### Abstract

We consider two standard models of surface-energy-driven coarsening: a constant-mobility Cahn-Hilliard equation, whose large-time behavior corresponds to Mullins-Sekerka dynamics; and a degenerate-mobility Cahn-Hilliard equation, whose large-time behavior corresponds to motion by surface diffusion. Arguments based on scaling suggest that the typical length scale should behave as l(t) ∼ t^{1/3} in the first case and l(t) ∼ t^{1/4} in the second. We prove a weak, one-sided version of this assertion - showing, roughly speaking, that no solution can coarsen faster than the expected rate. Our result constrains the behavior in a time-averaged sense rather than pointwise in time, and it constrains not the physical length scale but rather the perimeter per unit volume. The argument is simple and robust, combining the basic dissipation relations with an interpolation inequality and an ODE argument.

Original language | English (US) |
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Pages (from-to) | 375-395 |

Number of pages | 21 |

Journal | Communications in Mathematical Physics |

Volume | 229 |

Issue number | 3 |

DOIs | |

State | Published - Sep 2002 |

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### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics

### Cite this

*Communications in Mathematical Physics*,

*229*(3), 375-395. https://doi.org/10.1007/s00220-002-0693-4

**Upper bounds on coarsening rates.** / Kohn, Robert; Otto, Felix.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 229, no. 3, pp. 375-395. https://doi.org/10.1007/s00220-002-0693-4

}

TY - JOUR

T1 - Upper bounds on coarsening rates

AU - Kohn, Robert

AU - Otto, Felix

PY - 2002/9

Y1 - 2002/9

N2 - We consider two standard models of surface-energy-driven coarsening: a constant-mobility Cahn-Hilliard equation, whose large-time behavior corresponds to Mullins-Sekerka dynamics; and a degenerate-mobility Cahn-Hilliard equation, whose large-time behavior corresponds to motion by surface diffusion. Arguments based on scaling suggest that the typical length scale should behave as l(t) ∼ t1/3 in the first case and l(t) ∼ t1/4 in the second. We prove a weak, one-sided version of this assertion - showing, roughly speaking, that no solution can coarsen faster than the expected rate. Our result constrains the behavior in a time-averaged sense rather than pointwise in time, and it constrains not the physical length scale but rather the perimeter per unit volume. The argument is simple and robust, combining the basic dissipation relations with an interpolation inequality and an ODE argument.

AB - We consider two standard models of surface-energy-driven coarsening: a constant-mobility Cahn-Hilliard equation, whose large-time behavior corresponds to Mullins-Sekerka dynamics; and a degenerate-mobility Cahn-Hilliard equation, whose large-time behavior corresponds to motion by surface diffusion. Arguments based on scaling suggest that the typical length scale should behave as l(t) ∼ t1/3 in the first case and l(t) ∼ t1/4 in the second. We prove a weak, one-sided version of this assertion - showing, roughly speaking, that no solution can coarsen faster than the expected rate. Our result constrains the behavior in a time-averaged sense rather than pointwise in time, and it constrains not the physical length scale but rather the perimeter per unit volume. The argument is simple and robust, combining the basic dissipation relations with an interpolation inequality and an ODE argument.

UR - http://www.scopus.com/inward/record.url?scp=0036025749&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036025749&partnerID=8YFLogxK

U2 - 10.1007/s00220-002-0693-4

DO - 10.1007/s00220-002-0693-4

M3 - Article

AN - SCOPUS:0036025749

VL - 229

SP - 375

EP - 395

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -