UNSPECIFIED. (2000) The geometry of the phase diffusion equation. JOURNAL OF NONLINEAR SCIENCE, 10 (2). pp. 223-274. ISSN 0938-8974

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Abstract

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 right arrow = del 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 "self-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.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics T Technology > TJ Mechanical engineering and machinery Q Science > QC Physics
Journal or Publication Title:	JOURNAL OF NONLINEAR SCIENCE
Publisher:	SPRINGER VERLAG
ISSN:	0938-8974
Date:	March 2000
Volume:	10
Number:	2
Number of Pages:	52
Page Range:	pp. 223-274
Publication Status:	Published
URI:	http://wrap.warwick.ac.uk/id/eprint/13640

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