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# 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: https://dx.doi.org/10.4310/PAMQ.2015.v11.n2.a5

#### Authors

#### 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

Published 24 August 2016