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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.
Gonzalo Emiliano Ruiz Stolowicz
Fabrizio Carbone, Giovanni Maria Vanacore, Ivan Madan, Ido Kaminer, Simone Gargiulo, Ebrahim Karimi