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Papers by Gung-Min Gie

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

Communications in Mathematical Sciences, 2014

Research paper thumbnail of Analysis of Mixed Elliptic and Parabolic Boundary Layers with Corners

International Journal of Differential Equations, 2013

Research paper thumbnail of Stability of vortex solutions to an extended Navier–Stokes system

Communications in Mathematical Sciences, 2016

Research paper thumbnail of Recent progresses in boundary layer theory

Discrete and Continuous Dynamical Systems, 2015

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

Research paper thumbnail of Cell centered finite volume methods using Taylor series expansion scheme without fictitious domains

Research paper thumbnail of Asymptotic analysis of the navier-stokes equations

Physica D: Nonlinear Phenomena, 1983

New bounds are established on the number of modes which determine the solutions of the Navier-Sto... more New bounds are established on the number of modes which determine the solutions of the Navier-Stokes equations in two dimensions. The best bound available at present is nearly proportional to the generalized Grashof number (defined in the paper), and less than logarithmically dependent on the spatial structure, or the shape of the force driving the flow. To the extent than

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

Numerische Mathematik, 2014

ABSTRACT We construct the cell-centered Finite Volume discretization of the two-dimensional invis... more ABSTRACT We construct the cell-centered Finite Volume discretization of the two-dimensional inviscid primitive equations in a domain with topography. To compute the numerical fluxes, the so-called Upwind Scheme (US) and the Central-Upwind Scheme (CUS) are introduced. For the time discretization, we use the classical fourth order Runge–Kutta method. We verify, with our numerical simulations, that the US (or CUS) is a robust first (or second) order scheme, regardless of the shape or size of the topography and without any mesh refinement near the topography.

Research paper thumbnail of Interior penalty discontinuous Galerkin methods with implicit time-integration techniques for nonlinear parabolic equations

Numerical Methods for Partial Differential Equations, 2013

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

Journal of Differential Equations, 2012

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

Applicable Analysis, 2010

ABSTRACT The goal of this article is to study the asymptotic behaviour of the solutions of linear... more ABSTRACT 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 3 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.

Research paper thumbnail of Boundary layers in smooth curvilinear domains: Parabolic problems

Discrete and Continuous Dynamical Systems, 2009

ABSTRACT The goal of this article is to study the boundary layer of the heat equation with therma... more ABSTRACT 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 Rd, d ≥ 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 ε1/2 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.

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

Communications in Mathematical Sciences, 2014

Research paper thumbnail of Analysis of Mixed Elliptic and Parabolic Boundary Layers with Corners

International Journal of Differential Equations, 2013

Research paper thumbnail of Stability of vortex solutions to an extended Navier–Stokes system

Communications in Mathematical Sciences, 2016

Research paper thumbnail of Recent progresses in boundary layer theory

Discrete and Continuous Dynamical Systems, 2015

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

Research paper thumbnail of Cell centered finite volume methods using Taylor series expansion scheme without fictitious domains

Research paper thumbnail of Asymptotic analysis of the navier-stokes equations

Physica D: Nonlinear Phenomena, 1983

New bounds are established on the number of modes which determine the solutions of the Navier-Sto... more New bounds are established on the number of modes which determine the solutions of the Navier-Stokes equations in two dimensions. The best bound available at present is nearly proportional to the generalized Grashof number (defined in the paper), and less than logarithmically dependent on the spatial structure, or the shape of the force driving the flow. To the extent than

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

Numerische Mathematik, 2014

ABSTRACT We construct the cell-centered Finite Volume discretization of the two-dimensional invis... more ABSTRACT We construct the cell-centered Finite Volume discretization of the two-dimensional inviscid primitive equations in a domain with topography. To compute the numerical fluxes, the so-called Upwind Scheme (US) and the Central-Upwind Scheme (CUS) are introduced. For the time discretization, we use the classical fourth order Runge–Kutta method. We verify, with our numerical simulations, that the US (or CUS) is a robust first (or second) order scheme, regardless of the shape or size of the topography and without any mesh refinement near the topography.

Research paper thumbnail of Interior penalty discontinuous Galerkin methods with implicit time-integration techniques for nonlinear parabolic equations

Numerical Methods for Partial Differential Equations, 2013

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

Journal of Differential Equations, 2012

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

Applicable Analysis, 2010

ABSTRACT The goal of this article is to study the asymptotic behaviour of the solutions of linear... more ABSTRACT 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 3 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.

Research paper thumbnail of Boundary layers in smooth curvilinear domains: Parabolic problems

Discrete and Continuous Dynamical Systems, 2009

ABSTRACT The goal of this article is to study the boundary layer of the heat equation with therma... more ABSTRACT 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 Rd, d ≥ 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 ε1/2 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.

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