Blow-up phenomena for critical nonlinear Schrödinger and Zakharov equations.
Recent progress on the blow-up problem of Zakharov equations
Asymptotics for minimal blow-up solutions of critical nonlinear Schrödinger equation
Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation
We consider the critical nonlinear Schrödinger equation with initial condition in dimension . For , local existence in time of solutions on an interval is known, and there exists finite time blow up solutions, that is such that . This is the smallest power in the nonlinearity for which blow up occurs, and is critical in this sense. The question we address is to understand the blow up dynamic. Even though there exists an explicit example of blow up solution and a class of initial data...
Multi solitary waves for nonlinear Schrödinger equations
Note on coupled linear systems related to two soliton collision for the quartic gKdV equation.
Dégénérescence du comportement linéaire pour l’équation des ondes semi-linéaire focalisante critique
C. Kenig et F. Merle ont montré que les solutions de l’équation des ondes focalisante quintique sur l’espace euclidien de dimension 3 ont un comportement linéaire en-dessous d’un certain seuil d’énergie. Ce comportement linéaire est caractérisé par la finitude de la norme dans les variables espace-temps. Dans cet exposé, je donnerai une estimation précise de cette norme globale pour les solutions dont l’énergie est proche de l’énergie seuil.
Existence de solutions explosives dans l’espace d’énergie pour l’équation de Korteweg–de Vries généralisée critique
Estimations uniformes à l’explosion pour les équations de la chaleur non linéaires et applications
Isolatedness of characteristic points at blow-up for a semilinear wave equation in one space dimension
We consider the semilinear wave equation with power nonlinearity in one space dimension. We first show the existence of a blow-up solution with a characteristic point. Then, we consider an arbitrary blow-up solution , the graph of its blow-up points and the set of all characteristic points and show that is locally finite. Finally, given , we show that in selfsimilar variables, the solution decomposes into a decoupled sum of (at least two) solitons, with alternate signs and that forms a...
On the collision of two solitons for the generalized KdV equation in the nonintegrable case
Blow up and near soliton dynamics for the critical gKdV equation
These notes present the main results of [, , ] concerning the mass critical (gKdV) equation for initial data in close to the soliton. These works revisit the blow up phenomenon close to the family of solitons in several directions: definition of the stable blow up and classification of all possible behaviors in a suitable functional setting, description of the minimal mass blow up in , construction of various exotic blow up rates in , including grow up in infinite time.
On the energy critical focusing non-linear wave equation
Two blow-up regimes for supercritical nonlinear Schrödinger equations
We consider the focusing nonlinear Schrödinger equations . We prove the existence of two finite time blow up dynamics in the supercritical case and provide for each a qualitative description of the singularity formation near the blow up time.
Blow up for the critical gKdV equation. II: Minimal mass dynamics
We consider the mass critical (gKdV) equation for initial data in . We first prove the existence and uniqueness in the energy space of a minimal mass blow up solution and give a sharp description of the corresponding blow up soliton-like bubble. We then show that this solution is the universal attractor of all solutions near the ground state which have a defocusing behavior. This allows us to sharpen the description of near soliton dynamics obtained in [29].
Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation
Consider the energy-critical focusing wave equation on the Euclidian space. A blow-up type II solution of this equation is a solution which has finite time of existence but stays bounded in the energy space. The aim of this work is to exhibit universal properties of such solutions. Let be the unique radial positive stationary solution of the equation. Our main result is that in dimension 3, under an appropriate smallness assumption, any type II blow-up radial solution is essentially the sum of...
Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case
Following our previous paper in the radial case, we consider type II blow-up solutions to the energy-critical focusing wave equation. Let W be the unique radial positive stationary solution of the equation. Up to the symmetries of the equation, under an appropriate smallness assumption, any type II blow-up solution is asymptotically a regular solution plus a rescaled Lorentz transform of concentrating at the origin.
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