Stable soliton resolution for equivariant wave maps exterior to a ball
Séminaire Laurent Schwartz — EDP et applications (2014-2015), Exposé no. 3, 11 p.

In this report we review the proof of the stable soliton resolution conjecture for equivariant wave maps exterior to a ball in ℝ 3 and taking values in the 3-sphere. This is joint work with Carlos Kenig, Baoping Liu, and Wilhelm Schlag.

Publié le :
DOI : https://doi.org/10.5802/slsedp.66 
@article{SLSEDP_2014-2015____A3_0,
     author = {Andrew Lawrie},
     title = {Stable soliton resolution for equivariant wave maps exterior to a ball},
     journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
     note = {talk:3},
     publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2014-2015},
     doi = {10.5802/slsedp.66},
     language = {en},
     url = {https://slsedp.centre-mersenne.org/articles/10.5802/slsedp.66/}
}
Lawrie, Andrew. Stable soliton resolution for equivariant wave maps exterior to a ball. Séminaire Laurent Schwartz — EDP et applications (2014-2015), Exposé no. 3, 11 p. doi : 10.5802/slsedp.66. https://slsedp.centre-mersenne.org/articles/10.5802/slsedp.66/

[1] H. Bahouri and P. Gérard. High frequency approximation of solutions to critical nonlinear wave equations. Amer. J. Math., 121:131–175, 1999.

[2] B. Balakrishna, V. Schechter Sanyuk, J., and A. Subbaraman. Cutoff quantization and the skyrmion. Physical Review D, 45(1):344–351, 1992.

[3] P. Bizoń, T. Chmaj, and M. Maliborski. Equivariant wave maps exterior to a ball. Nonlinearity, 25(5):1299–1309, 2012.

[4] T. Duyckaerts, C. Kenig, and F. Merle. Universality of the blow-up profile for small radial type II blow-up solutions of the energy critical wave equation. J. Eur math. Soc. (JEMS), 13(3):533–599, 2011.

[5] T. Duyckaerts, C. Kenig, and F. Merle. Profiles of bounded radial solutions of the focusing, energy-critical wave equation. Geom. Funct. Anal., 22(3):639–698, 2012.

[6] T. Duyckaerts, C. Kenig, and F. Merle. Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case. J. Eur. Math. Soc. (JEMS), 14(5):1389–1454, 2012.

[7] T. Duyckaerts, C. Kenig, and F. Merle. Classification of radial solutions of the focusing, energy critical wave equation. Cambridge Journal of Mathematics, 1(1):75–144, 2013.

[8] T. Duyckaerts, C. Kenig, and F. Merle. Scattering for radial, bounded solutions of focusing supercritical wave equations. To appear in I.M.R.N, Preprint, 2012.

[9] K. Hidano, J. Metcalfe, H. Smith, C. Sogge, and Y. Zhou. On abstract Strichartz estimates and the Strauss conjecture for nontrapping obstacles. Trans. Amer. Math. Soc., 362(5):2789–2809, 2010.

[10] C. Kenig, A. Lawrie, B. Liu, and W. Schlag. Channels of energy for the linear radial wave equation. Preprint, 2014.

[11] C. Kenig, A. Lawrie, B. Liu, and W. Schlag. Stable soliton resolution for exterior wave maps in all equivariance classes. Preprint, 2014.

[12] C. Kenig, A. Lawrie, and W. Schlag. Relaxation of wave maps exterior to a ball to harmonic maps for all data. Geom. Funct. Anal., 24(2):610–647, 2014.

[13] C. Kenig and F. Merle. Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case. Invent. Math., 166(3):645–675, 2006.

[14] C. Kenig and F. Merle. Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation. Acta Math., 201(2):147–212, 2008.

[15] A. Lawrie and W. Schlag. Scattering for wave maps exterior to a ball. Advances in Mathematics, 232(1):57–97, 2013.

[16] J. Shatah. Weak solutions and development of singularities of the SU (2) σ-model. Comm. Pure Appl. Math., 41(4):459–469, 1988.

[17] J. Shatah and M. Struwe. Geometric wave equations. Courant Lecture notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York. American Mathematical Society, Providence RI, 1998.