Local and global results for wave maps. I. (English) Zbl 0914.35083

Summary: We consider the initial value problem for wave-maps corresponding to constant coefficient second order hyperbolic equations in \(n + 1\) dimensions, \(n \geq 4\). We prove that this problem is globally well-posed for initial data which is small in the homogeneous Besov space \(\dot B^{2,1}_{n/2} \times\dot B^{2,1}_{n/2-1}\).Our second result deals with more regular solutions; it essentially says that if in addition the initial data is in \(H^s \times H^{s-1}\), \(s> n/2\), then the solutions stay bounded in the same space. In part II of this work we shall prove that the same result holds in dimensions \(n = 2,3\).


35L70 Second-order nonlinear hyperbolic equations
58J45 Hyperbolic equations on manifolds
35L15 Initial value problems for second-order hyperbolic equations
35B65 Smoothness and regularity of solutions to PDEs
Full Text: DOI


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