Ordinary differential equations Constant coefficient linear dispersive equations Semilinear dispersive equations The Korteweg de Vries equation Energy-critical semilinear dispersive equations Wave maps Tools from harmonic analysis Construction of ground states Bibliography.

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Nonlinear dispersive equations : local and global analysis

We prove, for the energy critical, focusing NLS, that for data whose energy is smaller than that of the standing wave, and whose homogeneous Sobolev norm H^1 is smaller than that of the standing wave and which is radial, we have global well-posedness and scattering in dimensions 3, 4 and 5. This is sharp since if the data is in the inhomogeneous Sobolev space H^1, of energy smaller than the standing wave but of larger homogeneous H^1 norm, we have blow-up in finite time. The result follows from a general method that we introduce into this type of critical problem. By concentration-compactness we produce a critical element, which modulo the symmetries of the equation is compact, has minimal energy among those which fail to have the conclusion of our theorem. In addition, we show that the dilation parameter in the symmetry, for this solution, can be taken strictly positive.We then establish a rigidity theorem that shows that no such compact, modulo symmetries, object can exist. It is only at this step that we use the radial hypothesis.The same analysis, in a simplified form, applies also to the defocusing case, giving a new proof of results of Bourgain and Tao.

Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case

Preliminaries.- Model Elliptic Problems.- Model Parabolic Problems.- Systems.- Equations with Gradient Terms.- Nonlocal Problems.

Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States

Physics of shock waves and high-temperature hydrodynamic phenomena - Ya.B. Zeldovich and Yu.P. Raizer (translated from the Russian and then edited by Wallace D. Hayes and Ronald F. Probstein); Dover Publications, New York, 2002, 944 pp., $34.

Reviews - Eigenfunction Expansions Associated with Second-Order Differential Equations. By E. C. Titchmarsh Pp. 175. 20s. 1946. (Oxford University)

We consider the critical nonlinear Schr?odinger equation iut = -.u-|u| 4 N u with initial condition u(0, x) = u0 in dimension N = 1. For u0 . H1, local existence in the time of solutions on an interval [0, T) is known, and there exist finite time blow-up solutions, that is, u0 such that limt.T 1.

http://annals.math.princeton.edu/wp-content/uploads/annals-v161-n1-p05.pdf

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

We consider finite time blow-up solutions to the critical nonlinear Schrodinger equation iut=-Δu-|u|4/Nu with initial condition u0∈H1. Existence of such solutions is known, but the complete blow-up dynamic is not understood so far. For a specific set of initial data, finite time blow-up with a universal sharp upper bound on the blow-up rate has been proved in [22], [23].

On universality of blow-up profile for L 2 critical nonlinear Schrödinger equation

We consider the critical nonlinear Schrodinger equation $iu_{t} = -\Delta u-|u|^{4/N}$ with initial condition u(0, x) = u0

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Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation

We consider finite time blow up solutions to the critical nonlinear Schrodinger equation Open image in new window For a suitable class of initial data in the energy space H1, we prove that the solution splits in two parts: the first part corresponds to the singular part and accumulates a quantized amount of L2 mass at the blow up point, the second part corresponds to the regular part and has a strong L2 limit at blow up time.

Profiles and Quantization of the Blow Up Mass for Critical Nonlinear Schrödinger Equation

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 critical equivariant SO(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.

Pierre Raphaël

Papers

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

On universality of blow-up profile for L 2 critical nonlinear Schrödinger equation

Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation

Profiles and Quantization of the Blow Up Mass for Critical Nonlinear Schrödinger Equation

Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems