We show that self-similar solutions for the mean curvature flow, surface diffusion, and Willmore flow of entire graphs are stable upon perturbations of initial data with small Lipschitz norm. Roughly speaking, the perturbed solutions are asymptotically self-similar as time tends to infinity. Our results are built upon the global analytic solutions constructed by Koch and Lamm in 2012, the compactness arguments adapted by Asai and Giga in 2014, and the spatial equi-decay properties on certain weighted function spaces. The proof for all of the above flows are achieved in a unified framework by utilizing the estimates of the linearized operator.
1
Vanderbilt University, Nashville, USA
2
Purdue University, West Lafayette, USA
Hengrong Du; N.K. Yip. Stability of self-similar solutions to geometric flows. Interfaces and free boundaries, Tome 25 (2023) no. 2, pp. 155-191. doi: 10.4171/ifb/488
@article{10_4171_ifb_488,
author = {Hengrong Du and N.K. Yip},
title = {Stability of self-similar solutions to geometric flows},
journal = {Interfaces and free boundaries},
pages = {155--191},
year = {2023},
volume = {25},
number = {2},
doi = {10.4171/ifb/488},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/488/}
}
TY - JOUR
AU - Hengrong Du
AU - N.K. Yip
TI - Stability of self-similar solutions to geometric flows
JO - Interfaces and free boundaries
PY - 2023
SP - 155
EP - 191
VL - 25
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.4171/ifb/488/
DO - 10.4171/ifb/488
ID - 10_4171_ifb_488
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