no. 2
On the growth of Sobolev norms of solutions of the fractional defocusing NLS equation on the circle
Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 2, pp. 509-531.

This paper is devoted to the study of large time bounds for the Sobolev norms of the solutions of the following fractional cubic Schrödinger equation on the torus:

i∂tu=|D|αu+|u|2u,u(0,⋅)=u0,
where α is a real parameter. We show that, apart from the case α=1, which corresponds to a half-wave equation with no dispersive property at all, solutions of this equation grow at a polynomial rate at most. We also address the case of the cubic and quadratic half-wave equations.

DOI : 10.1016/j.anihpc.2016.02.002
Classification : 37K40, 35B45
Mots clés : Hamiltonian systems, Fractional nonlinear Schrödinger equation, Nonlinear wave equation, Dispersive properties 
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     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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Thirouin, Joseph. On the growth of Sobolev norms of solutions of the fractional defocusing NLS equation on the circle. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 2, pp. 509-531. doi : 10.1016/j.anihpc.2016.02.002. http://www.numdam.org/articles/10.1016/j.anihpc.2016.02.002/

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