Oh, Tadahiro

Blue, Pieter

Liu, Ruoyuan

2024-08-07T09:23:17Z

2024-08-07T09:23:17Z

2024-08-07

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.

https://hdl.handle.net/1842/42063

http://dx.doi.org/10.7488/era/4785

en

The University of Edinburgh

R. Liu, On the probabilistic well-posedness of the two-dimensional periodic nonlinear Schr¨odinger equation with the quadratic nonlinearity |u| 2 , J. Math. Pures Appl. 171 (2023), 75-101

R. Liu, T. Oh, Sharp local well-posedness of the two-dimensional periodic nonlinear Schr¨odinger equation with a quadratic nonlinearity |u| 2 , to appear in Math. Res. Lett.

R. Liu, Local well-posedness of the periodic nonlinear Schr¨odinger equation with a quadratic nonlinearity u 2 in negative Sobolev spaces, to appear in J. Dyn. Diff. Equat.

nonlinear Schrodinger equations

partial differential equations

randomness

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

Thesis or Dissertation

Doctoral

PhD Doctor of Philosophy