Geodesic
On the formation of singularities in the critical $O(3)$ $\sigma$-model
Igor Rodnianski1 ; Jacob Sterbenz2
1 Department of Mathematics, Fine Hall, Washington Road, Princeton University, Princeton, NJ 08544-1000, >United States
2 University of California at San Diego, Department of Mathematics, 9500 Gilman Drive, Dept. 0112, La Jolla, 92093-0112, United States
Annals of mathematics, Tome 172 (2010) no. 1, pp. 187-242.

Voir la notice de l'article provenant de la source Annals of Mathematics website

We study the phenomena of energy concentration for the critical $O(3)$ sigma model, also known as the wave map flow from $\mathbb{R}^{2+1}$ Minkowski space into the sphere $\mathbb{S}^2$. We establish rigorously and constructively existence of a set of smooth initial data resulting in a dynamic finite time formation of singularities. The construction and analysis are done in the context of the $k$-equivariant symmetry reduction, and we restrict to maps with homotopy class $k\geqslant 4$. The concentration mechanism we uncover is essentially due to a resonant self-focusing (shrinking) of a corresponding harmonic map. We show that the phenomenon is generic (e.g. in certain Sobolev spaces) in that it persists under small perturbations of initial data, while the resulting blowup is bounded by a log-modified self-similar asymptotic.
DOI : 10.4007/annals.2010.172.187

Igor Rodnianski 1 ; Jacob Sterbenz 2

1 Department of Mathematics, Fine Hall, Washington Road, Princeton University, Princeton, NJ 08544-1000, >United States
2 University of California at San Diego, Department of Mathematics, 9500 Gilman Drive, Dept. 0112, La Jolla, 92093-0112, United States 
@article{10_4007_annals_2010_172_187,
     author = {Igor Rodnianski and Jacob Sterbenz},
     title = {On the formation of singularities in the critical $O(3)$ $\sigma$-model},
     journal = {Annals of mathematics},
     pages = {187--242},
     publisher = {mathdoc},
     volume = {172},
     number = {1},
     year = {2010},
     doi = {10.4007/annals.2010.172.187},
     mrnumber = {2680419},
     zbl = {1213.35392},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.172.187/}
}
TY  - JOUR
AU  - Igor Rodnianski
AU  - Jacob Sterbenz
TI  - On the formation of singularities in the critical $O(3)$ $\sigma$-model
JO  - Annals of mathematics
PY  - 2010
SP  - 187
EP  - 242
VL  - 172
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.172.187/
DO  - 10.4007/annals.2010.172.187
LA  - en
ID  - 10_4007_annals_2010_172_187
ER  - 
%0 Journal Article
%A Igor Rodnianski
%A Jacob Sterbenz
%T On the formation of singularities in the critical $O(3)$ $\sigma$-model
%J Annals of mathematics
%D 2010
%P 187-242
%V 172
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.172.187/
%R 10.4007/annals.2010.172.187
%G en
%F 10_4007_annals_2010_172_187
Igor Rodnianski; Jacob Sterbenz. On the formation of singularities in the critical $O(3)$ $\sigma$-model. Annals of mathematics, Tome 172 (2010) no. 1, pp. 187-242. doi : 10.4007/annals.2010.172.187. http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.172.187/

Cité par Sources :