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### Open Access Journals

NHM

We consider the Navier-Stokes equations of an incompressible fluid in a three dimensional curved domain with permeable walls in the limit of small viscosity.
Using a curvilinear coordinate system, adapted to the boundary, we construct a corrector function at order $ε^j$, $j=0,1$, where $ε$ is the (small) viscosity parameter.
This allows us to obtain an asymptotic expansion of the Navier-Stokes solution at order $ε^j$, $j=0,1$, for $ε$ small .
Using the asymptotic expansion, we prove that the Navier-Stokes solutions converge, as the viscosity parameter tends to zero, to the corresponding Euler solution in the natural energy norm.
This work generalizes earlier results in [14] or [26], which discussed the case of a channel domain, while here the domain is curved.

CPAA

In this article, we establish the asymptotic behavior, when the
viscosity goes to zero, of the solutions of the Linearized Primitive
Equations (LPEs) in space dimension $2$. More precisely, we prove
that the LPEs solution behaves like the corresponding inviscid
problem solution inside the domain plus an explicit corrector
function in the neighborhood of some parts of the boundary. Two
cases are considered, the subcritical and supercritical modes
depending on the fact that the frequency mode is less or greater
than the ratio between the reference stratified flow (around which
we linearized) and the buoyancy frequency. The problem of boundary
layers for the LPEs is of a new type since the corresponding limit
problem displays a set of (unusual) nonlocal boundary conditions.

DCDS-S

In this article, we give an asymptotic expansion, with respect to
the viscosity which is considered here to be small, of the solutions
of the $3D$ linearized Primitive Equations (EPs) in a channel with
lateral periodicity. A rigorous convergence result, in some
physically relevant space, is proven. This allows, among other
consequences, to confirm the natural choice of the

*non-local*boundary conditions for the non-viscous PEs.
DCDS

The goal of this article is to study the boundary layer of the heat equation with thermal diffusivity in a general (curved), bounded and smooth domain in $\mathbb{R}^{d}$, $d \geq 2$, when the diffusivity parameter ε is small. Using a curvilinear coordinate system fitting the boundary, an asymptotic expansion, with respect to ε, of the heat solution is obtained at all orders.
It appears that unlike the case of a straight boundary, because of the curvature of the boundary, two correctors in powers of ε and ε

^{1/2}must be introduced at each order. The convergence results, between the exact and approximate solutions, seem optimal. Beside the intrinsic interest of the results presented in the article, we believe that some of the methods introduced here should be useful to study boundary layers for other problems involving curved boundaries.## Year of publication

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