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.

For the viscous and heat-conductive fluids governed by the compressible NavierStokes equations with an external potential force, there exist non-trivial stationary solutions with zero velocity. By combining the L^p - L^q estimates for the linearized equations and an elaborate energy method, the convergence rates are obtained in various norms for the solution to the stationary profile in the whole space when the initial perturbation of the stationary solution and the potential force are small in some Sobolev norms. More precisely, the optimal convergence rates of the solution and its first order derivatives in L²-norm are obtained when the L¹-norm of the perturbation is bounded.

We study the convergence rate of Glimm scheme for general systems of hyperbolic conservation laws without the assumption that each characteristic field is either genuinely nonlinear or linearly degenerate. We first give a sharper estimate of the error arising from the wave tracing argument by a careful analysis of the interaction between small waves. With this key estimate, the convergence rate is shown to be o (1) s 1 3| ln s| 1+ , which is sharper compared to o (1) s 1 4| ln s| given in [T. Yang, Convergence rate of Glimm scheme for general systems of hyperbolic conservation laws, Taiwanese J. Math. 7 (2)(2003) 195205]. However, it is still slower than o (1) s 1 2| ln s| given in [A. Bressan, A. Marson, Error bounds for a deterministic version of the Glimm scheme, Arch. Ration. Mech. Anal. 142 (2)(1998) 155176] for systems with each characteristic field being genuinely nonlinear or linearly degenerate. Here s is the

A semi-explicit formula of solution to the boundary layer system for thermal layer derived from the compressible Navier-Stokes equations with the non-slip boundary condition when the viscosity coefficients vanish is given, in particular in three space dimension. In contrast to the inviscid Prandtl system studied by [7] in two space dimension, the main difficulty comes from the coupling of the velocity field and the temperature field through a degenerate parabolic equation. The convergence of these boundary layer equations to the inviscid Prandtl system is justified when the initial temperature goes to a constant. Moreover, the time asymptotic stability of the linearized system around a shear flow is given, and in particular, it shows that in three space dimension, the asymptotic stability depends on whether the direction of tangential velocity field of the shear flow is invariant in the normal direction respective to the boundary.

The existence of solution for the 2D-Keller-Segel system in the subcritical case, i.e. when the initial mass is less than 8π, is reproved. Instead of using the entropy in the free energy and free energy dissipation, which was used in the proofs (Blanchet et al. in SIAM J. Numer. Anal. 46:691–721, 2008; Electron. J. Differ. Equ. Conf. 44:32, 2006 (electronic)), the potential energy term is fully utilized by adapting Delort’s theory on 2D incompressible Euler equation (Delort in J. Am. Math. Soc. 4:553–386, 1991).