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.
