### Abstract

The Cross-Newell phase diffusion equation, τ(|k^{→}|)Θ_{T} = -∇ · (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.

Original language | English (US) |
---|---|

Pages (from-to) | 223-274 |

Number of pages | 52 |

Journal | Journal of Nonlinear Science |

Volume | 10 |

Issue number | 2 |

State | Published - Mar 2000 |

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### ASJC Scopus subject areas

- Computational Mechanics
- Mechanics of Materials
- Mathematics(all)
- Applied Mathematics
- Mathematical Physics
- Statistical and Nonlinear Physics

### Cite this

*Journal of Nonlinear Science*,

*10*(2), 223-274.

**The geometry of the phase diffusion equation.** / Ercolani, Nicholas M; Indik, Robert A; Newell, Alan C; Passot, T.

Research output: Contribution to journal › Article

*Journal of Nonlinear Science*, vol. 10, no. 2, pp. 223-274.

}

TY - JOUR

T1 - The geometry of the phase diffusion equation

AU - Ercolani, Nicholas M

AU - Indik, Robert A

AU - Newell, Alan C

AU - Passot, T.

PY - 2000/3

Y1 - 2000/3

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

M3 - Article

VL - 10

SP - 223

EP - 274

JO - Journal of Nonlinear Science

JF - Journal of Nonlinear Science

SN - 0938-8974

IS - 2

ER -