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Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation

Frank MerlePierre Raphael — 2002

Journées équations aux dérivées partielles

We consider the critical nonlinear Schrödinger equation i u t = - Δ u - | u | 4 N u with initial condition u ( 0 , x ) = u 0 in dimension N . For u 0 H 1 , local existence in time of solutions on an interval [ 0 , T ) is known, and there exists finite time blow up solutions, that is u 0 such that lim t T < + | u x ( t ) | L 2 = + . 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...

Stable blow up dynamics for the critical co-rotational Wave Maps and equivariant Yang-Mills Problems

Pierre RaphaëlIgor Rodnianski

Séminaire Équations aux dérivées partielles

This note summarizes the results obtained in []. We exhibit stable finite time blow up regimes for the energy critical co-rotational Wave Map with the 𝕊 2 target in all homotopy classes and for the equivariant critical S O ( 4 ) Yang-Mills problem. We derive sharp asymptotics on the dynamics at blow up time and prove quantization of the energy focused at the singularity.

Blow up and near soliton dynamics for the L 2 critical gKdV equation

Yvan MartelFrank MerlePierre Raphaël

Séminaire Laurent Schwartz — EDP et applications

These notes present the main results of [, , ] concerning the mass critical (gKdV) equation u t + ( u x x + u 5 ) x = 0 for initial data in H 1 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 H 1 , construction of various exotic blow up rates in H 1 , including grow up in infinite time.

Blow up for the critical gKdV equation. II: Minimal mass dynamics

Yvan MartelFrank MerlePierre Raphaël — 2015

Journal of the European Mathematical Society

We consider the mass critical (gKdV) equation u t + ( u x x + u 5 ) x = 0 for initial data in H 1 . 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].

Local estimation of the Hurst index of multifractional brownian motion by increment ratio statistic method

Pierre Raphaël BertrandMehdi FhimaArnaud Guillin — 2013

ESAIM: Probability and Statistics

We investigate here the central limit theorem of the increment ratio statistic of a multifractional Brownian motion, leading to a CLT for the time varying Hurst index. The proofs are quite simple relying on Breuer–Major theorems and an original strategy. A simulation study shows the goodness of fit of this estimator.

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