Asymptotic model for free surface flow of an electrically conducting fluid in a high-frequency magnetic field

We study the behaviour of a layer of an electrically conducting inviscid incompressible fluid in a high-frequency alternating magnetic field. We derive nonlinear asymptotic equations governing the evolution of the fluid layer in the high-frequency limit. As a test for the model, we consider the linearised stability problem for an infinite planar free surface of a layer of finite depth.     

Asymptotic analysis of a boundary optimal control problem on a general branched structure

We consider an optimal control problem posed on a domain with a highly oscillating smooth boundary where the controls are applied on the oscillating part of the boundary. There are many results on domains with oscillating boundaries where the oscillations are pillar‐type (non‐smooth) while the literature on smooth oscillating boundary is very few. In this article, we use appropriate scaling on the controls acting on the oscillating boundary leading to different limit control problems, namely, boundary optimal control and interior optimal control problem. In the last part of the article, we visualize the domains as a branched structure, and we introduce unfolding operators to get contributions from each level at every branch.     

Navier边界条件下湖方程的边界层问题

考虑光滑区域上二维粘性湖方程在Navier边界条件下的无粘极限问题,证明了具有Navier边界条件粘性湖方程的边界层在Sobolev空间中是非线性稳定的,验证了具有较弱强度的边界层的渐近展开的合理性.     

On the Inviscid Limit Problem of the Vorticity Equations for Viscous Incompressible Flows in the Half‐Plane

We consider the Navier‐Stokes equations for viscous incompressible flows in the half‐plane under the no‐slip boundary condition. By using the vorticity formulation we prove the local‐in‐time convergence of the Navier‐Stokes flows to the Euler flows outside a boundary layer and to the Prandtl flows in the boundary layer in the inviscid limit when the initial vorticity is located away from the boundary. © 2014 Wiley Periodicals, Inc.     

Quasineutral Limit of the Vlasov-Poisson System with Massless Electrons

In this paper, we study the quasineutral limit (in other words the limit when the Debye length tends to zero) of Vlasov-Poisson like equations describing the behavior of ions in a plasma. We consider massless electrons, with a charge density following a Maxwell-Boltzmann law. For cold ions, using the relative entropy method, we derive the classical Isothermal Euler or the (inviscid) Shallow Water systems from fluid mechanics. We then study the combined quasineutral and strong magnetic field regime for such plasmas.     

An elementary approach to the inviscid limits for the 3D Navier–Stokes equations with slip boundary conditions and applications to the 3D Boussinesq equations

In this note we consider the inviscid limit for the 3D Boussinesq equations without diffusion, under slip boundary conditions of Navier’s type. We first study more closely the Navier–Stokes equations, to better understand the problem. The role of the initial data is also emphasized in connection with the vanishing viscosity limit.     

La limite pour la viscosité zéro des équations de Stokes dans un demi-espace

The time-dependent Stokes equations are considered. It is shown that the solution is the sum of an inviscid solution, a boundary layer solution, and a small (in the zero viscosity limit) correction. Bounds on these solutions are given, in the appropriate Sobolev spaces, in terms of the norms of the initial and boundary data.     

Eulerian limit for 2D Navier-Stokes equation and damped/driven KdV equation as its model

We discuss the inviscid limits for the randomly forced 2D Navier-Stokes equation (NSE) and the damped/driven KdV equation. The former describes the space-periodic 2D turbulence in terms of a special class of solutions for the free Euler equation, and we view the latter as its model. We review and revise recent results on the inviscid limit for the perturbed KdV and use them to suggest a setup which could be used to make a next step in the study of the inviscid limit of 2D NSE. The proposed approach is based on an ergodic hypothesis for the flow of the 2D Euler equation on iso-integral surfaces. It invokes a Whitham equation for the 2D Navier-Stokes equation, written in terms of the ergodic measures.     

The compressible Gortler problem in two-dimensional boundary layers

In this paper the authors investigate the growth rates of Görtlervortices in a compressible flow in the inviscid limit of largeGörtler number. Numerical solutions are obtained for O(1)wavenumbers. The further limits of (i) large Mach number and(ii) large wavenumber with O(1) Mach number are considered.It is shown that two different types of disturbance mode canappear in this problem. The first is a wall layer mode, so namedas it has its eigenfunctions trapped in a thin layer near thewall. The other mode investigated is confined to a thin layeraway from the wall and termed a trapped-layer mode for largewavenumbers and an adjustment-layer mode for large Mach numbers,since then this mode has its eigenfunctions concentrated inthe temperature adjustment layer. It is possible to investigatethe near crossing of the modes which occurs in each of the limitsmentioned. The inviscid limit does not predict a fastest growingmode, but does enable a most dangerous mode to be identifiedfor O(1) Mach number. For hypersonic flow the most dangerousmode depends on the size of the Görtler number.     

