On the global regularity of wave maps in the critical Sobolev norm

By S. Klainerman, Igor Rodnianski. 2000

80 citations recorded by the source corpus.

We extend the recent result of T.Tao (5) to wave maps defined from the Minkowski space R n+1 , n � 5, to a target manifold N which pos- sesses a "bounded parallelizable" structure. This is the case of Lie groups, homogeneous spaces as well as the hyperbolic spaces H N. General compact Riemannian manifolds can be imbedded as totally geodesic submanifolds in bounded parallelizable manifolds, see (1), and therefore are also covered, in principle, by our result. Compactness of the target manifold, which seemed to play an important role in (5), turns out however to play no role in our discus- sion. Our proof follows closely that of (5) and is based, in particular, on its remarkable microlocal gauge renormalization idea.

Subfields: Applied Mathematics, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/28149dfe8f0e5cc33cfa900f5f807f902a7e3e33

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In Applied Mathematics

1. D. Tataru
2. Carlos E. Kenig
3. A. Ionescu
4. Jacob Sterbenz
5. Terence Tao
6. J. Krieger
7. I. Bejenaru
8. Zihua Guo

In Mathematical Physics

1. D. Tataru
2. Carlos E. Kenig
3. A. Ionescu
4. Jacob Sterbenz
5. Terence Tao
6. J. Krieger
7. I. Bejenaru
8. Zihua Guo

In Nuclear and High Energy Physics

1. Carlos E. Kenig
2. Jacob Sterbenz
3. J. Krieger
4. Terence Tao
5. D. Tataru
6. A. Ionescu
7. S. Aretakis
8. András Vasy

Citations in the corpus

• Global Schrödinger maps in dimensions $d≥ 2$: Small data in the critical Sobolev spaces (2011, background): “See also [23], [25], [47], [48], [42], [24], [38], [35], [28], [29], and [49] for other local and global regularity (or well-posedness) theorems for wave maps.”
• GLOBAL WELL-POSEDNESS OF THE BENJAMIN–ONO EQUATION IN H1(R) (2003, background): “the gauge transformations for wave maps in [25], [26], [13], [28]).”
• Low-regularity global solutions to nonlinear dispersive equations (2002, methodology): “Klainerman and Rodnianski[ 22 ] have reinterpreted the above transform in the following appealing way.”
• Global Regularity of Wave Maps from R2+1 to H2. Small Energy (2004, background): “Proceeding as in [12, 18], one deduces the following divergence-curl system, provided u is a Wave Map:”
• Global Regularity of Wave Maps¶II. Small Energy in Two Dimensions (2000, background): “4 In dimensions n ≥ 5 this has recently been achieved [ 19 ] in the case when the target manifold is boundedly parallelizable.”
• On the well-posedness of the wave map problem in high dimensions (2001, background): “and Rodniansky [ 7 ] for target manifolds that admit a bounded parallelizable structure.”
• Rough solutions for the wave maps equation (2005, unknown): “This includes work of Klainerman-Rodnianskii [10] (n ≥ 5), Nahmod-Stefanov-Uhlenbeck [13] and Shatah-Struve [16] (n ≥ 4) and Krieger (n = 3, hyperbolic space).”
• Energy Dispersed Large Data Wave Maps in 2 + 1 Dimensions (2009, background): “There is also an extensive literature devoted to the more tractable higher dimensional case; we refer the reader to [31], [28], [8], [14], [17], and [20] for more information.”
• Concentration Compactness for Critical Wave Maps (2009, unknown): “Later results by Klainerman, Rodnianski [18], Nahmod, Stephanov, Uhlenbeck [34], Tataru [59], [58], and Krieger [22], [23], [24] considered other cases of targets by using similar methods as in Tao’s work.”
• The wave maps equation (2004, background): “Klainerman and Rodnianski [17] did this for n ≥ 5, Shatah-Struwe [27] and Nahmod-Stefanov-Uhlenbeck [23] for n ≥ 4 and Krieger [19] for the hyperbolic space and n = 3.”