Report

Teaser, summary, work performed and final results

Periodic Reporting for period 3 - SINGWAVES (Singularity formation in nonlinear evolution equations)

Teaser

The SingWave project is project addressing fundamental questions in the description of solutions to problems of modern physics. More specifically, we address the question of the singularity formation, and more generally energy concentration in the propagation of nonlinear...

Summary

The SingWave project is project addressing fundamental questions in the description of solutions to problems of modern physics. More specifically, we address the question of the singularity formation, and more generally energy concentration in the propagation of nonlinear waves. The singularity formation problem in the context of incompressible fluid mechanics is one of the famous Clay Millenium problems, and our aim is to develop the knowledge of such phenomenons on other probably simpler models to begin with, in particular nonlinear systems like the nonlinear schrodinger equation. Our aim is clearly to push much further the knowledge on these subjects, relying on tremendous progress done for the past 10 years, and we hope to be able to start connections with fluid like models within the end of the project. The understanding of the propagation of non-linear waves and energy concentration mechanisms can have tremendous impacts both on the understanding of wave propagation phenomenons in nonlinear optics, turbulence in fluid mechanics and more generally the formation of extreme events. The scope of our project is to focus so far on special type of bubble of energies, solitons, and their interaction.

Work performed

The main achievements of the ERC project Singwaves which started in august 2015 are the followings:

[Martel, Raphaël, Strongly interacting blow up bubbles for the mass critical NLS, to appear in Ann. Sci. Eco Norm Super.]: we construct of a new class of minimal blow up solutions for the mass critical nonlinear Schrödinger equation. A new and unexpected strong interaction mechanism of solitary waves is exhibited which is very promising for other models.

[Hadzic, Raphaël, Metling and freezing for the 2d radial Stefan problem, to appear in Jour. Eur. Math. Soc] This work is a breakthrough which designs a new functional framework for the study of type II blow up bubbles in the parabolic setting for energy critical and super critical problems, here applied to the classical problem of melting of an ice ball.

[Collot, Type II blow up manifolds for a super critical semi linear wave equation, to appear in Mem. Amer. Math. Soc], [Collot, Non radial type II blow up for the energy super critical semilinear heat equation, Anal. PDE 10 (2017), no 1, 127-252] Charles Collot is one of the main young collaborators of the projects and has defended his Phd in October 2016. In this series of works, he has extended the construction of type II super critical bubbles to the wave equation, and obtained the first truly non radial type II blow up for the energy super critical heat equation, hence showing the robustness of the method and is applicability beyond classical radial solutions.

[Collot, Merle, Raphaël, Dynamics near the ground state for the energy critical heat equation in large dimension, to appear in Comm. Math, Phys,], [Collot, Merle, Raphael, Stability of the ODE blow up for the energy critical semilinear heat equation, to appear in Compte Rendu Acad. Sciences]. We give a complete classification of the flow for small perturbations in the sharp energy topology of the ground state for the energy critical nonlinear heat equation in large dimension d>6. We prove that only three scenario are possible: dissipation and finite time type I blow up which are stable, and the threshold soliton dynamics.

[Collot, Raphaël, Szeftel, On the stability of type I blow up for the energy super critical heat equation, to appear in Mem. Amer. Math. Soc]. We construct using a bifurcation argument a family of self similar solutions for the energy super critical heat equation, and prove that the corresponding solution is the blow up profile of finite energy blow up solutions, with an associated finite codimensional stability. This is a fundamental first step towards a better understanding of the super critical self similar blow up which is essential in some fluid mechanics models.

[Lenzman, Gerard, Pocovnicu, Raphaël, A two soliton with transient turbulent regime for the cubic half wave equation on the real line, to appear in Annals of PDE\'s] In this monumental work, we construct a turbulent solution to the half wave equation for which a growth of Sobolev norm occurs which saturates after an explicit interaction time. The analysis proposes a new functional framework for the construction of turbulent bubbles through a two soliton interaction scenario.

[Naumkin I. and Raphäel P., On small travelling waves to the mass critical fractional NLS] We construct a generalised class of travelling wave solutions for the fractional NLS along with a complete description of the associated profile in the small mass limit.

[Collot, Merle, Raphael, Szeftel, On strongly anisotropic type II blow up for the energy super critical nonlinear heat equation, submitted] In this work, we provide a completely new functional setting for the construction of energy super critical blow up bubbles in the parabolic setting. We in particular understand a completely anisotropic phenomenon, and the fact that finite energy implies a point (and not a line) singularity.

Final results

The PI is developing in connexion with leading experts in the field (Rodnianski (Princeton), Merle (IHES), Szeftel (Paris 6)) a new approach for the description of completely new type of blow up bubbles, in direct connection with fluid mechanics. We hope to be able to report more specifically on these progress in the next scientific report.