TY - JOUR

T1 - The geometry of the phase diffusion equation

AU - Ercolani, N. M.

AU - Indik, R.

AU - Newell, A. C.

AU - Passot, T.

N1 - Funding Information:
The authors would like to express their appreciation to J. Lega for helpful discussions and a critical reading of the manuscript. The authors would like to acknowledge the support received from NSF Grants DMS-9302013 and DMS-9626306, and AFOSR Grant F49620-94-1-0144. T. Passot gratefully acknowledges the hospitality of the Arizona Center for Mathematical Sciences.

PY - 2000

Y1 - 2000

N2 - The Cross-Newell phase diffusion equation, τ(|k→|)ΘT = -∇ · (B(|k→|) · k→), k→ = ∇ Θ, 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.

AB - The Cross-Newell phase diffusion equation, τ(|k→|)ΘT = -∇ · (B(|k→|) · k→), k→ = ∇ Θ, 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.

UR - http://www.scopus.com/inward/record.url?scp=0000486391&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000486391&partnerID=8YFLogxK

U2 - 10.1007/s003329910010

DO - 10.1007/s003329910010

M3 - Article

AN - SCOPUS:0000486391

SN - 0938-8974

VL - 10

SP - 223

EP - 274

JO - Journal of Nonlinear Science

JF - Journal of Nonlinear Science

IS - 2

ER -