ABSTRACT We investigate a kinetic model for the sedimentation of dilute suspensions of rod-like particles under gravity, deduced by Helzel, Otto, and Tzavaras (2011), which couples the impressible (Navier-)Stokes equation with the Fokker-Planck equation. With a no-flux boundary condition for the distribution function, we establish the existence and uniqueness of a global weak solution to the two dimensional model involving the Stokes equation.

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

In the paper, we study the three-dimensional Prandtl equations, and prove that if one component of the tangential velocity field satisfies the monotonicity assumption in the normal direction, then the system is locally well-posed in the Gevrey function space with Gevrey index in ] 1, 2]. The proof relies on some new cancellation mechanism in the system in addition to those observed in the two-dimensional setting.

We investigate Lamé systems in periodically perforated domains, and establish quantitative homogenization results in the setting where the domain is clamped at the boundary of the holes. Our method is based on layer potentials and it provides a unified proof for various regimes of hole-cell ratios (the ratio between the size of the holes and the size of the periodic cells), and, more importantly, it yields natural correctors that facilitate error estimates. A key ingredient is the asymptotic analysis for the rescaled cell problems, and this is studied by exploring the convergence of the periodic layer potentials for the Lamé system to those in the whole space when the period tends to infinity.

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