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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 \(n = 2,3\).

MSC:
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
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References:
[1] DOI: 10.1007/BF01896020 · Zbl 0787.35097
[2] DOI: 10.1007/BF01215290 · Zbl 0321.35052
[3] Cazenave T., to appear, Ann. IHP, Physique Theorique
[4] DOI: 10.1002/cpa.3160460705 · Zbl 0744.58071
[5] DOI: 10.1006/jfan.1995.1119 · Zbl 0849.35064
[6] DOI: 10.1215/S0012-7094-93-07101-3 · Zbl 0787.35090
[7] Klainerman S., Appl. Math. 23 pp 293– (1986)
[8] DOI: 10.1002/cpa.3160460902 · Zbl 0803.35095
[9] DOI: 10.1215/S0012-7094-97-08718-4 · Zbl 0878.35075
[10] DOI: 10.1155/S1073792896000529 · Zbl 0909.35095
[11] DOI: 10.1080/03605309708821288 · Zbl 0884.35102
[12] Klainerman S. Tataru D. On the optimal local regularity for Yang-Mills equations zn R4+1 · Zbl 0924.58010
[13] DOI: 10.1002/cpa.3160410405 · Zbl 0686.35081
[14] DOI: 10.1080/03605309608821210 · Zbl 0853.35017
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