We show a regularity criterion to the harmonic heat flow from 2-dimensional Riemannian manifold M into a sphere. It is shown that a weak solution of the harmonic heat flow from 2-dimensional manifold into a sphere is regular under the criterion ∫0T ||∇u(τ)|| BMOr2dτ where BMOr is the space of bounded mean oscillations on M. A sharp version of the Sobolev inequality of the Brezis-Gallouet type is introduced on M. A monotonicity formula by the mean oscillation is established and applied for proving such a regularity criterion for weak solutions as above.
|Number of pages||25|
|Journal||Calculus of Variations and Partial Differential Equations|
|Publication status||Published - 2008 Dec|
ASJC Scopus subject areas
- Applied Mathematics