A remark on norm inflation for nonlinear wave equations

In this note, we study the ill-posedness of nonlinear wave equations (NLW). Namely, we show that NLW experiences norm inflation at every initial data in negative Sobolev spaces. This result covers a gap left open in a paper of Christ, Colliander, and Tao (2003) and extends the result by Oh, Tzvetkov, and the second author (2019) to non-cubic integer nonlinearities. In particular, for some low dimensional cases, we obtain norm inflation above the scaling critical regularity. We also prove ill-posedness for NLW, via norm inflation at general initial data, in negative regularity Fourier-Lebesgue and Fourier-amalgam spaces.Comment: 20 pages. Published in Dyn. Partial Differ. Eq

On the transport of Gaussian measures under the one-dimensional fractional nonlinear Schrödinger equations

Under certain regularity conditions, we establish quasi-invariance of Gaussian measures on periodic functions under the flow of cubic fractional nonlinear Schr\"{o}dinger equations on the one-dimensional torus.Comment: 42 pages; typos corrected, added Remark 1.8. To appear in Ann. Inst. H. Poincar\'e Anal. Non Lin\'eair

Intermediate long wave equation in negative Sobolev spaces

We study the intermediate long wave equation (ILW) in negative Sobolev spaces. In particular, despite the lack of scaling invariance, we identify the regularity $s = -\frac 12$ as the critical regularity for ILW with any depth parameter, by establishing the following two results. (i) By viewing ILW as a perturbation of the Benjamin-Ono equation (BO) and exploiting the complete integrability of BO, we establish a global-in-time a priori bound on the $H^s$-norm of a solution to ILW for $- \frac 12 < s < 0$. (ii) By making use of explicit solutions, we prove that ILW is ill-posed in $H^s$ for $s < - \frac 12$. Our results apply to both the real line case and the periodic case.Comment: 15 page

On the deterministic and probabilistic Cauchy problem of nonlinear dispersive partial differential equations

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

On the unique ergodicity for a class of 2 dimensional stochastic wave equations

We study the global-in-time dynamics for a stochastic semilinear wave equation with cubic defocusing nonlinearity and additive noise, posed on the 2-dimensional torus. The noise is taken to be slightly more regular than space-time white noise. In this setting, we show existence and uniqueness of an invariant measure for the Markov semigroup generated by the flow over an appropriately chosen Banach space. This extends a result of the second author [Comm. Math. Phys. 377 (2020), pp. 1311–1347] to a situation where the invariant measure is not explicitly known.</p