On the growth of Sobolev norms for the cubic Szegő equation
Séminaire Laurent Schwartz — EDP et applications (2014-2015), Talk no. 11, 20 p.

We report on a recent result establishing that trajectories of the cubic Szegő equation in Sobolev spaces with high regularity are generically unbounded, and moreover that, on solutions generated by suitable bounded subsets of initial data, every polynomial bound in time fails for high Sobolev norms. The proof relies on an instability phenomenon for a new nonlinear Fourier transform describing explicitly the solutions to the initial value problem, which is inherited from the Lax pair structure enjoyed by the equation.

DOI: 10.5802/slsedp.70
Gérard, Patrick 1; Grellier, Sandrine 2

1 Université Paris-Sud Laboratoire de Mathématiques d’Orsay CNRS, UMR 8628 France
2 MAPMO-UMR 6628 Département de Mathématiques Université d’Orleans 45067 Orléans Cedex 2 France
     author = {G\'erard, Patrick and Grellier, Sandrine},
     title = {On the growth of {Sobolev} norms for the cubic {Szeg\H{o}} equation},
     journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
     note = {talk:11},
     publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2014-2015},
     doi = {10.5802/slsedp.70},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/slsedp.70/}
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Gérard, Patrick; Grellier, Sandrine. On the growth of Sobolev norms for the cubic Szegő equation. Séminaire Laurent Schwartz — EDP et applications (2014-2015), Talk no. 11, 20 p. doi : 10.5802/slsedp.70. http://archive.numdam.org/articles/10.5802/slsedp.70/

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