We provide formal matched asymptotic expansions for ancient convex
solutions to MCF. The formal analysis leading to the solutions is
analogous to that for the generic MCF neck pinch in
[1].
For any $p, q$ with $p+q=n$, $p\geq1$, $q\geq2$ we find a formal ancient
solution which is a small perturbation of an ellipsoid. For
$t\to-\infty$ the solution becomes increasingly astigmatic: $q$ of its
major axes have length $\approx\sqrt{2(q-1)(-t)}$, while the other $p$ axes
have length $\approx \sqrt{-2t\log(-t)}$.
We conjecture that an analysis similar to that in [2] will
lead to a rigorous construction of ancient solutions to MCF with the
asymptotics described in this paper.
Citation: Sigurd Angenent. Formal asymptotic expansions for symmetric ancient ovalsin mean curvature flow[J]. Networks and Heterogeneous Media, 2013, 8(1): 1-8. doi: 10.3934/nhm.2013.8.1
Related Papers:
| [1] |
Sigurd Angenent .
Formal asymptotic expansions for symmetric ancient ovals
in mean curvature flow. Networks and Heterogeneous Media, 2013, 8(1): 1-8.
doi: 10.3934/nhm.2013.8.1
|
| [2] |
Dimitra Antonopoulou, Georgia Karali .
A nonlinear partial differential equation for the volume preserving mean curvature flow. Networks and Heterogeneous Media, 2013, 8(1): 9-22.
doi: 10.3934/nhm.2013.8.9
|
| [3] |
Ken-Ichi Nakamura, Toshiko Ogiwara .
Periodically growing solutions in a class of strongly monotone semiflows. Networks and Heterogeneous Media, 2012, 7(4): 881-891.
doi: 10.3934/nhm.2012.7.881
|
| [4] |
Iryna Pankratova, Andrey Piatnitski .
Homogenization of convection-diffusion equation in infinite cylinder. Networks and Heterogeneous Media, 2011, 6(1): 111-126.
doi: 10.3934/nhm.2011.6.111
|
| [5] |
Leonid Berlyand, Mykhailo Potomkin, Volodymyr Rybalko .
Sharp interface limit in a phase field model of cell motility. Networks and Heterogeneous Media, 2017, 12(4): 551-590.
doi: 10.3934/nhm.2017023
|
| [6] |
Leonid Berlyand, Giuseppe Cardone, Yuliya Gorb, Gregory Panasenko .
Asymptotic analysis of an array of closely spaced absolutely conductive inclusions. Networks and Heterogeneous Media, 2006, 1(3): 353-377.
doi: 10.3934/nhm.2006.1.353
|
| [7] |
Marie Henry .
Singular limit of an activator-inhibitor type model. Networks and Heterogeneous Media, 2012, 7(4): 781-803.
doi: 10.3934/nhm.2012.7.781
|
| [8] |
Natalia O. Babych, Ilia V. Kamotski, Valery P. Smyshlyaev .
Homogenization of spectral problems in bounded domains with doubly high contrasts. Networks and Heterogeneous Media, 2008, 3(3): 413-436.
doi: 10.3934/nhm.2008.3.413
|
| [9] |
Gung-Min Gie, Makram Hamouda, Roger Temam .
Asymptotic analysis of the Navier-Stokes equations in a curved domain with a non-characteristic boundary. Networks and Heterogeneous Media, 2012, 7(4): 741-766.
doi: 10.3934/nhm.2012.7.741
|
| [10] |
Grigory Panasenko, Ruxandra Stavre .
Asymptotic analysis of the Stokes flow with variable viscosity in a thin elastic channel. Networks and Heterogeneous Media, 2010, 5(4): 783-812.
doi: 10.3934/nhm.2010.5.783
|
Abstract
We provide formal matched asymptotic expansions for ancient convex
solutions to MCF. The formal analysis leading to the solutions is
analogous to that for the generic MCF neck pinch in
[1].
For any $p, q$ with $p+q=n$, $p\geq1$, $q\geq2$ we find a formal ancient
solution which is a small perturbation of an ellipsoid. For
$t\to-\infty$ the solution becomes increasingly astigmatic: $q$ of its
major axes have length $\approx\sqrt{2(q-1)(-t)}$, while the other $p$ axes
have length $\approx \sqrt{-2t\log(-t)}$.
We conjecture that an analysis similar to that in [2] will
lead to a rigorous construction of ancient solutions to MCF with the
asymptotics described in this paper.
References
|
[1]
|
S. B. Angenent and J. J. L. Velázquez, Degenerate neckpinches in mean curvature flow, J. Reine Angew. Math., 482 (1997), 15-66.
|
|
[2]
|
Sigurd Angenent, Cristina Caputo and Dan Knopf, Minimally invasive surgery for Ricci flow singularities, J. Reine Angew. Math., 672 (2012), 39–87(to appear).
|
|
[3]
|
Sigurd Angenent, Shrinking doughnuts, Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989), Progr. Nonlinear Differential Equations Appl., 7 Birkhäuser Boston, Boston, MA, (1992), 21-38.
|
|
[4]
|
Panagiota Daskalopoulos, Richard Hamilton and Natasa Sesum, Classification of compact ancient solutions to the curve shortening flow, J. Differential Geom., 84 (2010), 455-464.
|
|
[5]
|
M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom., 23 (1986), 69-96.
|
DownLoad: