Vanishing viscosity limit for the 3D magnetohydrodynamic system with a slip boundary condition

Xiao, Yuelong; Xin, Zhouping; Wu, Jiahong

doi:10.1016/j.jfa.2009.09.010

Cited by 64 publications

(65 citation statements)

References 31 publications

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“…See also [23] for the magnetohydrodynamic system and [6] for a different approach to the inviscid limit for the slip-type boundary value problem.…”

Section: Introduction and Some Resultsmentioning

confidence: 99%

A missed persistence property for the Euler equations and its effect on inviscid limits

Veiga¹,

Crispo

2012

Nonlinearity

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We consider the problem of the strong convergence, as the viscosity goes to zero, of the solutions to the three-dimensional evolutionary Navier-Stokes equations under a Navier slip-type boundary condition to the solution of the Euler equations under the zero-flux boundary condition. In spite of the arbitrarily strong convergence results proved in the flat boundary case, see [4], it was shown in reference [5] that the result is false in general, by constructing an explicit family of smooth initial data in the sphere, for which the result fails. Our aim here is to present a more general, simpler and incisive proof. In particular, counterexamples can be displayed in arbitrary, smooth, domains. As in [5], the proof is reduced to the lack of a suitable persistence property for the Euler equations. This negative result is proved by a completely different approach.Mathematics Subject Classification 35Q30, 76D05, 76D09.

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“…See also [23] for the magnetohydrodynamic system and [6] for a different approach to the inviscid limit for the slip-type boundary value problem.…”

Section: Introduction and Some Resultsmentioning

confidence: 99%

A missed persistence property for the Euler equations and its effect on inviscid limits

Veiga¹,

Crispo

2012

Nonlinearity

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“…Recently, the vanishing viscosity limits problems was established on different physical models with the slip boundary conditions, such as [5,6,17,26,27,29].…”

Section: )mentioning

confidence: 99%

Vanishing viscosity limit for the 3D nonhomogeneous incompressible Navier–Stokes equations with a slip boundary condition

Chen

Xiao

Zhang

2017

Math Methods in App Sciences

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In this paper, we investigate the vanishing viscosity limit for the 3D nonhomogeneous incompressible Navier-Stokes equations with a slip boundary condition. We establish the local well-posedness of the strong solutions for initial boundary value problems for such systems. Furthermore, the vanishing viscosity limit process is established and a strong rate of convergence is obtained as the boundary of the domain is flat. In addition, it is needed to add some additional condition for density to match well the boundary condition.

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“…The discussion here will be confined to the classical case of the vanishing viscosity limit for incompressible, Newtonian fluids, although some partial results are available in the important cases of compressible flows [150,145,131], MHD [148,149], convection in porous media [82], and for non-Newtonian (secondgrade) fluids [19,101]. In the ensuing discussion, the case of unsteady flows in bounded, simply connected domains or a half space will be the main focus.…”

Section: Inviscid Limit Problem With Physical Boundarymentioning

confidence: 99%

The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

Maekawa¹,

Mazzucato²

2016

Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

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The validity of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations modeling viscous incompressible flows converge to solutions of the Euler equations modeling inviscid incompressible flows as viscosity approaches zero, is one of the most fundamental issues in mathematical fluid mechanics. The problem is classified into two categories: the case when the physical boundary is absent, and the case when the physical boundary is present and the effect of the boundary layer becomes significant. The aim of this article is to review recent progress on the mathematical analysis of this problem in each category.

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Vanishing viscosity limit for the 3D magnetohydrodynamic system with a slip boundary condition

Abstract: This work investigates the solvability, regularity and vanishing viscosity limit of the 3D viscous magnetohydrodynamic system in a class of bounded domains with a slip boundary condition.

Cited by 64 publications

References 31 publications

A missed persistence property for the Euler equations and its effect on inviscid limits

A missed persistence property for the Euler equations and its effect on inviscid limits

Vanishing viscosity limit for the 3D nonhomogeneous incompressible Navier–Stokes equations with a slip boundary condition

The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

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