Regularization by noise for some nonlinear dispersive PDEs
[Régularisation par le bruit pour certaines EDP dispersives]
Tristan Robert1
1 Université de Lorraine, CNRS, IECL, F-54000 Nancy, France
Journées équations aux dérivées partielles (2023), Exposé no. 9, 12 p.

In the context of ODEs or transport PDEs, there are examples where adding a rough stochastic perturbation to the equation at hand actually improves the well-posedness theory. In these notes, we review some results showing how a distributional modulation of the dispersion can also produce a regularization by noise effect for a rather large class of nonlinear dispersive PDEs.

Publié le :
DOI : 10.5802/jedp.680
Classification : 60H50, 35A01
Keywords: Nonlinear dispersive PDEs, Nonlinear Young integral
Tristan Robert 1

1 Université de Lorraine, CNRS, IECL, F-54000 Nancy, France 
@incollection{JEDP_2023____A9_0,
     author = {Tristan Robert},
     title = {Regularization by noise for some nonlinear dispersive {PDEs}},
     booktitle = {},
     series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     note = {talk:9},
     pages = {1--12},
     publisher = {R\'eseau th\'ematique AEDP du CNRS},
     year = {2023},
     doi = {10.5802/jedp.680},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/jedp.680/}
}
TY  - JOUR
AU  - Tristan Robert
TI  - Regularization by noise for some nonlinear dispersive PDEs
JO  - Journées équations aux dérivées partielles
N1  - talk:9
PY  - 2023
SP  - 1
EP  - 12
PB  - Réseau thématique AEDP du CNRS
UR  - https://proceedings.centre-mersenne.org/articles/10.5802/jedp.680/
DO  - 10.5802/jedp.680
LA  - en
ID  - JEDP_2023____A9_0
ER  - 
%0 Journal Article
%A Tristan Robert
%T Regularization by noise for some nonlinear dispersive PDEs
%J Journées équations aux dérivées partielles
%Z talk:9
%D 2023
%P 1-12
%I Réseau thématique AEDP du CNRS
%U https://proceedings.centre-mersenne.org/articles/10.5802/jedp.680/
%R 10.5802/jedp.680
%G en
%F JEDP_2023____A9_0
Tristan Robert. Regularization by noise for some nonlinear dispersive PDEs. Journées équations aux dérivées partielles (2023), Exposé no. 9, 12 p. doi : 10.5802/jedp.680. https://proceedings.centre-mersenne.org/articles/10.5802/jedp.680/

[1] Anne de Bouard; Arnaud Debussche The nonlinear Schrödinger equation with white noise dispersion, J. Funct. Anal., Volume 259 (2010) no. 5, pp. 1300-1321 | DOI | MR | Zbl

[2] Bjoern Bringmann; Yu Deng; Andrea R. Nahmod; Haitian Yue Invariant Gibbs measures for the three dimensional cubic nonlinear wave equation (2022) | arXiv

[3] Enguerrand Brun; Guopeng Li; Ruoyuan Liu; Younes Zine Global well-posedness of one-dimensional cubic fractional nonlinear Schrödinger equations in negative Sobolev spaces (2023) | arXiv

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

[5] Rémi Catellier; Massimiliano Gubinelli Averaging along irregular curves and regularisation of ODEs, Stochastic Processes Appl., Volume 126 (2016) no. 8, pp. 2323-2366 | DOI | MR | Zbl

[6] Thierry Cazenave Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, Courant Institute; American Mathematical Society, 2003 no. 10, xiv+323 pages | MR

[7] Khalil Chouk; Massimiliano Gubinelli Nonlinear PDEs with modulated dispersion II: Korteweg–de Vries equation (2014) | arXiv

[8] Khalil Chouk; Massimiliano Gubinelli Nonlinear PDEs with modulated dispersion I: Nonlinear Schrödinger equations, Commun. Partial Differ. Equations, Volume 40 (2015) no. 11, pp. 2047-2081 | DOI | MR | Zbl

[9] Alexander M. Davie Uniqueness of solutions of stochastic differential equations, Int. Math. Res. Not., Volume 2007 (2007) no. 24, rnm124, 26 pages | MR | Zbl

[10] Arnaud Debussche; Yoshio Tsutsumi 1D quintic nonlinear Schrödinger equation with white noise dispersion, J. Math. Pures Appl., Volume 96 (2011) no. 4, pp. 363-376 | DOI | MR | Zbl

[11] Yu Deng; Andrea R. Nahmod; Haitian Yue Random tensors, propagation of randomness, and nonlinear dispersive equations, Invent. Math., Volume 228 (2022) no. 2, pp. 539-686 | DOI | MR | Zbl

[12] Romain Duboscq; Anthony Réveillac On a stochastic Hardy–Littlewood–Sobolev inequality with application to Strichartz estimates for a noisy dispersion, Ann. Henri Lebesgue, Volume 5 (2022), pp. 263-274 | DOI | Numdam | MR | Zbl

[13] Franco Flandoli; Massimiliano Gubinelli; Enrico Priola Well-posedness of the transport equation by stochastic perturbation, Invent. Math., Volume 180 (2010) no. 1, pp. 1-53 | DOI | MR

[14] Lucio Galeati Pathwise methods in regularisation by noise, Ph. D. Thesis, Rheinische Friedrich-Wilhelms-Universität Bonn (2022) https://hdl.handle.net/20.500.11811/10089

[15] Donald Geman; Joseph Horowitz Occupation densities, Ann. Probab., Volume 8 (1980) no. 1, pp. 1-67 | MR | Zbl

[16] Martin Hairer A theory of regularity structures, Invent. Math., Volume 198 (2014) no. 2, pp. 269-504 | DOI | MR | Zbl

[17] Nicolai V. Krylov; Michael Roeckner Strong solutions of stochastic equations with singular time dependent drift, Probab. Theory Relat. Fields, Volume 131 (2005) no. 2, pp. 154-196 | DOI | MR | Zbl

[18] Felipe Linares; Gustavo Ponce Introduction to nonlinear dispersive equations. Second edition, Universitext, Springer, 2015, xiv+301 pages | DOI | MR

[19] T. Robert Noiseless regularization by noise for some modulated dispersive PDEs (In preparation)

[20] Marco Romito; Leonardo Tolomeo Yet another notion of irregularity through small ball estimates (2022) | arXiv

[21] Gigliola Staffilani; Daniel Tataru Strichartz estimates for a Schrödinger operator with nonsmooth coefficients, Commun. Partial Differ. Equations, Volume 27 (2002) no. 7-8, pp. 1337-1372 | DOI | MR | Zbl

[22] Gavin Stewart On the wellposedness for periodic nonlinear Schrödinger equations with white noise dispersion (2022) | arXiv

[23] Alexander Y. Veretennikov On strong solutions and explicit formulas for solutions of stochastic integral equations, Math. USSR, Sb., Volume 39 (1981), pp. 387-403 | DOI | Zbl

Cité par Sources :