox{\bf $\Theta$},$ : and its regularization describes natural patterns and defects far from onset in large aspect ratio systems with rotational symmetry. In this paper we construct explicit solutions of the unregularized equation 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 minimizers of a free energy, and we show these minimizers are remarkably well-approximated by a second-order 's'sself-dual'' equation. Moreover, the self-dual solutions give upper bounds for the free energy which imply the existence of weak limits for the asymptotic minimizers. In certain cases, some recent results of Jin and Kohn [28] combined with these upper bounds enable us to demonstrate that the energy of the asymptotic minimizers converges to that of the self-dual solutions in a viscosity limit. Springer-Verlag New York Inc., 2000 Ercolani N. M. Department of Mathematics, University of Arizona, Tucson, AZ 85719, USA, US aut Indik R. Department of Mathematics, University of Arizona, Tucson, AZ 85719, USA, US aut Newell A. C. Mathematical Institute, University of Warwick, Coventry CV4 7AL, UK, GB aut Passot T. Mathematical Institute, University of Warwick, Coventry CV4 7AL, UK, GB aut Journal of Nonlinear Science Springer-Verlag 10/2(2000-04-01), 223-274 0938-8974 10:2<223 2000 10 332 https://doi.org/10.1007/s003329910010 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s003329910010 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Ercolani N. M. Department of Mathematics, University of Arizona, Tucson, AZ 85719, USA, US aut NATIONALLICENCE 700 1- Indik R. Department of Mathematics, University of Arizona, Tucson, AZ 85719, USA, US aut NATIONALLICENCE 700 1- Newell A. C. Mathematical Institute, University of Warwick, Coventry CV4 7AL, UK, GB aut NATIONALLICENCE 700 1- Passot T. Mathematical Institute, University of Warwick, Coventry CV4 7AL, UK, GB aut NATIONALLICENCE 773 0- Journal of Nonlinear Science Springer-Verlag 10/2(2000-04-01), 223-274 0938-8974 10:2<223 2000 10 332 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer