On the Well-posedness of the Wave Map Problem in High Dimensions

We construct a gauge theoretic change of variables for the wave map from R × R into a compact group or Riemannian symmetric space, prove a new multiplication theorem for mixed Lebesgue-Besov spaces, and show the global well-posedness of a modified wave map equation n ≥ 4 for small critical initial data. We obtain global existence and uniqueness for the Cauchy problem of wave maps into compact Lie groups and symmetric spaces with small critical initial data and n ≥ 4. 0. Introduction The wave map equation between two Riemannian manifoldsthe wave equation version of the evolution equations which are derived from the same geometric considerations as the harmonic map equation between two Riemannian manifoldshas been studied by a number of mathematicians in the last decade. The work of Klainerman and Machedon and Klainerman and Selberg [5] [6] [8] studying the Cauchy problem for regular data is probably the best known. The more recent work of Tataru [15], [16] and Tao [13] [14] relies and further develops deep ideas from harmonic analysis in Tao’s case in conjunction with gauge theoretic geometric methods and thus seems very promising. Keel and Tao studied the one (spatial) dimensional case in [4]. In [13], Tao established the global regularity for wave maps from R × R into the sphere S when n ≥ 5. Similar results to those of Tao were obtained by Klainerman and Rodniansky [7] for target manifolds that admit a bounded parallelizable structure. In this paper we are interested in revisiting this work. We study the Cauchy problem for wave maps from R × R into a (compact) Lie group (or Riemannian symmetric 1991 Mathematics Subject Classification. Primary 35J10, Secondary 45B15, 42B35.