On the deterministic and probabilistic Cauchy problem of nonlinear dispersive partial differential equations
Abstract
In this thesis, we study well-posedness of nonlinear dispersive partial differential
equations (PDEs). We investigate the corresponding initial value problems in low-regularity from two perspectives: deterministic and probabilistic.
In the deterministic setting, we present two works. First, with T. Oh, we consider the one-dimensional cubic nonlinear Schrodinger equation (NLS) on the real-line. Adapting Kwon-Oh-Yoon (2018) and Kishimoto (2019), we apply an infinite
iteration of normal form reductions to construct solutions in almost critical Fourier-amalgam spaces. We also investigate the unconditional uniqueness of these solutions. In the second work, with M. Okamoto, we consider ill-posedness of nonlinear
wave equations, with integer power nonlinearity, in negative Sobolev spaces. More
precisely, using the approach by Oh (2017), we establish norm inflation at general
initial data, closing a gap left in the work of Christ-Colliander-Tao (2003).
Within the probabilistic setting, we present three works examining well-posedness
and dynamical properties of nonlinear dispersive PDEs, with either random initial
data or stochastic forcing. We first consider well-posedness of the Benjamin-Bona-Mahony equation with random initial data distributed according to Gaussian measures supported in negative Sobolev spaces. Namely, we construct almost surely
local-in-time solutions and, by adapting the I-method approach of Gubinelli-Koch-Oh-Tolomeo (2020), global-in-time solutions. Secondly, with T. Oh and Y. Wang,
we prove local well-posedness of the one-dimensional stochastic cubic NLS (SNLS)
with almost space-time white noise forcing. Given that the well-posedness of SNLS
with full space-time white noise forcing is a longstanding open problem, this result is
almost optimal. Finally, with W. J. Trenberth, we explore the relationship between
dispersion in nonlinear dispersive PDEs and the transport property of Gaussian
measures supported on periodic functions. We employ energy methods to prove the
quasi-invariance of these Gaussian measures under the flow of the cubic fractional
NLS, under certain conditions on the strength of the dispersion.