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Fourth order nonlinear Schrödinger equation : almost sure well-posedness and soliton resolution conjecture

Equação de Schrödinger não linear de quarta ordem: boa colocação quase certa e conjectura de resolução de solitão

Reference
2024.13582.PEX
Project Start Date
2026-02-02
Project End Date
2027-08-01
Principal Investigator
Scientific Area
Exact sciences
Funding Program
Concurso para Projetos de Investigação de caráter Exploratório em Todos os Domínios Científicos 2024

Abstract

The goal of this research program is to make theoretical advances in several deep and challenging questions in PDEs. More specifically, most of this program concerns the fourth order nonlinear mixed diffusion Schrödinger equation (4NLS). The fourth order term was introduced by Karpman and Shagalov as part of a nonparaxial correction to NLS to prevent finite-time blow-up in low dimension. This research program will be split into two parts. The first one concerns the almost sure well-posedness of this equation. The main goal would be to develop a dispersive counterpart to Martin Hairer's regularity structures for SPDEs. It is well-known since the work of Christ, Colliander and Tao that NLS is deterministically ill-posed if the initial data is too irregular. On the other hand, thanks to the works of Bourgain, it is possible to show probabilistic well-posedness results for initial data below the deterministic regularity threshold. Our main focus will be to consider the almost sure well-posedness (ASWP) of 4NLS on the whole space. In this setting, two different types of approaches can be used. The first one, introduced by Bényi, Oh and Pocovnicu, consists in decomposing the initial data in frequency into unit cubes and multiplying each cube by a Gaussian random variable. This randomization called Wiener randomization allows to improve the integrability of the initial data without changing its differentiability. Our main goal in this setting is to obtain ASWP local and global for a large class of equations (changing the operator and the nonlinearity) for initial data in H^s for any s>0. The second approach to obtain ASWP is the use of invariant measures as introduced by Bourgain. He considered mostly equations on compact spaces, mainly on the torus. Bourgain's idea was to construct an invariant Gibbs measure which is obtained by using the formal time invariance of the energy functional. Such measure is necessarily supported on rough spaces (the energy is a priori not well-defined at such low regularity). It is known that such measure can only exist in low dimension, namelystrictly less than 4. Another way to construct invariant measure is to use the so-called fluctuation-dissipation method introduced by Kuksin and Shirikyan. The main interest of this method is the fact that one can control the regularity of the support of the measure. So one is able to use this method to obtain invariant measures in situations where Gibbs measures do not exist. However the drawback is that this fluctuation-dissipation measure is not explicit so it is hard to study its qualitative informations. As already written, these two constructions are performed on compact base spaces like the torus (or on the whole space with a confining potential) mainly in order to use Galerkin's approximation. Our main goal in this context would be to consider such measures on the torus of period L and pass to the limit as L goes to infinity. This will allow us to obtain ASWP results on the whole space. In this setting and as explained later, our main focus will be given to equations involving Hartree nonlinearity namely (V\ast |u|^2) u where V roughly acts as an anti-derivative of order k. The second part of this projet is devoted to the construction and the analysis of qualitative behavior of blowing-up solutions in the energy-critical regime. In this setting a famous conjecture is the soliton resolution one. It asserts that a solution with finite energy should decompose asymptotically as a sum of finitely many bubbles and a radiation term. The conjecture was proved to hold true very recently by Jendrej and Lawrie for the energy critical wave equation in a radial setting. For the Schrödinger equation, this conjecture is totally open. The main focus of this part of the project is to classify all possible solutions behaving asymptotically like a sum of bubbles multiplied by some phase plus a radiation term. We will focus on the radial case first. We will start by constructing solutions of the form sum of two bubbles or sum of a bubble and a radiation term. Then, we will prove that all the parameters in our construction are 'rigid' in the sense that if one is looking for instance at a solution decomposing asymptotically as a sum of two bubbles centered at 0 then the blow-up speed is unique and the phase between the two bubbles forms necessarily a right angle. Concerning the solution looking like a bubble plus a radiation, we should be able to obtain a continuum of blow-up speeds. This phenomenon is new even for the classical Schrödinger equation. The blow-up speeds in this case should also be rigid in the sense that they should only depend on the behavior of the radiation at the origin. Once the radial case is understood, we will deal with the general case. Most of the questions we consider in this program are already new for the classical NLS equation.

Institutions

Main Institutions

  • FCiênciasID Associação para a Investigação e Desenvolvimento de Ciências (Fciências.ID)

Funding 59.973,60 €

Fundação para a Ciência e a Tecnologia (FCT) - Portugal

59.973,60 €