Harmonic map heat flow with rough boundary data
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Abstract:
Let $B_1$ be the unit open disk in $\mathbb {R}^2$ and $M$ a closed Riemannian manifold. In this note, we first prove the uniqueness for weak solutions of the harmonic map heat flow in $H^1([0,T]\times B_1,M)$ whose energy is non-increasing in time, given initial data $u_0\in H^1(B_1,M)$ and boundary data $\gamma =u_0|_{\partial B_1}$. Previously, this uniqueness result was obtained by Rivière (when $M$ is the round sphere and the energy of initial data is small) and Freire (when $M$ is an arbitrary closed Riemannian manifold), given that $u_0\in H^1(B_1,M)$ and $\gamma =u_0|_{\partial B_1}\in H^{3/2}(\partial B_1)$. The point of our uniqueness result is that no boundary regularity assumption is needed. Second, we prove the exponential convergence of the harmonic map heat flow, assuming that energy is small at all times.References
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Additional Information
- Lu Wang
- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
- Address at time of publication: Department of Mathematics, Johns Hopkins University, 3400 N. Charles Street, Baltimore, Maryland 21218
- Email: luwang@math.mit.edu, lwang@math.jhu.edu
- Received by editor(s): July 22, 2010
- Received by editor(s) in revised form: September 11, 2010
- Published electronically: May 29, 2012
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 5265-5283
- MSC (2010): Primary 58E20; Secondary 35K55
- DOI: https://doi.org/10.1090/S0002-9947-2012-05473-0
- MathSciNet review: 2931329