# Registered Data

## [01229] Cauchy problem for Deterministic and Stochastic nonlinear dispaersive equations

**Session Date & Time**: 5C (Aug.25, 13:20-15:00)**Type**: Proposal of Minisymposium**Abstract**: We consider Cauchy problem for Deterministic and Stochastic nonlinear dispersive equations. For the Deterministic nonlinear Schrödinger equation (NLS), some global dynamics will be discussed, that is scattering, blowing up, growing up, or uniform bound for the solutions of NLS. For the Stochastic NLS, we will deal with nonlinear equation with a multiplicable noise. We also intoroduce and discuss paracontrolled calculus to prove local well-posedness of a renormalized version of the stochastic nonlinear wave equation.**Organizer(s)**: Shuji Machihara**Classification**:__35Q55__,__60H15__,__60H40__**Speakers Info**:- Guopeng Li (Maxwell Institute for Mathematical Sciences)
- Masahiro Ikeda (Riken)
- Masaru Hamano (Waseda university)
- Shunya Hashimoto (Saitama university)

**Talks in Minisymposium**:**[03584] Well-posedness for the fourth-order Schrödinger equation with third order derivative nonlinearities****Author(s)**:**Masahiro Ikeda**(RIKEN)

**Abstract**: We study the Cauchy problem to the semilinear fourth-order Schr\"odinger equations with third order derivative nonlinearities. The purpose of this presentation is to prove well-posedness of the problem in the lower order Sobolev space $H^s(\R)$ or with more general nonlinearities than previous results. Our proof of the main results is based on the contraction mapping principle on a suitable function space employed by D. Pornnopparath (2018). To obtain the key linear and bilinear estimates, we construct a suitable decomposition of the Duhamel term introduced by I. Bejenaru, A. D. Ionescu, C. E. Kenig, and D. Tataru (2011). Moreover we discuss scattering of global solutions and the optimality for the regularity of our well-posedness results, namely we prove that the flow map is not smooth in several cases.

**[03890] The well-posedness of the stochastic nonlinear Schrödinger equations in H^2****Author(s)**:**Shunya Hashimoto**(Saitama university)

**Abstract**: We consider the well-posedness of $H^2$-solutions in initial value problems for the stochastic nonlinear Schrödinger equations with power-type nonlinear terms with multiplicative noise. For the proof, we use the rescaling approach, which transforms the stochastic equation into a random equation in which no white noise appears. Unlike $L^2$- and $H^1$-solution, there are two difficulties in the $H^2$-solution: first, the lack of the smoothness of nonlinear functions, and second, the treatment of white noise that reappears.

**[04966] Time beharior of solutions to nonlinear Schr\"odinger equation with a potential****Author(s)**:**Masaru Hamano**(Waseda University)

**Abstract**: In this talk, we deal with the Cauchy problem of a nonlinear Schr\"odinger equation with a potential. In particular, we consider time behavior of solutions to the equation with initial data, whose energy is equal to that of the Talenti function.

**[05020] Convergence of the intermediate long wave equation from a statistical perspective****Author(s)**:**Guopeng Li**(The Maxwell Institute for Mathematical Sciences)

**Abstract**: The intermediate long wave equation (ILW) models water waves of finite depth, connecting the Benjamin-Ono equation (deep-water limit) and the KdV equation (shallow-water limit). Convergence problems of ILW (in both the deep-water and shallow-water limits) have attracted attention from both the applied and theoretical points of view. In this talk, I will discuss convergence problems from a statistical viewpoint. I first consider the convergence problem of the Gibbsian ensembles. In this case, I establish convergence of the Gibbs measures and then also show convergence of invariant Gibbs dynamics to that of the Benjamin-Ono and KdV equations (without uniqueness). ILW is known to be completely integrable and thus possesses infinitely many conservation laws. In the second part of the talk, I consider invariant dynamics for ILW associated with higher order conservation laws. Due to a complicated nature of the dispersion, even the construction of measures associated with higher order conservation laws turns out to be highly non-trivial. By considering a suitable combination of higher order conservation laws, I overcome this issue and construct invariant dynamics for ILW with a fixed depth parameter. In the final part, I will discuss convergence of the invariant dynamics associated with higher order conservation laws. This talk is based on a joint work with Tadahiro Oh (Edinburgh), Guangqu Zheng (Liverpool), and Andreia Chapouto (UCLA).