Gung-Min Gie

Asymptotic analysis of the Stokes problem on general bounded domains: the case of a characteristic boundary

The goal of this article is to study the asymptotic behaviour of the solutions of linearized Navier–Stokes equations (LNSE), when the viscosity is small, in a general (curved) bounded and smooth domain in ℝ3 with a characteristic boundary. To handle the difficulties due to the curvature of the boundary, we first introduce a curvilinear coordinate system which is adapted to the boundary. Then we prove the existence of a strong corrector for the LNSE. More precisely, we show that the solution of LNSE behaves like the corresponding Euler solution except in a thin region, near the boundary, where a certain heat solution is added as a corrector.

Singular perturbation problems in a general smooth domain

The goal of this article is to study the boundary layer of a reaction-diffusion equation with a small viscosity in a general (curved), bounded and smooth domain in R, n 2. To the best of our knowledge, the classical expansion in the case of a bounded interval or of a channel is not valid for a general domain. Using the techniques of differential geometry, a new asymptotic expansion proposed in this article recovers the optimal convergence rate of the remainder at all orders.

Vorticity layers of the 2D Navier-Stokes equations with a slip type boundary condition

We study the asymptotic behavior, at small viscosity e, of the Navier-Stokes equations in a 2D curved domain. The Navier-Stokes equations are supplemented with the slip boundary condition, which is a special case of the Navier friction boundary condition where the friction coefficient is equal to two times the curvature on the boundary. We construct an artificial function, which is called a corrector, to balance the discrepancy on the boundary of the Navier-Stokes and Euler vorticities. Then, performing the error analysis on the corrected difference of the Navier-Stokes and Euler vorticity equations, we prove convergence results in the L 2 norm in space uniformly in time, and in the norm of H 1 in space and L 2 in time with rates of order e 3/4 and e 1/4 , respectively. In addition, using the smallness of the corrector, we obtain the convergence of the Navier-Stokes solution to the Euler solution in the H 1 norm uniformly in time with rate of order e 1/4 .

Asymptotic analysis of the Navier-Stokes equations in a curved domain with a non-characteristic boundary

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

Analysis of Mixed Elliptic and Parabolic Boundary Layers with Corners

e^j

A higher order Finite Volume resolution method for a system related to the inviscid primitive equations in a complex domain

,

Asymptotic analysis of the Stokes equations in a square at small viscosity

j=0,1

Boundary layer analysis of the Navier–Stokes equations with generalized Navier boundary conditions

, where

Boundary layers in smooth curvilinear domains: Parabolic problems

e

Asymptotic expansion of the stokes solutions at small viscosity: The case of non-compatible initial data

is the (small) viscosity parameter. This allows us to obtain an asymptotic expansion of the Navier-Stokes solution at order