Hydrodynamic limit for attractive particle systems on 417-1417-1417-1

We study the hydrodynamic behavior of asymmetric simple exclusions and zero range processes in several dimensions. Under Euler scaling, a nonlinear conservation law is derived for the time evolution of the macroscopic particle density.Research supported by NSF Grant No. DMS-8806731. Present address: Department of Mathematics, University of California at Berkeley, Berkeley, CA 94720, USA     

A Stochastic Model Associated with Enskog Equation and Its Kinetic Limit

 We examine a system of particles in which the particles travel deterministically in between stochastic collisions. The collisions are elastic and occur with probability ɛ ^d when two particles are at a distance σ. When the number of particles N goes to infinity and Nɛ ^d goes to a nonzero constant, we show that the particle density converges to a solution of the Enskog Equation. Received: 29 January 2002 / Accepted: 30 July 2002 Published online: 14 November 2002 RID="*" ID="*" Research supported in part by NSF Grant DMS-0072666     

Kinetic Limit for a System of Coagulating Planar Brownian Particles

We study a model of mass-bearing coagulating planar Brownian particles. The coagulation occurs when two particles are within a distance of order ε. We assume that the initial number of particles N is of order |logε|. Under suitable assumptions of the initial distribution of particles and the microscopic coagulation propensities, we show that the macroscopic particle densities satisfy a Smoluchowski-type equation.     

Moment Bounds for the Smoluchowski Equation and their Consequences

We prove bounds on moments of the Smoluchowski coagulation equations with diffusion, in any dimension d ≥ 1. If the collision propensities α(n, m) of mass n and mass m particles grow more slowly than , and the diffusion rate is non-increasing and satisfies for some b ₁ and b ₂ satisfying 0 ≤ b ₂ < b ₁ < ∞, then any weak solution satisfies for every and T ∈(0, ∞), (provided that certain moments of the initial data are finite). As a consequence, we infer that these conditions are sufficient to ensure uniqueness of a weak solution and its conservation of mass. This work was performed while A.H. held a postdoctoral fellowship in the Department of Mathematics at U.B.C. This work is supported in part by NSF grant DMS0307021.     

Pointwise Bounds for the Solutions of the Smoluchowski Equation with Diffusion

We prove various decay bounds on solutions (f _n : n > 0) of the discrete and continuous Smoluchowski equations with diffusion. More precisely, we establish pointwise upper bounds on n ^? f _n in terms of a suitable average of the moments of the initial data for every positive ?. As a consequence, we can formulate sufficient conditions on the initial data to guarantee the finiteness of ${L^p(\mathbb{R}^d \times [0, T])}$ norms of the moments ${X_a(x, t) := \sum_{m\in\mathbb{N}}m^a f_m(x, t)}$ , ( ${\int_0^{\infty} m^a f_m(x, t)dm}$ in the case of continuous Smoluchowski’s equation) for every ${p \in [1, \infty]}$ . In previous papers [11] and [5] we proved similar results for all weak solutions to the Smoluchowski’s equation provided that the diffusion coefficient d(n) is non-increasing as a function of the mass. In this paper we apply a new method to treat general diffusion coefficients and our bounds are expressed in terms of an auxiliary function ${\phi(n)}$ that is closely related to the total increase of the diffusion coefficient in the interval (0, n].     

Metabelian Q1-groups

A finite group G is called a ${\mathbb{Q}}_1$-group if all of its non-linear irreducible characters are rational valued. In this paper, we will find the general structure of a metabelian ${\mathbb{Q}}_1$-group.     

Homogenization for¶Stochastic Hamilton-Jacobi Equations

Homogenization asks whether average behavior can be discerned from partial differential equations that are subject to high-frequency fluctuations when those fluctuations result from a dependence on two widely separated spatial scales. We prove homogenization for certain stochastic Hamilton-Jacobi partial differential equations; the idea is to use the subadditive ergodic theorem to establish the existence of an average in the infinite scale-separation limit. In some cases, we also establish a central limit theorem. Accepted: (April 23, 1999)     

Hydrodynamic limit for a system with finite range interactions

We study a system of interacting diffusions. The variables present the amount of charge at various sites of a periodic multidimensional lattice. The equilibrium states of the diffusion are canonical Gibbs measures of a given finite range interaction. Under an appropriate scaling of lattice spacing and time, we derive the hydrodynamic limit for the evolution of the macroscopic charge density.     

Hydrodynamic limit for particle systems with nonconstant speed parameter

We establish the hydrodynamic limit for a class of particle systems on ℤ ^d with nonconstant speed parameter, assuming that the speed parameter is continuously differentiable in the spatial variable. If the particle system is on the one-dimensional latticeℤ and totally asymmetric, we derive the hydrodynamic equation for continuous speed parameters. We obtain nontrivial upper and lower bounds when either the speed parameter is discontinuous or there is a blockage at a fixed site.