## Periodic Reporting for period 2 - ProbDynDispEq (Probabilistic and Dynamical Study of Nonlinear Dispersive Equations)

**Reporting period:**2016-09-01

**to**2018-02-28

## Summary of the context and overall objectives of the project

Nonlinear dispersive partial differential equations (PDEs) such as the nonlinear Schrödinger equations (NLS) and the Korteweg-de Vries equation (KdV) appear ubiquitously as models describing wave phenomena in various branches of physics and engineering. Over the last several decades, multilinear harmonic analysis has played a crucial role in the development of the theoretical understanding of the subject. Furthermore, in recent years, a non-deterministic point of view has been incorporated into the study of nonlinear dispersive PDEs, enabling us to study typical behavior of solutions in a probabilistic manner and go beyond the limit of deterministic analysis.

The main objective of this project is to develop novel mathematical ideas and techniques, and make significant progress on some of the central problems related to the fundamental nonlinear dispersive PDEs such as NLS and KdV from both the deterministic and probabilistic points of view. On the one hand, such a study is of importance from the theoretical points of view since it promotes development of analytical tools, motivating intradisciplinary interactions across various fields in mathematics. On the other hand, some of these problems are motivated by physics and engineering and hence they are also of importance from the applied points of view.

There are several objectives in this project.

1. The first project involves invariant Gibbs measures for nonlinear Hamiltonian PDEs. One project to study the limiting behavior of the Gibbs measure as the mass cutoff size approaches the threshold mass. The other project aims to make a progress in understanding how the invariant state is related at different times.

2. The second project involves well-posedness issues (i.e. existence of unique solutions) to NLS and KdV with random initial data and/or random forcing. The random noise we consider here is white noise, which appears ubiquitously in the physics literature.

Our main goal is to construct dynamics for NLS and KdV either (i) with white noise initial data and/or (ii) with additive space-time white noise forcing. The main difficulty of this project is the roughness of the white noise and a novel approach is anticipated in overcoming this difficulty.

3. The third project is to develop novel analytical techniques and construct the local-in-time dynamics for the cubic NLS on the circle in a low regularity. The construction of solution to the one-dimensional cubic NLS remains as an open question over twenty-five years and the PI intends to address this problem.

4. The PI aims to advance the understanding some energy-critical NLS with non-vanishing boundary conditions.

What is common among these projects is to deepen our understanding of the behavior of solutions to nonlinear dispersive PDEs.

The main objective of this project is to develop novel mathematical ideas and techniques, and make significant progress on some of the central problems related to the fundamental nonlinear dispersive PDEs such as NLS and KdV from both the deterministic and probabilistic points of view. On the one hand, such a study is of importance from the theoretical points of view since it promotes development of analytical tools, motivating intradisciplinary interactions across various fields in mathematics. On the other hand, some of these problems are motivated by physics and engineering and hence they are also of importance from the applied points of view.

There are several objectives in this project.

1. The first project involves invariant Gibbs measures for nonlinear Hamiltonian PDEs. One project to study the limiting behavior of the Gibbs measure as the mass cutoff size approaches the threshold mass. The other project aims to make a progress in understanding how the invariant state is related at different times.

2. The second project involves well-posedness issues (i.e. existence of unique solutions) to NLS and KdV with random initial data and/or random forcing. The random noise we consider here is white noise, which appears ubiquitously in the physics literature.

Our main goal is to construct dynamics for NLS and KdV either (i) with white noise initial data and/or (ii) with additive space-time white noise forcing. The main difficulty of this project is the roughness of the white noise and a novel approach is anticipated in overcoming this difficulty.

3. The third project is to develop novel analytical techniques and construct the local-in-time dynamics for the cubic NLS on the circle in a low regularity. The construction of solution to the one-dimensional cubic NLS remains as an open question over twenty-five years and the PI intends to address this problem.

4. The PI aims to advance the understanding some energy-critical NLS with non-vanishing boundary conditions.

What is common among these projects is to deepen our understanding of the behavior of solutions to nonlinear dispersive PDEs.

## Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

During this reporting period, the PI mainly worked on the first three projects. One of the main goals in the first project is to study the limiting behavior of the Gibbs measure as the mass cutoff size approaches the critical mass threshold. By combining harmonic analytic approach with stochastic analysis and variational techniques, the PI essentially solved the proposed problem.