Eulerian limit for 2D Navier-Stokes equation and damped/driven KdV equation as its model

We discuss the inviscid limits for the randomly forced 2D Navier-Stokes equation (NSE) and the damped/driven KdV equation. The former describes the space-periodic 2D turbulence in terms of a special class of solutions for the free Euler equation, and we view the latter as its model. We review and revise recent results on the inviscid limit for the perturbed KdV and use them to suggest a setup which could be used to make a next step in the study of the inviscid limit of 2D NSE. The proposed approach is based on an ergodic hypothesis for the flow of the 2D Euler equation on iso-integral surfaces. It invokes a Whitham equation for the 2D Navier-Stokes equation, written in terms of the ergodic measures. Dedicated to Vladimir Igorevich Arnold on his 70th birthday     

ASYMPTOTIC LIMITS FOR QUANTUM TRAJECTORY MODELS

ABSTRACT

The semi-classical and the inviscid limit in quantum trajectory models given by a one-dimensional steady-state hydrodynamic system for quantum fluids are rigorously performed. The model consists of the momentum equation for the particle density in a bounded domain, with prescribed current density, and the Poisson equation for the electrostatic potential. The momentum equation can be written as a dispersive third-order differential equation which may include viscous terms. It is shown that the semi-classical and inviscid limit commute for sufficiently small data (i.e. current density) corresponding to subsonic states, where the inviscid non-dispersive solution is regular. In addition, we show that these limits do not commute in general. The proofs are based on a reformulation of the problem as a singular second-order elliptic system and on elliptic and W ^1,1 estimates.     

The Nonlinear Critical-Wall Layer in a Parallel Shear Flow

The nonlinear critical layer theory is developed for the case where the critical point is close enough to a solid boundary so that the critical layer and viscous wall layers merge. It is found that the flow structure differs considerably from the symmetric “eat's eye” pattern obtained by Benney and Bergeron [1] and Haberman [2]. One of the new features is that higher harmonics generated by the critical layer are in some cases induced in the outer flow at the same order as the basic disturbance. As a consequence, the lowest-order critical layer problem must be solved numerically. In the inviscid limit, on the other hand, a closed-form solution is obtained. It has continuous vorticity and is compared with the solutions found by Bergeron [3], which contain discontinuities in vorticity across closed streamlines.     

Vanishing viscosity limits and boundary layers for circularly symmetric 2D flows

We continue the work of Lopes Filho, Mazzucato and Nussenzveig Lopes [10] on the vanishing viscosity limit of circularly symmetric viscous flow in a disk with rotating boundary, shown there to converge to the inviscid limit in L ²-norm as long as the prescribed angular velocity α(t) of the boundary has bounded total variation. Here we establish convergence in stronger L ² and L ^p -Sobolev spaces, allow for more singular angular velocities α, and address the issue of analyzing the behavior of the boundary layer. This includes an analysis of concentration of vorticity in the vanishing viscosity limit. We also consider such flows on an annulus, whose two boundary components rotate independently. Supported in part by NSF grant DMS-0456861.     

Barotropic Instability of the Bickley Jet at High Reynolds Numbers

We investigate the linear stability of the Bickley jet in the framework of the beta-plane approximation. Because singular inviscid neutral modes exist in the retrograde case     , it is necessary to add viscosity to interpret them. One of these modes was found in closed form by Howard and Drazin [1] . However, its critical point is at the center of the jet and it was therefore not possible for these authors to ascertain the relationship of this mode to the stability problem or to discuss how to continue the eigenfunction across the singularity.
The viscous critical layer problem associated with this singularity is considerably more difficult than the usual one (which leads to integrals of the Airy function) because     and, consequently, a second-order turning point is involved. Our analysis shows that the Howard–Drazin mode is degenerate in the domain where it is valid as a limit of the viscous problem (wavenumber  α²≤ 9/2  ), that is, it corresponds to both an odd and an even mode. This conclusion is confirmed by direct numerical solution of the Orr–Sommerfeld equation which shows, in addition, that viscosity is destabilizing along portions of the stability boundary. For a retrograde jet, instability is found to occur beyond the inviscid critical value of β, that is, in the region where the flow would be stable according to the Rayleigh–Kuo condition.     

