In this paper, we give a systematic study of the boundary layer behavior for linear convection-diffusion equation in the zero viscosity limit. We analyze the boundary layer structures in the viscous solution and derive the boundary condition satisfied by the viscosity limit as a solution of the inviscid equation. The results confirm that the Neumann type of far-field boundary condition is preferred in the outlet and characteristic boundary condition. Under some appropriate regularity and compatibility conditions on the initial and boundary data, we obtain optimal error estimates between the full viscous solution and the inviscid solution with suitable boundary layer corrections. These results hold in arbitrary space dimensions and similar statements also hold for the strip problem. This model well describes the behavior at the far-field for many physical and engineering systems such as fluid dynamical equation and electro-magnetic equation. The results obtained here should provide some theoretical guidance for designing effective far-field boundary conditions.

This article is a sequel to our previous work [13] concerned with the derivation of high-order homogenized models for the Stokes equation in a periodic porous medium. We provide an improved asymptotic analysis of the coefficients of the higher order models in the low-volume fraction regime whereby the periodic obstacles are rescaled by a factor η which converges to zero. By introducing a new family of order k corrector tensors with a controlled growth as η→0 uniform in k∈\N, we are able to show that both the infinite order and the finite order models converge in a coefficient-wise sense to the three classical asymptotic regimes. Namely, we retrieve the Darcy model, the Brinkman equation or the Stokes equation in the homogeneous cubic domain depending on whether η is respectively larger, proportional to, or smaller than the critical size η_{crit}∼ϵ2/(d−2). For completeness, the paper first establishes the analogous results for the perforated Poisson equation, considered as a simplified scalar version of the Stokes system.

We obtain the existence and uniqueness of solutions for a class of retarded parabolic equations with superlinear perturbations. The asymptotic behavior result is studied by using the pullback attractor framework.

In this paper, we consider the three-dimensional Schrdinger operator with a -interaction of strength > 0 supported on an unbounded surface parametrized by the mapping R2x(x,f(x)), where 0, and f:R2R, f 0, is a C2-smooth, compactly supported function. The surface supporting the interaction can be viewed as a local deformation of the plane. It is known that the essential spectrum of this Schrdinger operator coincides with 142,+. We prove that for all sufficiently small > 0, its discrete spectrum is non-empty and consists of a unique simple eigenvalue. Moreover, we obtain an asymptotic expansion of this eigenvalue in the limit 0+. In particular, this eigenvalue tends to 142 exponentially fast as 0+.

In this paper, we study the existence of stationary solutions to the VlasovPoissonBoltzmann system when the background density function tends to a positive constant with a very mild decay rate as| x|. In fact, the stationary VlasovPoissonBoltzmann system can be written into an elliptic equation with exponential nonlinearity. Under the assumption on the decay rate being (ln (e+| x|)) for some > 0, it is shown that this elliptic equation has a unique solution. This result generalizes the previous work [R. Glassey, J. Schaeffer, Y. Zheng, Steady states of the VlasovPoissonFokkerPlanck system, J. Math. Anal. Appl. 202 (1996) 10581075] where the decay rate (1+| x|) 1 2 is assumed.