### Abstract

We investigate self-similar solutions of the dipole problem for the one-dimensional thin film equation on the half-line {x ≥ 0}. We study compactly supported solutions of the linear moving boundary problem and show how they relate to solutions of the nonlinear problem. The similarity solutions are generally of the second kind, given by the solution of a nonlinear eigenvalue problem, although there are some notable cases where first-kind solutions also arise. We examine the conserved quantities connected to these first-kind solutions. Difficulties associated with the lack of a maximum principle and the non-self-adjointness of the fundamental linear problem are also considered. Seeking similarity solutions that include sign changes yields a surprisingly rich set of (coexisting) stable solutions for the intermediate asymptotics of this problem. Our results include analysis of limiting cases and comparisons with numerical computations.

Original language | English |
---|---|

Pages (from-to) | 1727-1748 |

Number of pages | 22 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 66 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2006 |

Externally published | Yes |

### Fingerprint

### Keywords

- Dipole problem
- Similarity solutions
- Thin-film equations

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*SIAM Journal on Applied Mathematics*,

*66*(5), 1727-1748. https://doi.org/10.1137/050637832

**The linear limit of the dipole problem for the thin film equation.** / Bowen, Mark; Witelski, Thomas P.

Research output: Contribution to journal › Article

*SIAM Journal on Applied Mathematics*, vol. 66, no. 5, pp. 1727-1748. https://doi.org/10.1137/050637832

}

TY - JOUR

T1 - The linear limit of the dipole problem for the thin film equation

AU - Bowen, Mark

AU - Witelski, Thomas P.

PY - 2006

Y1 - 2006

N2 - We investigate self-similar solutions of the dipole problem for the one-dimensional thin film equation on the half-line {x ≥ 0}. We study compactly supported solutions of the linear moving boundary problem and show how they relate to solutions of the nonlinear problem. The similarity solutions are generally of the second kind, given by the solution of a nonlinear eigenvalue problem, although there are some notable cases where first-kind solutions also arise. We examine the conserved quantities connected to these first-kind solutions. Difficulties associated with the lack of a maximum principle and the non-self-adjointness of the fundamental linear problem are also considered. Seeking similarity solutions that include sign changes yields a surprisingly rich set of (coexisting) stable solutions for the intermediate asymptotics of this problem. Our results include analysis of limiting cases and comparisons with numerical computations.

AB - We investigate self-similar solutions of the dipole problem for the one-dimensional thin film equation on the half-line {x ≥ 0}. We study compactly supported solutions of the linear moving boundary problem and show how they relate to solutions of the nonlinear problem. The similarity solutions are generally of the second kind, given by the solution of a nonlinear eigenvalue problem, although there are some notable cases where first-kind solutions also arise. We examine the conserved quantities connected to these first-kind solutions. Difficulties associated with the lack of a maximum principle and the non-self-adjointness of the fundamental linear problem are also considered. Seeking similarity solutions that include sign changes yields a surprisingly rich set of (coexisting) stable solutions for the intermediate asymptotics of this problem. Our results include analysis of limiting cases and comparisons with numerical computations.

KW - Dipole problem

KW - Similarity solutions

KW - Thin-film equations

UR - http://www.scopus.com/inward/record.url?scp=33750192169&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33750192169&partnerID=8YFLogxK

U2 - 10.1137/050637832

DO - 10.1137/050637832

M3 - Article

AN - SCOPUS:33750192169

VL - 66

SP - 1727

EP - 1748

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 5

ER -