On the growth of Sobolev norms for NLS on $2d$ and $3d$ manifolds

29 Jul 2016 · Planchon Fabrice, Tzvetkov Nikolay, Visciglia Nicola ·

Using suitable modified energies we study higher order Sobolev norms' growth in time for the nonlinear Schr\"odinger equation (NLS) on a generic $2d$ or $3d$ compact manifold. In $2d$ we extend earlier results that dealt only with cubic nonlinearities, and get polynomial in time bounds for any higher order nonlinearities. In $3d$, we prove that solutions to the cubic NLS grow at most exponentially, while for sub-cubic NLS we get polynomial bounds on the growth of the $H^2$-norm.

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On the growth of Sobolev norms for NLS on $2d$ and $3d$ manifolds

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