Dispersive Day 2022

Schedule:

9:30 - 10:00		Coffee and breakfast
10:00 - 10:20		Ruoyuan Liu (University of Edinburgh) slides
Title:		Deterministic and probabilistic well-posedness of the nonlinear Schrödinger equation with a quadratic nonlinearity on the two-dimensional torus
Abstract:		In this talk, I will present results on deterministic and probabilistic well-posedness of the nonlinear Schrödinger equation (NLS) with a quadratic nonlinearity \|u\|², posed on the two-dimensional torus. I will first go over the deterministic local well-posedness result of the quadratic NLS in L². Next, I will talk about almost sure local well-posedness of a renormalized quadratic NLS with random initial data, which is distributed according to a fractional derivative of the Gaussian free field. I will also talk about the case of very rough initial data, where the second Picard second iterate diverges. The first part is based on a joint work with Tadahiro Oh (Edinburgh).
10:25 - 10:45		Luigi Forcella (Università degli Studi dell'Aquila) slides
Title:		A finite time blow-up result for a Davey-Stewartson system
Abstract:		In this talk, we consider a three dimensional elliptic-elliptic Davey-Stewartson system, and we give sufficient conditions for formation of singularities in finite time. The studied model is a non-local NLS-type equation, and the proof of the main result is based on general results on distributions defined via homogeneous symbols, which allow us to employ a convexity argument.
10:50 - 11:10		Rui Liang (University of Birmingham) slides
Title:		Gibbs measures for fractional NLS on the multi-dimensional torus
Abstract:		We will talk about the construction of the Gibbs measures for the focusing mass-critical fractional nonlinear Schrödinger equation on the multi-dimensional torus. We identify the sharp mass threshold for normalizability and non-normalizability of the focusing Gibbs measures, which generalizes the influential works of Lebowitz-Rose-Speer (1988), Bourgain (1994), and Oh-Sosoe-Tolomeo (2021) on the one-dimensional nonlinear Schrödinger equations. For this purpose, we establish an almost sharp fractional Gagliardo-Nirenberg-Sobolev inequality on the torus, where the main difficulty is the non-local nature of the fractional derivative. This is based on a joint work with Yuzhao Wang (Birmingham).
11:10 - 11:30		Coffee break
11:30 - 12:10		Plenary talk: Bjoern Bringmann (IAS/Princeton University) slides
Title:		Invariant Gibbs measures for the three-dimensional cubic nonlinear wave equation.
Abstract:		In this talk, we prove the invariance of the Gibbs measure for the three-dimensional cubic nonlinear wave equation. Since the audience consists of experts on random dispersive equations, this talk focuses on specific aspects of our argument. First, we discuss a caloric representation of the Gibbs measure, which is obtained using the cubic stochastic heat equation. Second, we discuss multi-linear dispersive estimates for wave equations and the corresponding lattice point counting estimates. If time permits, we also discuss linear and bilinear random tensor estimates. This is joint work with Y. Deng, A. Nahmod, and H. Yue.
12:15 - 12:35		Guopeng Li (University of Edinburgh) slides
Title:		Convergence on the finite-depth fluid equation in the shallow water surface and infinitely deep water limits.
Abstract:		The finite-depth fluid equation (FDF) is a mathematical model that is of physical interest and characterizes the wave propagation in the fluid of two layers systems. In particular, its dispersion phenomenon varies along with the depth parameters, and at the equation level, it is known to converge to the Korteweg-de Vries equation (KdV) and Benjamin-Ono equation (BO) as the depth parameters go to zero and infinity, respectively. In this talk, we will discuss this convergence problem first from the deterministic viewpoint. We show that the solutions to the FDF converge to the solutions to the KdV and BO, respectively. Such convergence results give the microscopic properties in the sense that deterministic convergence holds for each given initial data. Next, we briefly present the probabilistic counterpart. We study (i) the convergence of the Gibbs measures of FDF; (ii) the associated convergence dynamical problem of FDF. Rather than looking at a single solution, we study the dynamics as the statistical ensembles, and hence such convergence properties are explained via the macroscopic limit. The second part is based on a joint work with Tadahiro Oh (Edinburgh) and Guangqu Zheng (Edinburgh).
12:40 - 13:00		Kihoon Seong (KAIST) slides
Title:		Quasi-invariance of Gaussian measures under the flow of the cubic fourth order NLS in negative Sobolev spaces
