Harmonic map heat flow with rough boundary data

2010/10/16 by Lu Wang, Wang, Lu
Mathematics · #35K55 #58E20 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1010.3313

openalex publication_date 2010/10/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let B₁ be the unit open disk in \Real² and M be a closed Riemannian manifold. In this note, we first prove the uniqueness for weak solutions of the harmonic map heat flow in H¹([0,T]× B₁,M) whose energy is non-increasing in time, given initial data u₀∈ H¹(B₁,M) and boundary data γ=u₀|_{∂ B1}. 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 u₀∈ H¹(B₁,M) and γ=u₀|_{∂ B1}∈ H^3/2(∂ B₁). 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.

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