The weak solutions to the evolution problems of harmonic maps (Q1826177)

The author proves the existence of global weak solution to the evolution problem of harmonic maps of a compact Riemannian manifold M into the Euclidean n-sphere \(S^ n\), i.e. the existence of a distribution solution of \(\partial_ tu-\Delta_ Mu+| u_*|^ 2u=0\), \(t>0\), \(| u|^ 2=1\), with \(u(0,.)=u_ 0\in H^{1,2}(M)\) that is \(L^{\infty}\)-bounded and weakly continuous in \(t>0\) with values in \(H^{1,2}(M)\).