Initial results from new calculations of interacting anti-parallel Euler vortices are presented with the objective of understanding the origins of singular scaling presented by Kerr (1993) and the lack thereof by Hou and Li (2006). Core profiles designed to reproduce the two results are presented, new more robust
analysis is proposed, and new criteria for when calculations should be terminated are given. Most of the analysis is on a $512\times 128 \times 2048$ mesh, with new analysis on a just completed $1024\times 256\times 2048$ used to confirm trends. The qualitative conclusions of Kerr (1993) are supported, but most of the proposed scaling laws will have to be modified. Assume enstrophy growth like $\Omega\sim (T_c-t)^{-\gamma_\Omega}$ and vorticity growth like $||\omega||_\infty \sim (T_c-t)^{-\gamma}$. Present results would support $\gamma_\Omega\rightarrow 1/4-1/2$ and $\gamma>$. The results are not conclusive since they require higher resolution calculations (work in progress) to further confirm the trends.
Classical ideal fluid motion is described by Euler and Navier-Stokes equations. For real fluids, their motions are more complicated and governed by Euler and Navier-Stokes equations coupled with various constitutive equations. We study viscoelastic models whose motions are
carried out by the competition between the kinetic energies and internal elastic energies. The deformation tensor plays an essential role in our studies. We will present how to use the heuristics coming from the special
structure of the deformation tensor to establish the global well-posedness results for several viscoelastic models, but will focus on a 2D Strain-Rotation model.
In this talk, I will present recent results on wave localization in nonlinear random media in the frame work of the stochastic Gross-Pitaevskii equation (describing Bose-Einstein condensation). In particular, it is shown numerically that the disorder average spatial extension of the stationary density profile decreases with
an increasing strength of the disordered potential both for repulsive and attractive interactions.
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.
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.
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.
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.