Well-posedness Issues For the Half-Wave Maps Equation With Hyperbolic And Spherical Targets

This thesis is a study of the global well-posedness of the Cauchy problems for half-wave maps from the Minkowski space of dimension n+1 to the 2-dimensional sphere and the hyperbolic plane. The work is mainly based on the results from Krieger-Sire 17' in the energy-supercritical case of n>=5, and the improved result from Kiesenhofer-Krieger 19' of n>=4 for sphere target with the small initial Besov normed data. The first result obtained by the authors is to extend the well-posedness of the sphere target to the hyperbolic plane with small initial Besov normed data in higher dimension n>=4. The work utilizes the intrinsic distance of the hyperbolic plane to maintain the geometric structure of the half-wave map. For future works, the authors would improve the initial data condition from the Besov space to the critical Sobolev space in higher dimension n>=4 for both the spherical and hyperbolic targets. The authors would reference the Hélein's moving frame techniques and the gauge construction for wave maps as in Tao 01' and Shatah-Struwe 02' to address the problem. Moreover, the authors would construct the weaker solution for the half-wave maps in the lower dimensional case when n=1,2. The lower dimension case requires the authors to build new tools since the Strichartz estimate used in the higher dimension case no longer available.