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Lamm, Tobias; Rivière, Tristan

Conservation laws for fourth order systems in four dimensions. (English) Zbl 1139.35328

Commun. Partial Differ. Equations 33, No. 2, 245-262 (2008).
Summary: Following an approach of the second author for conformally invariant variational problems in two dimensions, we show in four dimensions the existence of a conservation law for fourth order systems, which includes both intrinsic and extrinsic biharmonic maps. With the help of this conservation law we prove the continuity of weak solutions of this system. Moreover we use the conservation law to derive the existence of a unique global weak solution of the extrinsic biharmonic map flow in the energy space.

Cited in 3 Reviews
Cited in 39 Documents

MSC:

35D10 Regularity of generalized solutions of PDE (MSC2000)
35J60 Nonlinear elliptic equations
58E20 Harmonic maps, etc.
58J35 Heat and other parabolic equation methods for PDEs on manifolds

Keywords:

biharmonic map heat flow; cLorentz spaces; Wente inequality; continuity of weak solutions
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References:

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