Randomization improved Strichartz estimates and global well-posedness for supercritical data
Annales de l'Institut Fourier, Volume 71 (2021) no. 5, pp. 1929-1961.

We introduce a novel data randomisation for the free wave equation which leads to the same range of Strichartz estimates as for radial data, albeit in a non-radial context. We then use these estimates to establish global well-posedness for a wave maps type nonlinear wave equation for certain supercritical data, provided the data are suitably small and randomised.

On introduit une nouvelle randomisation des données initiales pour l’équation des ondes, telle que les solutions satisfont les mêmes estimations de Strichartz que dans le cadre radial, au dépit de leur caractère non-radial. Nous utilisons ces estimations pour montrer que certaines équations modèles similaires aux applications d’ondes sont bien-posées pour des données initiales surcritiques.

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DOI: 10.5802/aif.3448
Classification: 35L05,  35B40
Keywords: wave equation, Strichartz estimates, randomised data
Burq, Nicolas 1; Krieger, Joachim 2

1 Laboratoire de Mathématiques d’Orsay Université Paris-Saclay Bat. 307, 91405 Orsay Cedex (France) and Institut Universitaire de France
2 Bâtiment des Mathématiques, EPFL Station 8, CH-1015 Lausanne, CH-1015 Lausanne (Switzerland)
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Burq, Nicolas; Krieger, Joachim. Randomization improved Strichartz estimates and global well-posedness for supercritical data. Annales de l'Institut Fourier, Volume 71 (2021) no. 5, pp. 1929-1961. doi : 10.5802/aif.3448. https://aif.centre-mersenne.org/articles/10.5802/aif.3448/

[1] Bringmann, Bjoern Almost sure local well-posedness for a derivative nonlinear wave equation (2018) (https://arxiv.org/abs/1809.00220)

[2] Bringmann, Bjoern Almost sure scattering for the energy critical nonlinear wave equation (2018) (https://arxiv.org/abs/1812.10187)

[3] Burq, Nicolas; Lebeau, Gilles Injections de Sobolev probabilistes et applications, Ann. Sci. Éc. Norm. Supér., Volume 46 (2013) no. 6, pp. 917-962 | Article | Numdam | MR: 3134684 | Zbl: 1296.46031

[4] Burq, Nicolas; Lebeau, Gilles Probabilistic Sobolev embeddings, applications to eigenfunctions estimates, Geometric and spectral analysis (Contemporary Mathematics), Volume 630, American Mathematical Society, 2014, pp. 307-318 | Article | MR: 3328548 | Zbl: 1357.46024

[5] Burq, Nicolas; Tzvetkov, Nikolay Random data Cauchy theory for supercritical wave equations. I. Local theory, Invent. Math., Volume 173 (2008) no. 3, pp. 449-475 | Article | MR: 2425133 | Zbl: 1156.35062

[6] Chanillo, Sagun; Czubak, Magdalena; Mendelson, Dana; Nahmod, Andrea; Staffilani, Gigliola Almost sure boundedness of iterates for derivative nonlinear wave equations (2017) (https://arxiv.org/abs/1710.09346)

[7] Klainerman, Sergiu; Machedon, Matei Space-time estimates for null forms and the local existence theorem, Commun. Pure Appl. Math., Volume 46 (1993) no. 9, pp. 1221-1268 | Article | MR: 1231427

[8] Klainerman, Sergiu; Tataru, Daniel On the optimal local regularity for Yang–Mills equations in 4+1 , J. Am. Math. Soc., Volume 12 (1999) no. 1, pp. 93-116 | Article | MR: 1626261 | Zbl: 0924.58010

[9] Krieger, Joachim; Schlag, Wilhelm Concentration compactness for critical wave maps, EMS Monographs in Mathematics, European Mathematical Society, 2012, vi+484 pages | Article | MR: 2895939

[10] Lührmann, Jonas; Mendelson, Dana On the almost sure global well-posedness of energy sub-critical nonlinear wave equations on 3 , New York J. Math., Volume 22 (2016), pp. 209-227 | MR: 3484682 | Zbl: 1408.35099

[11] Murphy, Jason Random data final-state problem for the mass-subcritical NLS in L 2 , Proc. Am. Math. Soc., Volume 147 (2019) no. 1, pp. 339-350 | Article | MR: 3876753 | Zbl: 1406.35366

[12] Poiret, Aurélien; Robert, Didier; Thomann, Laurent Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator, Anal. PDE, Volume 7 (2014) no. 4, pp. 997-1026 | Article | MR: 3254351 | Zbl: 1322.35190

[13] Stein, Elias M.; Weiss, Guido Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, 32, Princeton University Press, 1971, x+297 pages | MR: 0304972

[14] Sterbenz, Jacob Angular regularity and Strichartz estimates for the wave equation, Int. Math. Res. Not. (2005) no. 4, pp. 187-231 (with an appendix by Igor Rodnianski) | Article | MR: 2128434 | Zbl: 1072.35048

[15] Tao, Terence Global regularity of wave maps. II. Small energy in two dimensions, Commun. Math. Phys., Volume 224 (2001) no. 2, pp. 443-544 | Article | MR: 1869874 | Zbl: 1020.35046

[16] Tataru, Daniel On global existence and scattering for the wave maps equation, Am. J. Math., Volume 123 (2001) no. 1, pp. 37-77 | Article | MR: 1827277 | Zbl: 0979.35100

[17] Tzvetkov, Nikolay Construction of a Gibbs measure associated to the periodic Benjamin-Ono equation, Probab. Theory Relat. Fields, Volume 146 (2010) no. 3-4, pp. 481-514 | Article | MR: 2574736 | Zbl: 1188.35183

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