On global existence and scattering for the wave maps equation

By D. Tataru. 2001

187 citations recorded by the source corpus.

Subfields: Control and Systems Engineering, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/ca325febf4f11085d027d80298bb6e19d535ed21

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In Control and Systems Engineering

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Concentration Compactness for Critical Wave Maps (2009, methodology): “Further, they thank Carlos Kenig, Daniel Tataru, and Jacob Sterbenz for helpful discussions.”
The wave maps equation (2004, methodology): “The spaces used by Tao [26] and also later in [27] are variations of the spaces introduced in [29].”
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).”
Energy Dispersed Large Data Wave Maps in 2 + 1 Dimensions (2009, methodology): “The N space we use is the same as in [32],[29].”
Bilinear space-time estimates for homogeneous wave equations (2000, background): “Such estimates have recently appeared also in [21].”
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 regularity of wave maps, I: small critical Sobolev norm in high dimension (2000, methodology): “Fortunately, the low dimensional case has been effectively handled (in the Besov space case) by the more sophisticated arguments of [30], using the additional ingredients of angular frequency decomposition and X-type spaces (to more effectively exploit the null form structure in (1)) and null frames”
Low-regularity global solutions to nonlinear dispersive equations (2002, methodology): “Wellposedness and regularity was demonstrated in the critical Besov space Ḃ 1 in [35] in the high-dimensional case n ≥ 4 and in [36] for n = 2, 3; in the one-dimensional case n = 1, a logarithmic cascade from high frequencies to low frequencies causes ill-posedness in the critical Besov and Sobolev ”
Global Regularity of Wave Maps from R2+1 to H2. Small Energy (2004, methodology): “Finally, the S[k, κ] satisfy a crucial orthogonality property, see [33, 30] or also [18]: Let ψκ be a Schwartz function whose Fourier support is contained in an angular sector κ of radius r .”
Global Regularity of Wave Maps¶II. Small Energy in Two Dimensions (2000, background): “(38) 14 We have decided to organize this paper as a tree as opposed to the more usual linear structure.”
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.”
Global regularity of wave maps IV. Absence of stationary or self-similar solutions in the energy class (2008, background): “It will also rely heavily on (slight extensions of) the delicate spacetime function spaces and estimates from [22], [14].”
Global Schrödinger maps in dimensions $d≥ 2$: Small data in the critical Sobolev spaces (2011, methodology): “The idea of using sums of spaces such as L2,∞e,Wk+40 as substitutes for missing estimates in the setting of wave maps is due to Tataru [48].”
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].”