Abstract:		Transport properties of Gaussian measures under different transformations have been studied in probability theory. In this talk, we discuss transport properties of Gaussian measures on periodic functions/distributions under nonlinear Hamiltonian PDEs, taking the cubic nonlinear Schrödinger equation with the fourth order dispersion (4NLS) as a primary example. Previously, Oh-Tzvetkov (2017) and Oh-Sosoe-Tzvetkov (2018) proved quasi-invariance of the Gaussian measures supported on L² under the 4NLS dynamics. In this talk, we will present an extension of this quasi-invariance result to the Gaussian measures supported on negative Sobolev spaces. This establishes the first quasi-invariance result for nonlinear Hamiltonian PDEs in negative Sobolev spaces. This talk is based on a joint work with Tadahiro Oh (University of Edinburgh).
13:00 - 14:20		Lunch
14:20 - 15:00		Plenary talk: Leonardo Tolomeo (Hausdorff Center for Mathematics) slides
Title:		DiPerna-Lions for dispersive PDEs: quasi-invariance and global well-posedness for fractional NLS in negative regularity.
Abstract:		In this talk, we consider the Cauchy problem for the fractional NLS with cubic nonlinearity (FNLS), posed on the one-dimensional torus T, subject to a gaussian random initial data with negative regularity. Exploiting the structure of the Liouville equation for the transport of the gaussian measure, we can show global-in-time bounds for the L^p norm of the density of the evolved measure. These bounds rely exclusively on the probabilistic local well posedness theory for FNLS. We can then use Bourgain's invariant measure argument to extend these bounds to the solution of FNLS emanating from almost every initial data. In a certain range of the parameters of this problem, the global well posedness holds for initial data which is rougher than what is allowed by the deterministic local well posedness theory. This is a joint work with Justin Forlano (UCLA).
15:05 - 15:35		Guangqu Zheng (University of Edinburgh) slides
Title:		Pathwise well-posedness of stochastic nonlinear Schrödinger equations with multiplicative noises
Abstract:		In this talk, we present a recent work on the stochastic nonlinear Schrödinger equation (SNLS) with a multiplicative noise. We will present the first pathwise local well-posedness result for SNLS with a multiplicative noise. The main difficulty of this problem comes from the deficiency of temporal regularities in making sense of the product of an unknown and a noise. In the parabolic setting, this problem was studied by Gubinelli-Tindel (Ann. Probab. 2010) via rough path analysis. In the dispersive setting, however, there is a significant additional difficulty due to the lack of the strong parabolic smoothing. In order to overcome this difficulty, we combine the algebraic framework introduced by Gubinelli-Tindel, leading to operator-valued rough paths adapted to the Schrödinger flow, with the random matrix/tensor estimates. Furthermore, when a noise is white in time, a certain random operator suffers a loss of regularity. In order to make up this loss of regularity, we implement an infinite iteration of the (partial) Duhamel formulation, which leads to the notion of perturbed controlled rough paths. This talk is based on a joint work with Tadahiro Oh (University of Edinburgh).
15:35 - 16:00		Coffee break
16:00 - 16:30		Tristan Robert (Universitá de Lorraine) slides
Title:		Invariant Gibbs measure for a Schrödinger equation with exponential nonlinearity
Abstract:		We consider the construction and invariance of the Gibbs measure associated with the fractional NLS on compact manifolds with an exponential nonlinearity. There are three parameters in the model: the dispersion parameter, the coupling constant in the nonlinearity, and the sign of the nonlinearity. We restrict to the non-singular case where the dispersion is bigger than the dimension, making the base Gaussian measure supported on functions of positive regularity. In the focusing case, we show that the Gibbs measure is ill-defined regardless of the dispersion and the coupling constant, even for arbitrarily small mass cut-off. In the defocusing case, the measure is always well-defined regardless of the (non-singular) dispersion and the coupling constant. As for the invariance, we show that even weak invariance (global existence without uniqueness) via a compactness argument requires smallness of the coupling constant. Finally, in the case where the manifold is the circle, we show almost sure local and global well-posedness for large enough dispersion and small enough coupling constant. This is done via Bourgain-Da Prato-Debussche trick and Bourgain's invariant measure argument, by using an appropriate choice of gauge transform to remove some resonances, then expanding the nonlinearity into an infinite sum of multilinear forms, and obtaining appropriate estimates on the latter.

Acknowledgements:

This event is supported by the European Research Council (grant no. 864138 "SingStochDispDyn") and the EPSRC New Investigator Awards (grant no. EP/S033157/1 and grant no. EP/V003178/1)