The second project involves the study of NLS and KdV with white noise initial data and/or space-time white noise forcing. This problem turned out to be critical, which is not approachable with the current technology. The PI proposed to study NLS with fourth order dispersion (4NLS) as a toy model. While the problem for 4NLS is easier, it still possesses part of the critical nature of the NLS problem. By employing a sophisticated combination of analysis and stochastic tools, the PI managed to construct global-in-time dynamics for 4NLS with white noise initial data and proved invariance of the white noise.

As for the stochastic nonlinear dispersive PDEs with singular noise, the PI constructed global-in-time dynamics for the stochastic KdV equation with singular space-time white noise forcing. The PI also constructed dynamics for the stochastic NLS with almost space-time white noise. While the stochastic NLS with space-time white noise seems within reach, it seems to require a significant novel idea to close this gap.

Under the PI's guidance, the team members (a Ph.D. student and a postdoc) studied the three-dimensional stochastic beam equations with space-time white noise and proved its well-posedness.

The main goal of the third project is to prove well-posedness of the one-dimensional cubic NLS in negative Sobolev spaces. The PI has successfully carried out the first step for this problem, namely he proved global-in-time existence of solutions

to the renormalized NLS in this low regularity setting. As a byproduct, a very strong form of ill-posedness of the non-renormalized NLS in negative Sobolev spaces was proven (namely, non-existence of solutions).

In an effort to study uniqueness property of solutions to the renormalized NLS, the PI considered 4NLS as a toy model and constructed global-in-time unique solutions in negative Sobolev spaces. Moreover, the PI introduced a new notion of enhanced uniqueness of solutions to evolution equations, which is essentially the strongest form of uniqueness in practice, and proved such enhanced uniqueness holds for the renormalized 4NLS in negative Sobolev spaces.

There are other research achievements, which do not directly ddress an issue toward a resolution of the aforementioned projects but are clearly related to the topics mentioned in the projects above. Hence, it is important to promote further advancement of ideas and tools in these works, which may allow the PI to find a more innovative approach to the proposed problems in the project.

(i) Further advancement to probabilistic well-posedness of nonlinear dispersive PDEs with random initial data. In particular, the PI proved the first almost sure global well-posedness of the energy-critical NLS with rough random initial data.

(ii) Transport properties of Gaussian measures under nonlinear Hamiltonian dynamics. In particular, the PI proved quasi-invariance of the Gaussian measure under the dynamics.

(iii) Well-posedness of stochastic dispersive PDEs with singular noise.

(iv) Further development of an infinite iteration of normal form reductions.

(v) Ill-posedness of nonlinear dispersive PDEs in a low regularity setting. In order to study well-posedness issues with rough (deterministic or random) initial data, it is important to understand bad behavior of solutions in a low regularity setting.

During this reporting period, the PI and the team members published over twenty papers and preprints and gave thirty talks including two mini-courses (26 by the PI including the twomini-courses and 4 by the Ph.D. student and the postdoc).

The second project involves the study of NLS and KdV with white noise initial data and/or space-time white noise forcing. This problem turned out to be critical, which is not approachable with the current technology. The PI proposed to study NLS with fourth order dispersion (4NLS) as a toy model. While the problem for 4NLS is easier, it still possesses part of the critical nature of the NLS problem. By employing a sophisticated combination of analysis and stochastic tools, the PI managed to construct global-in-time dynamics for 4NLS with white noise initial data and proved invariance of the white noise.

As for the stochastic nonlinear dispersive PDEs with singular noise, the PI constructed global-in-time dynamics for the stochastic KdV equation with singular space-time white noise forcing. The PI also constructed dynamics for the stochastic NLS with almost space-time white noise. While the stochastic NLS with space-time white noise seems within reach, it seems to require a significant novel idea to close this gap.

Under the PI's guidance, the team members (a Ph.D. student and a postdoc) studied the three-dimensional stochastic beam equations with space-time white noise and proved its well-posedness.

The main goal of the third project is to prove well-posedness of the one-dimensional cubic NLS in negative Sobolev spaces. The PI has successfully carried out the first step for this problem, namely he proved global-in-time existence of solutions

to the renormalized NLS in this low regularity setting. As a byproduct, a very strong form of ill-posedness of the non-renormalized NLS in negative Sobolev spaces was proven (namely, non-existence of solutions).

