A remark on norm inflation for nonlinear Schrödinger equations
arXiv:1806.10066
Abstract
We consider semilinear Schrödinger equations with nonlinearity that is a polynomial in the unknown function and its complex conjugate, on or on the torus. Norm inflation (ill-posedness) of the associated initial value problem is proved in Sobolev spaces of negative indices. To this end, we apply the argument of Iwabuchi and Ogawa (2012), who treated quadratic nonlinearities. This method can be applied whether the spatial domain is non-periodic or periodic and whether the nonlinearity is gauge/scale-invariant or not.
arXiv comment28 pages. Minor changes; results unchanged
References in corpus (10)
- Ill-posedness for nonlinear Schrodinger and wave equations
- On the one-dimensional cubic nonlinear Schrodinger equation below L^2
- Instability of the periodic nonlinear Schrodinger equation
- Geometric optics and instability for NLS and Davey-Stewartson models
- Norm-inflation for periodic NLS equations in negative Sobolev spaces
- Norm inflation for the generalized Boussinesq and Kawahara equations
- Global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation in almost critical spaces
- Some local wellposedness results for nonlinear Schroedinger equations below L^2
- Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation
- Low regularity conservation laws for integrable PDE