1,112 research outputs found
The Inviscid Limit and Boundary Layers for Navier-Stokes Flows
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.Comment: To appear in "Handbook of Mathematical Analysis in Mechanics of
Viscous Fluids", Y. Giga and A. Novotn\'y Ed., Springer. The final
publication is available at http://www.springerlink.co
On the inviscid Boussinesq system with rough initial data
We deal with the local well-posedness theory for the two-dimensional inviscid
Boussinesq system with rough initial data of Yudovich type. The problem is in
some sense critical due to some terms involving Riesz transforms in the
vorticity-density formulation. We give a positive answer for a special
sub-class of Yudovich data including smooth and singular vortex patches. For
the latter case we assume in addition that the initial density is constant
around the singular part of the patch boundary.Comment: 26 page
Planar inviscid flows in a channel of finite length : washout, trapping and self-oscillations of vorticity
The paper addresses the nonlinear dynamics of planar inviscid incompressible flows in the straight channel of a finite length. Our attention is focused on the effects of boundary conditions on vorticity dynamics. The renowned Yudovich's boundary conditions (YBC) are the normal component of velocity given at all boundaries, while vorticity is prescribed at an inlet only. The YBC are fully justified mathematically: the well posedness of the problem is proven. In this paper we study general nonlinear properties of channel flows with YBC. There are 10 main results in this paper: (i) the trapping phenomenon of a point vortex has been discovered, explained and generalized to continuously distributed vorticity such as vortex patches and harmonic perturbations; (ii) the conditions sufficient for decreasing Arnold's and enstrophy functionals have been found, these conditions lead us to the washout property of channel flows; (iii) we have shown that only YBC provide the decrease of Arnold's functional; (iv) three criteria of nonlinear stability of steady channel flows have been formulated and proven; (v) the counterbalance between the washout and trapping has been recognized as the main factor in the dynamics of vorticity; (vi) a physical analogy between the properties of inviscid channel flows with YBC, viscous flows and dissipative dynamical systems has been proposed; (vii) this analogy allows us to formulate two major conjectures (C1 and C2) which are related to the relaxation of arbitrary initial data to C1: steady flows, and C2: steady, self-oscillating or chaotic flows; (viii) a sufficient condition for the complete washout of fluid particles has been established; (ix) the nonlinear asymptotic stability of selected steady flows is proven and the related thresholds have been evaluated; (x) computational solutions that clarify C1 and C2 and discover three qualitatively different scenarios of flow relaxation have been obtained
On the Yudovich solutions for the ideal MHD equations
In this paper, we address the problem of weak solutions of Yudovich type for
the inviscid MHD equations in two dimensions. The local-in-time existence and
uniqueness of these solutions sound to be hard to achieve due to some terms
involving Riesz transforms in the vorticity-current formulation. We shall prove
that the vortex patches with smooth boundary offer a suitable class of initial
data for which the problem can be solved. However this is only done under a
geometric constraint by assuming the boundary of the initial vorticity to be
frozen in a magnetic field line.
We shall also discuss the stationary patches for the incompressible Euler
system and the MHD system. For example, we prove that a stationary simply
connected patch with rectifiable boundary for the system is necessarily
the characteristic function of a disc.Comment: 40 page
Interaction of vortices in viscous planar flows
We consider the inviscid limit for the two-dimensional incompressible
Navier-Stokes equation in the particular case where the initial flow is a
finite collection of point vortices. We suppose that the initial positions and
the circulations of the vortices do not depend on the viscosity parameter \nu,
and we choose a time T > 0 such that the Helmholtz-Kirchhoff point vortex
system is well-posed on the interval [0,T]. Under these assumptions, we prove
that the solution of the Navier-Stokes equation converges, as \nu -> 0, to a
superposition of Lamb-Oseen vortices whose centers evolve according to a
viscous regularization of the point vortex system. Convergence holds uniformly
in time, in a strong topology which allows to give an accurate description of
the asymptotic profile of each individual vortex. In particular, we compute to
leading order the deformations of the vortices due to mutual interactions. This
allows to estimate the self-interactions, which play an important role in the
convergence proof.Comment: 39 pages, 1 figur
On the inviscid limit of the Navier-Stokes equations
We consider the convergence in the norm, uniformly in time, of the
Navier-Stokes equations with Dirichlet boundary conditions to the Euler
equations with slip boundary conditions. We prove that if the Oleinik
conditions of no back-flow in the trace of the Euler flow, and of a lower bound
for the Navier-Stokes vorticity is assumed in a Kato-like boundary layer, then
the inviscid limit holds.Comment: Improved the main result and fixed a number of typo
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