In an effort to study uniqueness property of solutions to the renormalized NLS, the PI considered 4NLS as a toy model and constructed global-in-time unique solutions in negative Sobolev spaces. Moreover, the PI introduced a new notion of enhanced uniqueness of solutions to evolution equations, which is essentially the strongest form of uniqueness in practice, and proved such enhanced uniqueness holds for the renormalized 4NLS in negative Sobolev spaces.

There are other research achievements, which do not directly ddress an issue toward a resolution of the aforementioned projects but are clearly related to the topics mentioned in the projects above. Hence, it is important to promote further advancement of ideas and tools in these works, which may allow the PI to find a more innovative approach to the proposed problems in the project.

(i) Further advancement to probabilistic well-posedness of nonlinear dispersive PDEs with random initial data. In particular, the PI proved the first almost sure global well-posedness of the energy-critical NLS with rough random initial data.

(ii) Transport properties of Gaussian measures under nonlinear Hamiltonian dynamics. In particular, the PI proved quasi-invariance of the Gaussian measure under the dynamics.

(iii) Well-posedness of stochastic dispersive PDEs with singular noise.

(iv) Further development of an infinite iteration of normal form reductions.

(v) Ill-posedness of nonlinear dispersive PDEs in a low regularity setting. In order to study well-posedness issues with rough (deterministic or random) initial data, it is important to understand bad behavior of solutions in a low regularity setting.

During this reporting period, the PI and the team members published over twenty papers and preprints and gave thirty talks including two mini-courses (26 by the PI including the twomini-courses and 4 by the Ph.D. student and the postdoc).

## Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

"1. One of the most notable progress beyond the state-of-art is the (essential) resolution of the first project on the study of the limiting behavior of the Gibbs measure as the mass cutoff size approaches the critical mass threshold. In this problem, the PI combined harmonic analytic approach with stochastic analysis in an unconventional manner. In particular, a concentration-compactness technique to study the limiting behavior of the Gibbs measure is novel.

2. The PI introduced a novel approach to exhibit a pathological behavior the cubic NLS in negative Sobolev spaces (namely non-existence of solutions). After appearance of the paper, a similar technique was used by Kappeler-Molnar (SIAM J. Math. Anal. 2017) to prove ill-posedness of modified KdV outside L^2.

3. The PI developed an infinite iteration scheme of normal form reductions at the level of energy functionals. In particular, this technique allowed the PI to introduce (and prove for 4NLS) a new notion of enhanced uniqueness of solutions, which seem to be the strongest possible notion of uniqueness of solutions in practice.

4. The PI advanced significantly probabilistic well-posedness theory of nonlinear dispersive PDEs with random initial data. In particular, he introduced a novel iterative approach based on a (modified) partial power series expansion in proving probabilistic local well-posedness, which seems to be widely applicable to other equations. Moreover, he incorporated a sophisticated deterministic energy-critical theory in the context of probabilistic well-posedness and proved the first almost sure global well-posedness of the energy-critical NLS with rough random initial data.

5. In studying the 4NLS with white noise initial data, the PI introduced a new decomposition of a solution called ""random-resonant / nonlinear decomposition'' and successfully constructed dynamics. This new decomposition was based on an infinite iterative scheme of identifying and eliminating the worst term at each iterative step.

"

2. The PI introduced a novel approach to exhibit a pathological behavior the cubic NLS in negative Sobolev spaces (namely non-existence of solutions). After appearance of the paper, a similar technique was used by Kappeler-Molnar (SIAM J. Math. Anal. 2017) to prove ill-posedness of modified KdV outside L^2.

3. The PI developed an infinite iteration scheme of normal form reductions at the level of energy functionals. In particular, this technique allowed the PI to introduce (and prove for 4NLS) a new notion of enhanced uniqueness of solutions, which seem to be the strongest possible notion of uniqueness of solutions in practice.

4. The PI advanced significantly probabilistic well-posedness theory of nonlinear dispersive PDEs with random initial data. In particular, he introduced a novel iterative approach based on a (modified) partial power series expansion in proving probabilistic local well-posedness, which seems to be widely applicable to other equations. Moreover, he incorporated a sophisticated deterministic energy-critical theory in the context of probabilistic well-posedness and proved the first almost sure global well-posedness of the energy-critical NLS with rough random initial data.

5. In studying the 4NLS with white noise initial data, the PI introduced a new decomposition of a solution called ""random-resonant / nonlinear decomposition'' and successfully constructed dynamics. This new decomposition was based on an infinite iterative scheme of identifying and eliminating the worst term at each iterative step.

"