Heat flow of harmonic maps whose gradients belong to \(L^{n}_{x}L^{\infty}_{t}\)
type: Item
instance of: scholarly article
title: Heat flow of harmonic maps whose gradients belong to \(L^{n}_{x}L^{\infty}_{t}\)
zbMATH Open document ID: 1156.35052
published in: Archive for Rational Mechanics and Analysis
publication date: 2008-06-30T00:00:00Z
review text: The author studies the \(L^n_x L^\infty_t\) norm estimate of the gradient of the heat flow from a compact Riemannian manifold \((M, g)\) without boundary to another compact Riemannian manifold \(N\subset\mathbb R^k\) without boundary. Let \(T\in(0,+\infty]\) be the maximal existence time, which depends on the initial data and is scaling invariant. Similar to the Navier-Stokes equation, the scaling invariant norm estimates play an important role in the regularity theory of the solutions. The author proves that for \(n\geq 4\), if \(u: M\times [0,T]\to N\) is a weak solution to the heat flow of harmonic maps satisfying \(\nabla u\in L^n_x L^\infty_t(M\times [0,T])\), then \(u\in C^\infty(M\times (0,T],N)\). As a consequence, he shows that for \(n\geq 3\), if \(0< T<\infty\) is the maximal time for the unique smooth solution \(u\in C^\infty(M\times(0, T], N)\) of the heat flow, then \(\|\nabla u(t)\|_n\) blows up as \(T\to\infty\).
reviewed by: Kung Ching Chang
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K45
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 58J47
zbMATH DE Number: 5294554
zbMATH Keywords: energy monotonicity inequality
zbMATH Keywords: gradient blow up
author: Changyou Wang
MaRDI profile type: Publication
full work available at URL: s00205-007-0102-4
OpenAlex ID: W2064598489
cites work: Finite-time blow-up of the heat flow of harmonic maps from surfaces
cites work: Heat flow of \(p\)-harmonic maps with values into spheres
cites work: Partial regularity for weak heat flows into spheres
cites work: Harmonic Mappings of Riemannian Manifolds
cites work: Backward uniqueness for parabolic equations
cites work: L3,∞-solutions of the Navier-Stokes equations and backward uniqueness
cites work: Partial regularity for harmonic maps of evolution into spheres
cites work: On functions of bounded mean oscillation
cites work: Q4354424
cites work: Harmonic and quasi-harmonic spheres
cites work: Q5589555
cites work: Un teorema di unicita per le equazioni di Navier-Stokes
cites work: Q5723392
cites work: On the evolution of harmonic maps in higher dimensions
DOI: 10.1007/S00205-007-0102-4
description: scientific article; zbMATH DE number 5294554