Modulation Spaces and Schr"odinger Equation

Abstract:

We discuss some ongoing interest (progress in the last decade) of modulation spaces from the PDEs point of view. We prove the recent results on composition operators and Hartree type equations(HTE)(Schr\"odinger equation with cubic convolution nonlinearity) on modulation spaces.

It is known that NLS(nonlinear Schr\"odinger equation) is locally well-posed in modulation spaces for the power-type nonlinearity $u|u|^{\alpha}, \alpha \in 2 \mathbb N.$ As an application to composition operators on the modulation spaces, we point out that the standard method for the evolution of NLS cannot be considered for the power-type nonlinearity $|u|^{\alpha}u, \alpha \in (0, \infty) \setminus 2 \mathbb N.$ We study the Cauchy problem for the HTE with a Cauchy data in modulation spaces, and obtain local and global well-posedness results for the HTE.