Pure and Applied Mathematics Quarterly

Volume 11 (2015)

Number 2

A note on the heat flow of harmonic maps whose gradients belong to $L^q_t L^p_x$

Pages: 283 – 292

DOI: http://dx.doi.org/10.4310/PAMQ.2015.v11.n2.a5

Authors

Junfei Dai (Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang, China)

Wei Luo (Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang, China)

Meng Wang (Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang, China)

Abstract

For any compact $n$-dimensional Riemannian manifold $(M,g)$ without boundary, a compact Riemannian manifold $N \subset R^k$ without boundary, and $0 \lt T \leq \infty$, we prove that for $n \geq 3$, if $u : M \times (0, T] \to N$ is a weak solution to the heat flow of harmonic maps such that $\nabla u \in L^q_t L^p_x (M \times (0, T]) (n/p + 2/q = 1 \textrm{ for some } p \gt n)$, then $u \in C^{\infty} (M \times (0, T),N)$. For $p = n$, we proved the regularity for the suitable weak solution defined in [1].

Keywords

heat flow, suitable solution, Lorentz space, blow up

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