# Recent progress on the L^{2}-critical, defocusing semilinear Schr\"odinger equation

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In this talk I will describe the progress that has been made so far

concerning the existence of global strong solutions to the

L^{2}-critical defocusing semilinear Schr\"odinger equation. A long

standing conjecture in the area is the existence

of a unique global strong L^{2}

solution to the equation that in addition scatters to a free solution as

time goes to infinity. I will demonstrate the proofs of partial results

towards an attempt for a final resolution of this conjecture.

I will concentrate on the low dimensions but give the flavor of the results in higher dimensions for general or spherically symmetric initial data in certain Sobolev spaces.

Many authors have contributed to the theory of this equation. I will convey my personal involment to the problem and the results that I have obtained recently. Part of my work is in collaboration with D. De Silva, N. Pavlovic, G.

Staffilani, J. Colliander and M. Grillakis.

# On a splitting scheme for the nonlinear Schroedinger equation in a random medium.

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We consider a nonlinear Schroedinger equation (NLS) with random coefficients, in a regime of separation of scales corresponding to diffusion approximation. Our primary goal is to propose and study an efficient numerical scheme in this framework. We use a pseudo-spectral splitting scheme and we establish the order

of the global error. In particular we show that we can take an integration step larger than the smallest scale of the problem, here the correlation length of the random medium. Then, we study the asymptotic behavior of the numerical solution in the diffusion approximation regime.

# On recent development on the Green's functions of the Boltzmann equations and the applications to nonlinear problems

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In this talk we will survey the development on the Green's functions of the Boltzmann equations. The talk will include the motivation from the field of hyperbolic conservation laws, the connection between the Boltzmann equation

and the hyperbolic conservation laws, and the particle-like and the wave-like duality in the Boltzmann equation. With all these components one can realize a clear layout of the Green's function of the Boltzmann equation. Finally we will present the application of the Green's function the an initial-boundary value problem in the half space domain.

# Global Existence in 3d Nonlinear Elastodynamics

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We will discuss the equations of motion for 3d homogeneous isotropic elastic materials, in the compressible and incompressible case. We will present results on global existence of solutions to the initial value problem, under the assumption of small deformations and with appropriate structural conditions.