Global Regularity of Wave Maps¶II. Small Energy in Two Dimensions

By Terence Tao. 2000

251 citations recorded by the source corpus.

Abstract: We show that wave maps from Minkowski space ℝ1+n to a sphere Sm−1 are globally smooth if the initial data is smooth and has small norm in the critical Sobolev space , in all dimensions n≥ 5. This generalizes the results in the prequel [40] of this paper, which addressed the high-dimensional case n≥ 5. In particular, in two dimensions we have global regularity whenever the energy is small, and global regularity for large data is thus reduced to demonstrating non-concentration of energy.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e27b743098bdbbda54504bddc8381a0aefd8841c

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In Applied Mathematics

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Citations in the corpus

Concentration Compactness for Critical Wave Maps (2009, methodology): “Basic L2-bounds To begin with, we present the following geometric lemma for cones, see [58] for a similar result.”
Global regularity of wave maps III. Large energy from $\R^{1+2}$ to hyperbolic spaces (2008, background): “Thus, by passing to a subsequence, we may assume that φn[0] converges in Ḣ1 to a limit Φ∞, which has a maximal Cauchy development φ∞ on an open neighbourhood of 0.”
On the well-posedness of the wave map problem in high dimensions (2001, unknown): “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.”
Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems (2009, background): “Very recently, in a series of works [38],[39],[34],[35],[17], this result has been remarkably extended to the full (WM) problem without the assumpion of corotational symmetry, hence completing the program developed in [14],[13],[40],[37],[16].”
The wave maps equation (2004, background): “The first step in this direction was carried out by Tao, initially in high dimension in [25] and then in low dimension [26].”
Renormalization and blow up for charge one equivariant critical wave maps (2006, background): “We note that a fairly satisfactory understanding has been achieved for small-energy wave maps from R 2+1 to general targets, see Tao [ 29 ], Tataru [30–32], and Krieger [11], as well as for rotationally invariant wave maps and general initial data by Christodoulou, Tahvildar-Zadeh [5], and Struwe [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, methodology): “The N space we use is the same as in [32],[29].”
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 existence and scattering for rough solutions of a nonlinear Schrödinger equation on ℝ3 (2003, background): “(See the remark in [3], page 276, and (2.26), (4.20) below.) There are of course more significant examples where low-regularity techniques have helped to solve open problems for smooth solutions, e.g. [2, 32 ].”
Global existence and uniqueness of Schrödinger maps in dimensions d... 4 (2007, background): “Its direct analogue in the setting of wave maps is the work of Tao [25] (see also [14], [16], [27], [28], [26], [15], [22], [18], and [29] for other local and global existence (or 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 .”
Regularity of Wave-Maps in Dimension 2 + 1 (2009, methodology): “Tao, T.: Global regulrity of wave maps VI. Minimal energy blowup solutions. http://arxiv.org/abs/0906.”
On the formation of singularities in the critical $O(3)$ $\sigma$-model (2006, background): “oblem, the only general understanding of the equations (3) that is yet available is for the local theory (see [18]) and the small data scale invariant (global) results of [37] (in the Besov case) and [36], [19], and [38] for the case of small energy. Large data global regularity has been conjectured”