Pierre Raphael

Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation

Abstract.We consider finite time blow up solutions to the critical nonlinear Schrödinger equation with initial condition u0 ∈ H1. Existence of such solutions is known, but the complete blow up dynamic is not understood so far. For initial data with negative energy, finite time blow up with a universal sharp upper bound on the blow up rate corresponding to the so-called log-log law has been proved in [10], [11]. We focus in this paper onto the positive energy case where at least two blow up speeds are known to possibly occur. We establish the stability in energy space H1 of the log-log upper bound exhibited in the negative energy case, and a sharp lower bound on blow up rate in the other regime which corresponds to known explicit blow up solutions.

Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS

We consider the 2-dimensional focusing mass critical NLS with an inhomogeneous nonlinearity:

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

i\partial_tu+\Delta u+k(x)|u|^{2}u=0

Quantized slow blow-up dynamics for the corotational energy-critical harmonic heat flow

. From standard argument, there exists a threshold

ON STABILITY OF PSEUDO-CONFORMAL BLOWUP FOR L 2 -CRITICAL HARTREE NLS

M_k>0

Stable self-similar blow up dynamics for the three dimensional relativistic gravitational Vlasov-Poisson system

such that

ON ONE BLOW UP POINT SOLUTIONS TO THE CRITICAL NONLINEAR SCHRÖDINGER EQUATION

H^1

Stable Ground States for the Relativistic Gravitational Vlasov–Poisson System

solutions with

On Stability of Pseudo-Conformal Blowup for L2-critical Hartree NLS

\|u\|_{L^2} M_k

Structure of the Linearized Gravitational Vlasov–Poisson System Close to a Polytropic Ground State

. In this paper, we consider the dynamics at threshold