An Lp-theory for almost sure local well-posedness of the nonlinear Schrödinger equations

Oana Pocovnicu; Yuzhao Wang

doi:10.1016/j.crma.2018.04.009

Partial differential equations

An L^p-theory for almost sure local well-posedness of the nonlinear Schrödinger equations

Presented by: Haïm Brézis

Oana Pocovnicu ¹; Yuzhao Wang ^2,³

¹ Department of Mathematics, Heriot-Watt University and The Maxwell Institute for the Mathematical Sciences, Edinburgh, EH14 4AS, United Kingdom
² School of Mathematics, University of Birmingham, Watson Building, Edgbaston, Birmingham B15 2TT, United Kingdom
³ School of Mathematics, The University of Edinburgh and The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King's Buildings, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom

Comptes Rendus. Mathématique, Volume 356 (2018) no. 6, pp. 637-643.

Abstract
Résumé

We consider the nonlinear Schrödinger equations (NLS) on $R^{d}$ with random and rough initial data. By working in the framework of $L^{p} (R^{d})$ spaces, $p > 2$ , we prove almost sure local well-posedness for rougher initial data than those considered in the existing literature. The main ingredient of the proof is the dispersive estimate.

Dans cet article, nous considérons l'équation de Schrödinger non linéaire (NLS) sur $R^{d}$ à données initiales aléatoires et surcritiques. En travaillant dans des espaces de $L^{p} (R^{d})$ , $p > 2$ , nous améliorons les résultats précédents de la littérature, en ce sens que nous prouvons que l'équation NLS est localement bien posée presque sûrement pour des données initiales à régularité plus basse. L'ingrédient principal de la preuve est l'estimation dispersive.

Received: 2018-01-21
Accepted: 2018-04-12
Published online: 2018-04-25

DOI: 10.1016/j.crma.2018.04.009

Author's affiliations:

Oana Pocovnicu ¹; Yuzhao Wang ^{2, 3}

¹ Department of Mathematics, Heriot-Watt University and The Maxwell Institute for the Mathematical Sciences, Edinburgh, EH14 4AS, United Kingdom
² School of Mathematics, University of Birmingham, Watson Building, Edgbaston, Birmingham B15 2TT, United Kingdom
³ School of Mathematics, The University of Edinburgh and The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King's Buildings, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom

@article{CRMATH_2018__356_6_637_0,
     author = {Oana Pocovnicu and Yuzhao Wang},
     title = {An {\protect\emph{L}\protect\textsuperscript{\protect\emph{p}}-theory} for almost sure local well-posedness of the nonlinear {Schr\"odinger} equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {637--643},
     publisher = {Elsevier},
     volume = {356},
     number = {6},
     year = {2018},
     doi = {10.1016/j.crma.2018.04.009},
     language = {en},
}

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Oana Pocovnicu; Yuzhao Wang. An L^p-theory for almost sure local well-posedness of the nonlinear Schrödinger equations. Comptes Rendus. Mathématique, Volume 356 (2018) no. 6, pp. 637-643. doi : 10.1016/j.crma.2018.04.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.04.009/

References
Cited by

[1] Á. Bényi; T. Oh; O. Pocovnicu Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS, Excursions in Harmonic Analysis, vol. 4, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, New York, 2015, pp. 3-25

[2] Á. Bényi; T. Oh; O. Pocovnicu On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on $R^{d}$ , $d \geq 3$ , Trans. Amer. Math. Soc. Ser. B, Volume 2 (2015), pp. 1-50

[3] Á. Bényi; T. Oh; O. Pocovnicu Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schrödinger equation on $R^{3}$ | arXiv

[4] J. Brereton Almost sure local well-posedness for the supercritical quintic NLS | arXiv

[5] N. Burq; N. Tzvetkov Random data Cauchy theory for supercritical wave equations. I. Local theory, Invent. Math., Volume 173 (2008) no. 3, pp. 449-475

[6] N. Burq; N. Tzvetkov Probabilistic well-posedness for the cubic wave equation, J. Eur. Math. Soc., Volume 16 (2014) no. 1, pp. 1-30

[7] T. Cazenave; F. Weissler Some remarks on the nonlinear Schrödinger equation in the critical case, Washington, DC, 1987 (Lect. Notes Math.), Volume vol. 1394, Springer, Berlin (1989), pp. 18-29

[8] M. Christ; J. Colliander; T. Tao Ill-posedness for nonlinear Schrödinger and wave equations | arXiv

[9] J. Ginibre; G. Velo Smoothing properties and retarded estimates for some dispersive evolution equations, Commun. Math. Phys., Volume 144 (1992) no. 1, pp. 163-188

[10] L. Grafakos; S. Oh The Kato–Ponce inequality, Commun. Partial Differ. Equ., Volume 39 (2014) no. 6, pp. 1128-1157

[11] H. Hirayama; M. Okamoto Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity, Discrete Contin. Dyn. Syst., Volume 36 (2016) no. 12, pp. 6943-6974

[12] J. Lührmann; D. Mendelson Random data Cauchy theory for nonlinear wave equations of power-type on $R^{3}$ , Commun. Partial Differ. Equ., Volume 39 (2014) no. 12, pp. 2262-2283

[13] T. Oh A remark on norm inflation with general initial data for the cubic nonlinear Schrödinger equations in negative Sobolev spaces, Funkc. Ekvacioj, Volume 60 (2017), pp. 259-277

[14] T. Oh, O. Pocovnicu, Y. Wang, On the stochastic nonlinear Schrödinger equations with non-smooth additive noise, preprint.

[15] M. Okamoto; T. Oh; O. Pocovnicu On the probabilistic well-posedness of the nonlinear Schrödinger equation with non-algebraic nonlinearities | arXiv

[16] O. Pocovnicu Probabilistic global well-posedness of the energy-critical defocusing cubic nonlinear wave equations on $R^{4}$ , J. Eur. Math. Soc., Volume 19 (2017) no. 8, pp. 2321-2375

[17] Y. Tsutsumi $L^{2}$ -solutions for nonlinear Schrödinger equations and nonlinear groups, Funkc. Ekvacioj, Volume 30 (1987) no. 1, pp. 115-125

[18] M. Vişan The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions, Duke Math. J., Volume 138 (2007) no. 2, pp. 281-374

[19] T. Zhang; D. Fang Random data Cauchy theory for the generalized incompressible Navier–Stokes equations, J. Math. Fluid Mech., Volume 14 (2012) no. 2, pp. 311-324

[20] Y. Zhou Cauchy problem of nonlinear Schrödinger equation with initial data in Sobolev space $W^{s, p}$ for $p < 2$ , Trans. Amer. Math. Soc., Volume 362 (2010) no. 9, pp. 4683-4694

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