The aim of these notes is to provide a brief overview of a large body recent work whose aim was to prove the Threshold Theorem for energy critical geometric nonlinear wave equations. Within the class of geometric wave equations we include nonlinear wave evolutions which have a geometric structure and origin, including a nontrivial gauge group. The problems discussed here include Wave Maps, Maxwell–Klein–Gordon, as well as the hyperbolic Yang–Mills flow. In a nutshell, the Threshold theorem asserts that these problems are globally well-posed for initial data below the ground state energy.
@incollection{JEDP_2018____A10_0,
author = {Tataru, Daniel},
title = {The threshold theorem for geometric nonlinear wave equations},
booktitle = {},
series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
note = {talk:10},
pages = {1--15},
publisher = {Groupement de recherche 2434 du CNRS},
year = {2018},
doi = {10.5802/jedp.670},
language = {en},
url = {https://www.numdam.org/articles/10.5802/jedp.670/}
}
TY - JOUR AU - Tataru, Daniel TI - The threshold theorem for geometric nonlinear wave equations JO - Journées équations aux dérivées partielles N1 - talk:10 PY - 2018 SP - 1 EP - 15 PB - Groupement de recherche 2434 du CNRS UR - https://www.numdam.org/articles/10.5802/jedp.670/ DO - 10.5802/jedp.670 LA - en ID - JEDP_2018____A10_0 ER -
%0 Journal Article %A Tataru, Daniel %T The threshold theorem for geometric nonlinear wave equations %J Journées équations aux dérivées partielles %Z talk:10 %D 2018 %P 1-15 %I Groupement de recherche 2434 du CNRS %U https://www.numdam.org/articles/10.5802/jedp.670/ %R 10.5802/jedp.670 %G en %F JEDP_2018____A10_0
Tataru, Daniel. The threshold theorem for geometric nonlinear wave equations. Journées équations aux dérivées partielles (2018), Exposé no. 10, 15 p. doi : 10.5802/jedp.670. https://www.numdam.org/articles/10.5802/jedp.670/
[1] Global solutions of nonlinear Schrödinger equations, Colloquium Publications, 46, American Mathematical Society, 1999 | Zbl
[2] On the Division Problem for the Wave Maps Equation (2018) (https://arxiv.org/abs/1807.02066)
[3] A conformally invariant gap theorem in Yang–Mills theory (2017) (https://arxiv.org/abs/1708.01157)
[4] Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math., Volume 201 (2008) no. 2, pp. 147-212 | Zbl
[5] Space-time estimates for null forms and the local existence theorem, Commun. Pure Appl. Math., Volume 46 (1993) no. 9, pp. 1221-1268 | Zbl
[6] Dispersive estimates for principally normal pseudodifferential operators, Commun. Pure Appl. Math., Volume 58 (2005) no. 2, pp. 217-284 | Zbl
[7] Global regularity of wave maps from
[8] Concentration compactness for the critical Maxwell–Klein–Gordon equation, Ann. PDE, Volume 1 (2015) no. 1, 5, 208 pages | Zbl
[9] Concentration compactness for critical wave maps, EMS Monographs in Mathematics, European Mathematical Society, 2012 | Zbl
[10] Invent. Math., 171 (2008) no. 3, pp. 543-615 | Zbl
[11] Renormalization and blow up for the critical Yang–Mills problem, Adv. Math., Volume 221 (2009) no. 5, pp. 1445-1521 | Zbl
[12] Global regularity for the Yang–Mills equations on high dimensional Minkowski space, Mem. Am. Math. Soc., Volume 223 (2013) no. 1047 | Zbl
[13] Global well-posedness for the Maxwell–Klein–Gordon equation in
[14] Global well-posedness for the Yang–Mills equation in
[15] Finite energy global well-posedness of the Yang–Mills equations on
[16] Global well-posedness and scattering of the
[17] Local well-posedness of the
[18] The hyperbolic Yang–Mills equation for connections in an arbitrary topological class (2017) (https://arxiv.org/abs/1709.08604)
[19] The hyperbolic Yang–Mills equation in the caloric gauge. Local well-posedness and control of energy dispersed solutions (2017) (https://arxiv.org/abs/1709.09332)
[20] The threshold conjecture for the energy critical hyperbolic Yang–Mills equation (2017) (https://arxiv.org/abs/1709.08606)
[21] The Yang–Mills heat flow and the caloric gauge (2017) (https://arxiv.org/abs/1709.08599)
[22] Energy dispersed solutions for the (4+1)-dimensional Maxwell–Klein–Gordon equation, Am. J. Math., Volume 140 (2018) no. 1, pp. 1-82 | Zbl
[23] Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang–Mills problems, Publ. Math., Inst. Hautes Étud. Sci.
[24] On the formation of singularities in the critical
[25] Global regularity for the Maxwell–Klein–Gordon equation with small critical Sobolev norm in high dimensions, Commun. Math. Phys., Volume 251 (2004) no. 2, pp. 377-426
[26] Energy dispersed large data wave maps in
[27] Regularity of wave-maps in dimension
[28] Global regularity of wave maps. II. Small energy in two dimensions, Commun. Math. Phys., Volume 224 (2001) no. 2, pp. 443-544
[29] Geometric renormalization of large energy wave maps, Journ. Équ. Dériv. Partielles, Volume 2004 (2004), XI, 32 pages | DOI | Zbl
[30] Global regularity of wave maps VII. Control of delocalised or dispersed solutions (2009) (https://arxiv.org/abs/0908.0776)
[31] On global existence and scattering for the wave maps equation, Am. J. Math., Volume 123 (2001) no. 1, pp. 37-77
[32] Rough solutions for the wave maps equation, Am. J. Math., Volume 127 (2005) no. 2, pp. 293-377
Cité par Sources :
