Final Report Summary - DISPEQ (Qualitative study of nonlinear dispersive equations)
The project Dispeq deals with three important aspects of the qualitative behavior of nonlinear partial differential equations of dispersive type. The first aspect is concerned with the propagation of random waves and the study of the weak turbulence phenomenon. The second aspect is the study of soliton type solutions. The third aspect is the study of the energy critical NLS in non-euclidean geometry.
Concerning the first aspect, there are three main achievements. The first one is the works by Deng-Tzvetkov-Visciglia establishing an infinite sequence of invariant measures for the Benjamin-Ono equation. In particular, these works prove a recurrence property for smooth and large solutions. Such type of results are very rare in the field. The second achievement is the work by Burq-Tzvetkov establishing probabilistic global well-posedness for non linear wave equations with data of super-critical regularity. For such super-critical regularity, the deterministic approach fails since the problem was known to be ill posed in the classical Hadamard sense. These type of results are close in spirit to the work by Hairer on nonlinear parabolic PDE's with a singular stochastic source. The third achievement is the first true weak turbulence result for the non linear Schroedinger equation. More precisely, the work by Hani-Pausader-Tzvetkov-Visciglia constructs solutions with growing higher Sobolev norms. This gives a partial solution of a problem posed by Bourgain in 2000.
Concerning the second aspect, there are three main achievements. The first one is the work by Duyckaerts-Kenig-Merle showing soliton resolution results for problems which are not tractable by the methodology of the integrable systems such as the inverse scattering transform. The second one is the proof by Mizumachi-Tzvetkov of the transverse stability of the line soliton for the KP-II equation with respect to periodic transverse perturbation. The third one is the work by Rousset-Tzvetkov giving the evidence of a critical threshold in the context of transverse stability problems.
Since the start of the project, our understanding on the energy critical nonlinear Schroedinger equations on compact spatial domains has been considerably sharpen. Previous work by Bourgain and Burq-Gerard-Tzvetkov gave some quite satisfactory results in the sub-critical cases. Thanks to the initial breakthrough by Herr-Tataru-Tzvetkov and the subsequent articles on the subject, we now have a satisfactory theory in the critical case when the problem is posed on the three dimensional sphere or the three dimensional torus.
Concerning the first aspect, there are three main achievements. The first one is the works by Deng-Tzvetkov-Visciglia establishing an infinite sequence of invariant measures for the Benjamin-Ono equation. In particular, these works prove a recurrence property for smooth and large solutions. Such type of results are very rare in the field. The second achievement is the work by Burq-Tzvetkov establishing probabilistic global well-posedness for non linear wave equations with data of super-critical regularity. For such super-critical regularity, the deterministic approach fails since the problem was known to be ill posed in the classical Hadamard sense. These type of results are close in spirit to the work by Hairer on nonlinear parabolic PDE's with a singular stochastic source. The third achievement is the first true weak turbulence result for the non linear Schroedinger equation. More precisely, the work by Hani-Pausader-Tzvetkov-Visciglia constructs solutions with growing higher Sobolev norms. This gives a partial solution of a problem posed by Bourgain in 2000.
Concerning the second aspect, there are three main achievements. The first one is the work by Duyckaerts-Kenig-Merle showing soliton resolution results for problems which are not tractable by the methodology of the integrable systems such as the inverse scattering transform. The second one is the proof by Mizumachi-Tzvetkov of the transverse stability of the line soliton for the KP-II equation with respect to periodic transverse perturbation. The third one is the work by Rousset-Tzvetkov giving the evidence of a critical threshold in the context of transverse stability problems.
Since the start of the project, our understanding on the energy critical nonlinear Schroedinger equations on compact spatial domains has been considerably sharpen. Previous work by Bourgain and Burq-Gerard-Tzvetkov gave some quite satisfactory results in the sub-critical cases. Thanks to the initial breakthrough by Herr-Tataru-Tzvetkov and the subsequent articles on the subject, we now have a satisfactory theory in the critical case when the problem is posed on the three dimensional sphere or the three dimensional torus.