On the well-posedness of the wave map problem in high dimensions

By Andrea R. Nahmod, Atanas Stefanov, Karen K. Uhlenbeck. 2001

82 citations recorded by the source corpus.

We construct a gauge theoretic change of variables for the wave map from R × R into a compact group or Riemannian symmetric space, prove a new multiplication theorem for mixed Lebesgue-Besov spaces, and show the global well-posedness of a modified wave map equation n ≥ 4 for small critical initial data. We obtain global existence and uniqueness for the Cauchy problem of wave maps into compact Lie groups and symmetric spaces with small critical initial data and n ≥ 4. 0. Introduction The wave map equation between two Riemannian manifoldsthe wave equation version of the evolution equations which are derived from the same geometric considerations as the harmonic map equation between two Riemannian manifoldshas been studied by a number of mathematicians in the last decade. The work of Klainerman and Machedon and Klainerman and Selberg [5] [6] [8] studying the Cauchy problem for regular data is probably the best known. The more recent work of Tataru [15], [16] and Tao [13] [14] relies and further develops deep ideas from harmonic analysis in Tao’s case in conjunction with gauge theoretic geometric methods and thus seems very promising. Keel and Tao studied the one (spatial) dimensional case in [4]. In [13], Tao established the global regularity for wave maps from R × R into the sphere S when n ≥ 5. Similar results to those of Tao were obtained by Klainerman and Rodniansky [7] for target manifolds that admit a bounded parallelizable structure. In this paper we are interested in revisiting this work. We study the Cauchy problem for wave maps from R × R into a (compact) Lie group (or Riemannian symmetric 1991 Mathematics Subject Classification. Primary 35J10, Secondary 45B15, 42B35.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/38bc34600cc49c7b4985656e7fe123e6f0171388

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In Applied Mathematics

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

In Control and Systems Engineering

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

In Mathematical Physics

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

Citations in the corpus

• Global Regularity of Wave Maps from R2+1 to H2. Small Energy (2004, background): “Further developments include an alternative proof of (5) (in more restrictive formulation) by Nahmod-Stefanov-Uhlenbeck [ 20 ] 5 , as well as extension of (5) to the case n ≥ 3 by the author in [17‐19].”
• Global Schrödinger maps in dimensions $d≥ 2$: Small data in the critical Sobolev spaces (2011, background): “1 is closer to that of [38], [35], [28], and [29], in the sense that we prove a priori bounds on the derivatives of the Schrödinger map φ, in a suitable gauge, rather than the Schrödinger map itself.”
• Global regularity of wave maps III. Large energy from $\R^{1+2}$ to hyperbolic spaces (2008, background): “On the other hand, the local and perturbative theory for d = 2 has analogues for d > 2 (replacing the energy norm by the critical Sobolev norm Ḣ × Ḣ), and in fact the theory simplifies in this setting, particularly for d ≥ 4 or d ≥ 5: see [23], [24], [27], [40], [50], [62], [63], [73], [74].”
• 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.”
• Global existence and uniqueness of Schrödinger maps in dimensions d... 4 (2007, methodology): “3 In the context of wave maps, orthonormal frames have been used in [4,14,18,21] etc.”
• Rough solutions for the wave maps equation (2005, methodology): “This includes work of Klainerman-Rodnianskii [10] (n ≥ 5), Nahmod-Stefanov-Uhlenbeck [ 14 ] and Shatah-Struwe [17] (n ≥ 4) and Krieger [12] (hyperbolic space, n = 3; also the case n = 2 is considered in a very recent manuscript).”
• 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 .”
• 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.”