Global regularity of wave maps, I: small critical Sobolev norm in high dimension

By Terence Tao. 2000

160 citations recorded by the source corpus.

We show that wave maps from Minkowski space $R^{1+n}$ to a sphere are globally smooth if the initial data is smooth and has small norm in the critical Sobolev space $\dot H^{n/2}$ in the high dimensional case $n \geq 5$. A major difficulty, not present in the earlier results, is that the $\dot H^{n/2}$ norm barely fails to control $L^\infty$, potentially causing a logarithmic divergence in the nonlinearity; however, this can be overcome by using co-ordinate frames adapted to the wave map by approximate parallel transport. In the sequel of this paper we address the more interesting two-dimensional case, which is energy-critical.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ade3d12646869b9fd864693020b6ff3201a5d504

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Scattering for the radial 3D cubic wave equation (2014, background): “Definition 3 [Tao 2001, Definition 1].”
Low-regularity Schrödinger maps, II: global well-posedness in dimensions d ≥ 3 (2007, background): “Its direct analogue in the setting of wave maps is the work of Tataru [23] (see also [10, 12, 24, 21 , 22, 11, 18], and [25] for other local and global well-posedness theorems for wave maps).”
Global well-posedness of the Benjamin–Ono equation in low-regularity spaces (2005, methodology): “These types of spaces have been used in the context of wave maps (see, for example, [11], [23], [24], [19], and [20]); we remark that for the physical space component we use a suitable normalization of the local smoothing space L 1 x L 2 t instead of the energy space L 1 t L 2 x .”
GLOBAL WELL-POSEDNESS OF THE BENJAMIN–ONO EQUATION IN H1(R) (2003, methodology): “To obtain all of these additional properties of the solution map S(t), we will revisit the a priori bound in Theorem 3.1, but this time we will introduce the machinery of frequency envelopes, as used recently in wave maps in [ 25 ], [26], [28].”
On the well-posedness of the wave map problem in high dimensions (2001, methodology): “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.”
On Schrödinger maps (2008, background): “The major contributions were brought by Tataru, see [18] and [19], and Tao, see [16] and [17].”
Asymptotic smoothing and attractors for the generalized Benjamin-Bona-Mahony equation on R 3 (2005, background): “the form [P ; ]g := P ( g) P g. We shall need the following result, essentially due to Tao [ 44 ].”
Global existence and uniqueness of Schrödinger maps in dimensions d... 4 (2007, background): “We would like to thank S. Klainerman, I. Rodnianski, J. Shatah, and T. Tao for several useful discussions.”
Low-regularity global solutions to nonlinear dispersive equations (2002, background): “See [23], [16], [26] for further discussion. In [ 33 ], [34] it was shown”
The wave maps equation (2004, methodology): “The rst step in this direction was carried out by Tao, initially in high dimension in [ 30 ] and then in low dimension [31].”
Global Regularity of Wave Maps¶II. Small Energy in Two Dimensions (2000, methodology): “Indeed, our paper shall largely be a (self-contained) combination of [42] and the prequel [ 40 ] to this paper, although there are some additional technical issues arising here which do not occur in the two papers just mentioned.”
Global regularity of wave maps III. Large energy from $\R^{1+2}$ to hyperbolic spaces (2008, background): “> 2 (replacing the energy norm by the critical Sobolev norm H˙ d/2 × H˙ d/2−1), 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]. 1.7. Claims. The purpose of this paper is to establish Conjecture 1.4, condi”
Concentration Compactness for Critical Wave Maps (2009, methodology): “As already in [23], we will partially rely on Tao’s trilinear estimate from [59] which states that (relative to our norms in the S[k]-spaces)”
Rough solutions for the wave maps equation (2005, unknown): “The next main step was achieved by Tao in [18] (n ≥ 5) and [19] (n ≥ 2).”
Uniqueness (2021, unknown): “those in the celebrated articles [23, 21, 22], to half-wave maps.”
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.”