Local and global results for wave maps I

Daniel Tataru
1998 Communications in Partial Differential Equations  
We consider the initial value problem for wave-maps corresponding to constant coefficient second order hyperbolic equations in n + 1 dimensions, n ≥ 4. We prove that this problem is globally well-posed for initial data which is small in the homogeneous Besov spaceḂ 2,1 n/2 ×Ḃ 2,1 n/2−1 . Our second result deals with more regular solutions; it essentially says that if in addition the initial data is in H s × H s−1 , s > n/2, then the solutions stay bounded in the same space. In part II of this
more » ... n part II of this work we shall prove that the same result holds in dimensions n = 2, 3.
doi:10.1080/03605309808821400 fatcat:4n3i737a55by7hstac32vdi7fm