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Existence and partial regularity results for the heat flow for harmonic maps. (English) Zbl 0652.58024
For \(M={\mathbb{R}}^ m \)or compact m-dimensional manifolds M, \(m>2\), and compact n-dimensional target manifolds N we establish the existence of a global, partially regular solution to the evolution problem (1.6-7) for harmonic maps from M into N. The solution is smooth off a singular set of co-dimension \(\geq 2\) and as \(t\to \infty\) converges to a partially regular harmonic map from M into N.
Reviewer: M.Struwe

58E20 Harmonic maps, etc.
58J35 Heat and other parabolic equation methods for PDEs on manifolds
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[1] Chen, Y.: Weak solutions to the evolution problem for harmonic maps into spheres. Preprint (1988)
[2] Eells, J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Ann. J. Math.86, 109-160 (1964) · Zbl 0122.40102
[3] Eells, J., Wood, R.: Restrictions on harmonic maps of surfaces. Topology15, 263-266 (1976) · Zbl 0328.58008
[4] Keller, J., Rubinstein, J., Sternberg, P.: Reaction ? diffusion processes and evolution to harmonic maps. Preprint (1988) · Zbl 0702.35128
[5] Ladyzenskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and quasi-linear equations of parabolic type. Transl. Math. Monogr.23 (1968)
[6] Mitteau, J.-C.: Sur les applications harmoniques. J. Differ. Geom.9, 41-54 (1974) · Zbl 0281.35034
[7] Struwe, M.: On the evolution of harmonic maps of Riemannian surfaces. Commun. Math. Helv.60, 558-581 (1985) · Zbl 0595.58013
[8] Struwe, M.: On the evolution of harmonic maps in higher dimension. J. Differ. Geom. (To appear) · Zbl 0631.58004
[9] Struwe, M.: Heat flow methods for harmonic maps of surfaces and applications to free boundary problems. Lect. Notes Math.1324, 293-319. Berlin Heidelberg New York: Springer 1988 · Zbl 0651.53045
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