Singularity formation in nonlinear evolution equations

Project Information

SINGWAVES

Grant agreement ID: 646650

DOI

10.3030/646650

Project closed

EC signature date 4 May 2015

Start date 1 August 2015

End date 31 January 2021

Funded under

• EXCELLENT SCIENCE - European Research Council (ERC)

Total cost

€ 1 211 055,00

EU contribution

€ 1 211 055,00

Coordinated by

THE CHANCELLOR MASTERS AND SCHOLARS OF THE UNIVERSITY OF CAMBRIDGE
United Kingdom

Periodic Reporting for period 4 - SINGWAVES (Singularity formation in nonlinear evolution equations)

Reporting period: 2019-03-01 to 2021-01-31

Summary of the context and overall objectives of the project

The SingWave project aims at 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 the 6th 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 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 from the beginning of the project to the end of the period covered by the report and main results achieved so far

The main achievements of the ERC project Singwaves 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 which relies on solitary waves collision
--[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.
--[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.
--[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.
--[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.
--[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] 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.
--[Naumkin I. and Raphäel P., On small travelling waves to the mass critical fractional NLS, to appear in Ann Henri Poincarre] 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, to appear in Jour Amer Math Soc] 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.
--[Bahri, Martel. Raphael, Self-similar blow-up profiles for slightly supercritical nonlinear Schrödinger equations, archive 2019] We construct the self similar solution to the slightly mass super critical NLS model which is a delicate ode problem and solves a classical problem in the field.
--[Merle, Raphael, Rodnisanski, Szeftel, On smooth self similar solutions to the compressible Euler equations, archiv 2019] We construct smooth self similar solutions to the compressible Euler equations in a suitable regime of parameters and derive repulsively properties for the associated linearized operator. Smoothness is the fundamental new ingredient which is central in the further dynamical study of these profiles.
--[Merle, Raphael, Rodnisanski, Szeftel,On blow up for the energy super critical defocusing non linear Schrödinger equations, archiv 2019] We prove the existence of blow up solutions for the defocusing NLS in a suitable energy critical range of parameters, hence answering a major open problem in the field. Blow up occurs by implosion in the associated hydrodynamical variables which yield a highly oscillatory singularity.
--[Merle, Raphael, Rodnisanski, Szeftel, On the implosion of a three dimensional compressible fluid, archiv 2019] This breakthrough work provides the first description of viscous compressible three dimensional shock wave.
--[Faou, Raphael, On weakly turbulent solutions to the perturbed linear Harmonic oscillator, archival 2020] We construct the first explicit example of smooth, well localised and asympotically vanishing in time potential which induces a growth of Sobolev norm for the corresonding perturbed harmonic oscillator. The proof uses a new approach based on pseudo conformal bubbles which is very promising for further applications to non linear problems.

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)

The two preprints [Merle, Raphael, Rodnisanski, Szeftel, On the implosion of a three dimensional compressible fluid, archiv 2019], [Merle, Raphael, Rodnisanski, Szeftel, On the implosion of a three dimensional compressible fluid, archiv 2019] are the main achievements of the SingWave project and go far beyond the state of art and the initial expectation of the scientific project. In the first paper, we prove the existence of blow up solutions for non linear Schrodinger models in the defocusing case which was completely unexpected and argues against a conjecture formulated in 2000 by Jean Bourgain. In the second preprint, we use a similar approach to obtain the first description of viscous shock waves for compressible fluids in three dimensions. It was widely believed that viscosity should have a fundamental impact on the singularity formation, as is one dimension, and we prove exactly the opposite: in suitable regimes, viscosity can be neglected, but this requires a deep and new understanding of the underlying Eulerian dynamics. This result illustrates beyond expectation again the universality of our approach and its applicability to fluid mechanics equations, and brings our field a step closer to the 6th Clay Millenium on singularity formation for incompressible fluids.

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