Asymptotic analysis of the linearized Navier-Stokes equations in a channel
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In this article we study and derive explicit formulas for the boundary layers occurring in the linearized channel flows in the limit of small viscosity. Our study is based on classical boundary layer techniques combined with a new global treatment of the pressure term. Introduction. In the limit of large Reynolds numbers (or small viscosity), the solutions to the incompressible Navier-Stokes equations display a turbulent behavior whose understanding is still a major open problem in mathematical physics. It seems that there are at least two important aspects of the problem; on the one hand the role played by the nonlinear terms which tend to mix the modes and propagate energy among them. On the other hand for physically realistic flows which involve boundaries, it is known that important phenomena occur at the boundary in a thin region called the boundary layer; in particular, the generation of vortices which propagate in the fluid and contribute to generate the motion. Although much remains to be done, there has been recently some progress in the mathematical aspects of nonlinear dynamics, in relation with the point of view of attractors, based on the dynamical system approach to turbulence of Smale and RuelleTakens. Concerning the mathematical study of boundary layers, a thorough study is available in the long article of M.I. Vishik and L.A. Lyusternik ([15]) and in the book of J.L. Lions ([6]); see also W. Eckhaus ([2]) and P. Lagerstrom ([5]) at the interface of mathematics and mechanics. All these references contain a thorough study of boundary layers; in particular, for time-independent problems. For time-dependent problems partial results appear in [5] and [6]; let us mention also the work of O. Oleinik ([8]) who studied the mathematical theory of the Prandtl’s equations ([9]) independently of their relation to the Navier-Stokes equations and the work of P. Fife who studied the validity of Prandtl’s equations in the stationary case with special type of pressure ([3]). Although the full study of the convergence of the Navier-Stokes equations to the Euler equations is an outstanding problem which may still be out of reach, we would like in this and in forthcoming articles ([13, 14]) to develop some tools for the study of boundary layers in flows. An essential aspect of our approach as compared to the classical studies on boundary layers in the references above is that we treat the pressure in a global manner, and not only locally. The global treatment of the pressure is an essential aspect in the numerical treatment of the Navier-Stokes; it plays also an important role for a di↵erent problem in [12]. Received for publication March 1995. AMS Subject Classifications: 35B25, 35B40, 35C20, 35Q30, 76D03, 76D07, 76D10, 76D30. 1591 1592 ROGER TEMAM AND XIAOMING WANG The type of problems that we treat here is physically simple and perhaps physically irrelevant, but we feel that the techniques introduced here will be necessary for physically more relevant problems. We study the boundary layers for a channel flow, dropping furthermore the nonlinear terms of the equations; hence we study an evolution Stokes equation in a channel with two fixed boundaries and periodicity in the directions parallel to the walls. The simplifications due to the flat wall will be overcome in a subsequent article ([13]) where we will consider curved boundaries. The article is organized as follows. In Section 1 we study the boundary layers for the heat equation first in space dimension one, then in space dimension two (or higher). Although these results may not be new, we did not find them available in the literature in the form in which we present and use them. Section 2 is devoted to the linearized channel flow. Section 3 concerns some results on the behavior of the solutions for large time and small viscosity parameter. Finally the Appendix contains the proof of technical results. 1. Boundary layer for the heat equation. In this section we shall study the corrector problem for the heat equation in a 2D periodic channel. This is one of the necessary ingredients for describing the corrector for channel flows. In Section 1.1 we shall study the 1D case, derive some explicit expressions for the solutions and for the boundary layers and then in Section 1.2 we present the corrector for the heat equation in the 2D channel. 1.1. A preliminary result: The heat equation in space dimension one. As indicated before we want here to derive suitable expressions of the one-dimensional singularly perturbed heat equation and its corrector. Although some form of these correctors may be found perhaps in the engineering literature, we did not find them available in the mathematical literature (see e.g. M.I. Vishik and L.A. Lyusternik ([15]), J.L. Lions ([6])) and to the best of our knowledge the mathematical results given in Section 1 are new. We need to start with the classical heat equation 8< : @u @t @ 2u @y2 = f, 0 < y < 1, t > 0, u|t=0 = g, u(0) = u(1) = 0. (1.1) for which we give an explicit formula of the solution of (1.1) using the heat kernel. More precisely, let us define the extension operator T : L2(0, 1)! L2loc(R 1), >>< >>: (T (g))(y) = ( g(y), y 2 (0, 1), g( y), y 2 ( 1, 0), g 2 L(0, 1) and T (g) is extended periodically outside ( 1, 1), with period 2. We then define ũ(t, y) = Z 1 1 1 p 4⇡t e (x y)2 4t (T (g))(x) dx, (1.2) and we have the following result: THE LINEARIZED NAVIER-STOKES EQUATIONS IN A CHANNEL 1593 Lemma 1.1. Suppose g 2 L2(0, 1). Then ũ|(0,1) defined in (1.2) solves the 1D heat equation (1.1) with initial data u0 = g and zero forcing term (f = 0), in the sense that ũ(t, ·)! T (g) in L2loc(R ), or kũ(t, ·) gkL2(0,1) ! 0 as t! 0, (1.3) and ũ|(0,1) 2 H 0 (0, 1), or ũ(t, 0) = ũ(t, 1) = 0, for all t > 0. For the sake of completeness we recall the proof of this well-known result in the Appendix. Similarly by applying Duhamel’s principle we can show that the solution to (1.1) (with general u0 and f) can be expressed as
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