Courant Institute of Mathematical Sciences

About: Courant Institute of Mathematical Sciences is a education organization based out in New York, New York, United States. It is known for research contribution in the topics: Nonlinear system & Boundary value problem. The organization has 2414 authors who have published 7759 publications receiving 439773 citations. The organization is also known as: CIMS & New York University Department of Mathematics.

Low Froude number limiting dynamics for stably stratified flow with small or finite Rossby numbers

Abstract: Recent numerical simulations reveal remarkably different behavior in rotating stably stratified fluids at low Froude numbers for finite Rossby numbers as compared with the behavior at both low Froude and Rossby numbers. Here the reduced low Froude number limiting dynamics in both of these situations is developed with complete mathematical rigor by applying the theory for fast wave averaging for geophysical flows developed recently by the authors. The reduced dynamical equations include all resonant triad interactions for the slow (vortical) modes, the effect of the slow (vortical) modes on the fast (inertial gravity) modes, and also the general resonant triad interactions among the fast (internal gravity) waves. The nature of the reduced dynamics in these two situations is compared and contrasted here. For example, the reduced slow dynamics for the vortical modes in the low Froude number limit at finite Rossby numbers includes vertically sheared horizontal motion while the reduced slow dynamics i...

Kinetic Equations and Density Expansions: Exactly Solvable One-Dimensional System

Abstract: We have made a detailed study of the time evolution of the distribution function $f(q,v,t)$ of a labeled (test) particle in a one-dimensional system of hard rods of diameter $a$. The system has a density $\ensuremath{\rho}$ and is in equilibrium at $t=0$. (Some properties of this system were studied earlier by Jepsen.) When the distribution function $f$ at $t=0$ corresponds to a delta function in position and velocity, then $f(q,v,t)$ is essentially the time-displaced self-distribution function ${f}_{s}$. This function ${f}_{s}$ (which can be found in an explicit closed form) and all of the system properties which can be derived from it depend on $\ensuremath{\rho}$ and $a$ only through the combination $n=\frac{\ensuremath{\rho}}{(1\ensuremath{-}\ensuremath{\rho}a)}$. In particular, the diffusion constant $D$ is given by ${D}^{\ensuremath{-}1}={\mathrm{lim}}_{s\ensuremath{\rightarrow}0}{[\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\psi}}(s)]}^{\ensuremath{-}1}={(2\ensuremath{\pi}\ensuremath{\beta}m)}^{\frac{1}{2}}n$, where $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\psi}}(s)$ is the Laplace transform of the velocity autocorrelation function $\ensuremath{\psi}(t)=〈v(t)v〉$. An expansion of ${[\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\psi}}(s)]}^{\ensuremath{-}1}$ in powers of $n$, on the other hand, has the form $\ensuremath{\Sigma}\frac{{B}_{l}{n}^{l}}{{s}^{l\ensuremath{-}1}}$, leading to divergence of the density coefficients for $l\ensuremath{\ge}2$ when $s\ensuremath{\rightarrow}0$. This is similar to the divergences found in higher dimensional systems. Similar results are found as well in the expansion of the collision operator describing the time evolution of $f(q,v,t)$. The lowest-order term in the expansion is the ordinary (linear) Boltzmann equation, while higher terms are $O({\ensuremath{\rho}}^{l}{t}^{l\ensuremath{-}1})$. Thus any attempt to write a Bogoliubov, Choh-Uhlenbeck-type Markoffian kinetic equation as a power series in the density leads to divergence in the terms beyond the Boltzmann equation. A Markoffian collision operator can, however, be constructed, without using a density expansion, which, e.g., describes the stationary distribution of a charged test particle in the system in the presence of a constant electric field. The distribution of the test particle in the presence of an oscillating external field is also found. Finally, the short- and long-time behavior of the self-distribution is examined.

Remarks on the global sobolev inequalities in the minkowski space Rn+1

Abstract: On demontre des inegalites de Sobolev globales sur l'espace de Minkowski R n+1 . Le theoreme principal est valable sur des tranches a temps fixe

Spectra of hyperbolic surfaces

Abstract: These notes attempt to describe some aspects of the spectral theory of modular surfaces. They are by no means a complete survey.