Measurable Semigroup Selection of the Heat Flow for Harmonic Maps

Abstract

J.-M. Coron proved in [5] that the global weak solutions of the heat flow from $M$ to $N$, starting at non-stationary weakly harmonic maps, are not unique when $M = B^3$ and $N = S^2$. Hence, the semigroup property of the solution map does not hold in general. The present short paper uses the techniques developed by J. Cardona and L. Kapitanski to show the existence of infinitely many measurable semigroups solving the heat flow in the same cases where non-uniqueness was shown by J.-M. Coron.