Life span of 2‐D irrotational compressible fluids in the halfplane

We consider the Euler equations of barotropic inviscid compressible fluids in the half plane. It is well known that, as the Mach number goes to zero, the compressible flows approximate the solution of the equations of motion of inviscid, incompressible fluids. In dimension two such limit solution exists on any arbitrary time interval, with no restriction on the size of the initial velocity. It is then natural to expect the same for the compressible solution, if the Mach number is sufficiently small. We consider smooth irrotational solutions. First, we study the life span, i.e. the largest time interval T(ε) of existence of classical solutions, when the initial data are a small perturbation of size εfrom a constant state. For the proof of this result we use a combination of energy and decay estimates. Then, the estimate of the life span allows to show, by a suitable scaling of variables, the existence of irrotational solutions on any arbitrary time interval, for any small enough Mach number. Copyright © 2002 John Wiley & Sons, Ltd.     

Zero dissipation limit of the 1D linearized Navier-Stokes equations for a compressible fluid

In this paper we study the asymptotic limiting behavior of the solutions to the initial boundary value problem for linearized one-dimensional compressible Navier-Stokes equations. We consider the characteristic boundary conditions, that is we assume that an eigenvalue of the associated inviscid Euler system vanishes uniformly on the boundary. The aim of this paper is to understand the evolution of the boundary layer, to construct the asymptotic ansatz which is uniformly valid up to the boundary, and to obtain rigorously the uniform convergence to the solution of the Euler equations without the weakness assumption on the boundary layer.     

An Analytic Approach for Calculating Absolutely Unstable Inviscid Modes of the Boundary Layer on a Rotating Disk

An analytical treatment of inviscidly absolutely unstable modes is pursued using the long-wavelength asymptotic approach. It is shown using the inviscid Rayleigh scalings in conjunction with the linear critical layer theory that the rotating-disk boundary layer flow undergoes a region of absolute instability for some small azimuthal wave numbers. The analytically calculated branch points for the absolute instability are found to be in good agreement with those obtained via a numerical solution of the inviscid Rayleigh equation.     

Asymptotic calculation of inviscidly absolutely unstable modes of the compressible boundary layer on a rotating disk

In this work a long-wavelength asymptotic approach is used to analyze the region of absolute instability in the compressible rotating disk boundary layer flow. Theoretically determined values of branch points for the occurrence of absolute instability in the compressible flow are shown to match onto the ones which are obtained via a numerical solution of the linear inviscid compressible Rayleigh equations.     

Rossby Solitary Waves in the Presence of a Critical Layer

This study considers the evolution of weakly nonlinear long Rossby waves in a horizontally sheared zonal current. We consider a stable flow so that the nonlinear time scale is long. These assumptions enable the flow to organize itself into a large‐scale coherent structure in the régime where a competition sets in between weak nonlinearity and weak dispersion. This balance is often described by a Korteweg‐de‐Vries equation. The traditional assumption of a weak amplitude breaks down when the wave speed equals the mean flow velocity at a certain latitude, due to the appearance of a singularity in the leading‐order equation, which strongly modifies the flow in a critical layer. Here, nonlinear effects are invoked to resolve this singularity, because the relevant geophysical flows have high Reynolds numbers. Viscosity is introduced in order to render the nonlinear‐critical‐layer solution unique, but the inviscid limit is eventually taken. By the method of matched asymptotic expansions, this inner flow is matched at the edges of the critical layer with the outer flow. We will show that the critical‐layer–induced flow leads to a strong rearrangement of the related streamlines and consequently of the potential‐vorticity contours, particularly in the neighborhood of the separatrices between the open and closed streamlines. The symmetry of the critical layer vis‐à‐vis the critical level is also broken. This theory is relevant for the phenomenon of Rossby wave breaking and eventual saturation into a nonlinear wave. Spatially localized solutions are described by a Korteweg‐de‐Vries equation, modified by new nonlinear terms; depending on the critical‐layer shape, this leads to depression or elevation waves. The additional terms are made necessary at a certain order of the asymptotic expansion while matching the inner flow on the dividing streamlines. The new evolution equation supports a family of solitary waves. In this paper we describe in detail the case of a depression wave, and postpone for further discussion the more complex case of an elevation wave.     

Sinc Approximation of the Heat Flux on the Boundary of a Two-Dimensional Finite Slab

We consider the two-dimensional problem of recovering globally in time the heat flux on the surface of a layer inside of a heat-conducting body from two interior temperature measurements. The problem is ill-posed. The approximation function is represented by a two-dimensional Sinc series and the error estimate is given.