Low regularity solutions for the wave map equation into the 2-D sphere

A class of weak wave map solutions with initial data in Sobolev space of order s<1 is studied. A non uniqueness result is proved for the case, when the target manifold is a two dimensional sphere. Using an equivariant wave map ansatz a family of self - similar solutions is constructed. This construction enables one to show ill - posedness of the inhomogeneous Cauchy problem for wave maps.Mathematics Subject Classification (2000): 35L70, 58J45.in final form: 17 January 2003