Nonlinear Schrödinger Equations with Rough Data

In this thesis we consider nonlinear Schrödinger equations with rough initial data. Roughness of the initial data in nonlinear Schrödinger equations can be understood as being of low regularity and as a lack of decay at infinity.

Firstly we prove low regularity a priori estimates for the derivative nonlinear Schrödinger equation in Besov spaces with positive regularity index. These a priori estimates are sharp at the level of regularity but are conditional upon small mass. The proof uses the operator determinant characterization of the transmission coefficient introduced by Killip-Visan-Zhang.

Secondly we show global wellposedness for the tooth problem of defocusing nonlinear Schrödinger equations, that is the Cauchy problem with initial data in the space $H^{s_1}(\mathbb{R}) + H^{s_2}(\mathbb{T})$. This result can be seen as an intermediate step between the wellposedness theory in the $L^2(\mathbb{T})$-based setting and more generic non-decaying behavior at infinity. In the case $s_1 = 1$ we obtain an at most exponentially growing energy, based on the Hamiltonian of the perturbed equation. For the cubic nonlinearity we may choose $s_2 > 3/2$ whereas for higher power nonlinearities our assumption is $s_2 > 5/2$.
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Finally we investigate the question of wellposedness of nonlinear Schrödinger equations with initial data in modulation spaces. Modulation spaces $M^s_{p,q}(\mathbb{R}^n)$ encode both regularity ($s$ and $q$) and decay ($p$) in their indices. By making use of multilinear interpolation we prove new local wellposedness results. The local wellposedness results we obtain are proven to be sharp with respect to the regularity index. Moreover we complement the local results by showing global wellposedness in several cases, including low regularity and very slow decay. This is done on the one hand by an extension of techniques developed by Oh-Wang to a broader range of modulation spaces, and on the other hand by applying calculations from Dodson-Soffer-Spencer to the modulation space setting.