Oh, Tadahiro

Blue, Pieter

Zine, Younes

European Research Council

2023-11-07T15:22:59Z

2023-11-07T15:22:59Z

2023-11-07

In this thesis, we investigate convergence problems for some nonlinear dispersive and parabolic PDEs in the singular stochastic setting. In the first part of the thesis, we study the so-called Smoluchowski-Kramers approximation on convergence of stochastic nonlinear wave equations (SNLW) to stochastic nonlinear heat equations (SNLH), with a polynomial nonlinearity. In particular, we prove that, in the over-damped regime, solutions of SNLW converge to those of the corresponding SNLH. This convergence is established for deterministic initial data.  In the second part of the work, we study the inviscid limit for the stochastic complex Ginzburg-Landau equation (SCGL) with the cubic nonlinearity. We prove that, for Gaussian free field initial data, the solution of SCGL converges to that of the cubic nonlinear Schrödinger equation.

https://hdl.handle.net/1842/41119

http://dx.doi.org/10.7488/era/3855

en

The University of Edinburgh

T. Oh, Y. Wang, Y. Zine, Three-dimensional stochastic cubic nonlinear wave equation with almost space-time white noise, Stoch. Partial Differ. Equ. Anal. Comput. 10 (2022), no. 3, 898–963

convergence problems

nonlinear dispersive PDEs

parabolic PDEs

Smoluchowski-Kramers approximation

stochastic nonlinear wave equations

stochastic nonlinear heat equations

polynomial nonlinearity

Schrodinger equation

Convergence problems for singular stochastic dynamics

Thesis or Dissertation

Doctoral

PhD Doctor of Philosophy