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.