We revisit the homogenization problem for the Poisson equation in periodically perforated domains with zero Neumann data at the boundary of the holes and prescribed Dirichlet data at the outer boundary. It is known that, if the periodicity of the holes goes to zero but their volume fraction remains fixed and positive, the limit problem is a Dirichlet boundary value problem posed in the domain without the holes, and the effective diffusion coefficients are non-trivially modified; if that volume fraction goes to zero instead, i.e. the holes are dilute, the effective operator remains the Laplacian (that is, unmodified). Our main results contain the study of a "continuity" in those effective models with respect to the volume fraction of the holes and some new convergence rates for homogenization in the dilute setting. Our method explores the classical two-scale expansion ansatz and relies on asymptotic analysis of the rescaled cell problems using layer potential theory.

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We study the critical dissipative quasi-geostrophic equations in $\bR^ 2$ with arbitrary H^ 1 initial data. After showing certain decay estimate, a global well-posedness result is proved by adapting the method in [11] with a suitable modification. A decay in time estimate for higher order homogeneous Sobolev norms of solutions is also discussed.

In this paper, we are concerned with the one-species VlasovPoissonBoltzmann system with a nonconstant background density in full space. There exists a stationary solution when the background density is a small perturbation of a positive constant state. We prove the nonlinear stability of solutions to the Cauchy problem near the stationary state in some Sobolev space without any time derivatives. This result is nontrivial even when the background density is a constant state. In the proof, the macroscopic balance laws are essentially used to deal with the a priori estimates on both the microscopic and macroscopic parts of the solution. Moreover, some interactive energy functionals are introduced to overcome difficulty stemming from the absence of time derivatives in the energy functional.

A computational method, based on ₁-minimization, is proposed for the problem of link flow correction, when the available traffic flow data on many links in a road network are inconsistent with respect to the flow conservation law. Without extra information, the problem is generally ill-posed when a large portion of the link sensors are unhealthy. It is possible, however, to correct the corrupted link flows accurately with the proposed method under a recoverability condition if there are only a few bad sensors which are located at certain links. We analytically identify the links that are robust to miscounts and relate them to the geometric structure of the traffic network by introducing the recoverability concept and an algorithm for computing it. The recoverability condition for corrupted links is simply the associated recoverability being greater than 1. In a more realistic setting, besides the unhealthy link sensors, small