Well-posedness of nonlinear Schrödinger equations from deterministic and probabilistic viewpoints

Abstract

In this thesis, we study the Cauchy problem for nonlinear Schrödinger equations (NLS) in various settings.

Firstly, we consider NLS with a quadratic nonlinearity |u|² on the two-dimensional torus. By separately estimating the contributions from the nearly resonant and highly non-resonant interactions, we prove its sharp deterministic local well-posedness, thus resolving an open problem of thirty years since Bourgain (1993).

Secondly, we investigate the well-posedness issues of NLS with a quadratic nonlinearity ū² in negative Sobolev spaces on the one-dimensional and the two-dimensional tori. By introducing modified versions of the Fourier restriction norm spaces and overcome the failure of the crucial bilinear estimates, we establish deterministic local well-posedness in negative Sobolev spaces.

Thirdly, we come back to study NLS with the quadratic nonlinearity |u|² on the twodimensional torus with random initial data distributed according to a fractional derivative of the Gaussian free field. We prove almost sure local well-posedness below the deterministic threshold and a probabilistic ill-posedness result when the random initial data becomes too irregular.

Finally, we consider the dispersive Anderson model, namely NLS with a multiplicative spatial white noise, on the two-dimensional Euclidean space. We prove its global well-posedness by using a gauge-transform and constructing the solution as a limit of solutions to a family of approximating equations.