Prandtl boundary layer expansions of steady Navier-Stokes flows over a moving plate arXiv:1411.6984v1 [math.AP] 25 Nov 2014 Yan Guo∗ Toan T Nguyen† December 1, 2014 Abstract This paper concerns the validity of the Prandtl boundary layer theory in the inviscid limit for steady incompressible Navier-Stokes flows The stationary flows, with small viscosity, are considered on [0, L] × R+ , assuming a no-slip boundary condition over a moving plate at y = We establish the validity of the Prandtl boundary layer expansion and its error estimates Contents Introduction 1.1 Boundary conditions 1.2 Main result and discussions Construction of the approximate solutions 2.1 Zeroth-order Prandtl layers 2.2 ε1/2 -order corrections 2.3 Euler correctors 2.3.1 Euler profiles 2.4 Prandtl correctors 2.4.1 Construction of Prandtl layers 2.4.2 Cut-off Prandtl layers 2.5 Proof of Proposition 2.1 13 14 17 18 19 25 27 Linear stability estimates 28 3.1 Energy estimates 29 3.2 Positivity estimates 31 3.3 Proof of Proposition 3.1 38 L∞ estimates 38 Proof of the main theorem 42 ∗ Division of Applied Mathematics, Brown University, 182 George street, Providence, RI 02912, USA Email: Yan Guo@Brown.edu † Department of Mathematics, Pennsylvania State University, State College, PA 16802, USA Email: nguyen@math.psu.edu 1 Introduction In this paper, we consider the stationary incompressible Navier-Stokes equations U UX + V UY + PX = εUXX + εUY Y U VX + V VY + PY = εVXX + εVY Y (1.1) U X + VY = posed in a two dimensional domain Ω = {(X, Y ) : ≤ X ≤ L, Y ≥ 0}, with a “no-slip” boundary condition U (X, 0) = ub > 0, V (X, 0) = (1.2) on the boundary at Y = The given constant ub can be viewed as the moving speed of the plate (that is, the boundary edge Y = 0) The case when ub ≡ refers to the classical no-slip boundary condition on a motionless boundary Y = The boundary conditions at x = 0, L will be prescribed explicitly in the text We are interested in the problem when ε → The study of the inviscid limit and asymptotic boundary layer expansions of Navier-Stokes equations (1.1) in the presence of no-slip boundary condition is one of the central problems in the mathematical analysis of fluid mechanics A formal limit ε → should lead the Euler flow [U0 , V0 ] inside Ω which satisfies only non-penetration condition at Y = : V0 (X, 0) = Throughout this paper, we assume that the outside Euler flow is a shear flow [U0 , V0 ] ≡ [u0e (Y ), 0], (1.3) for some smooth functions u0e (Y ) We note that there is no pressure pe for this Euler flow Generically, there is a mismatch between the tangential velocities of the Euler flow ue ≡ U0 (X, 0) ≡ u0e (0) and the prescribed Navier-Stokes flows U (X, 0) = ub on the boundary Due to the mismatch on the boundary, Prandtl in 1904 proposed a thin fluid boundary layer of √ size ε to connect different velocities ue and ub We shall work with the scaled boundary layer, or Prandtl’s, variables: Y x = X, y=√ ε In these variables, we express the solution of the NS equation [U, V ] via [U ε , V ε ] as √ √ V (x, εy) √ √ √ [U (X, Y ), V (X, Y )] = [U (x, εy), ε ] = [U ε (x, y), εV ε (x, y)] ε in which we note that the scaled normal velocity V ε is √1ε of the original velocity V Similarly, P (X, Y ) = P ε (x, y) In these new variables, the Navier-Stokes equations (1.1) now read ε ε U ε Uxε + V ε Uyε + Pxε = Uyy + εUxx Pyε ε ε = Vyy + εVxx U ε Vxε + V ε Vyε + ε Uxε + Vyε = (1.4) Throughout the paper, we perform our analysis directly on these scaled equations together with the same no-slip boundary conditions (1.2) Prandtl then hypothesized that the Navier-Stokes flow can be approximately decomposed into two parts: √ [U ε , V ε ] ≈ [u0e ( εy), 0] + [¯ u(x, y), v¯(x, y)], (1.5) ε P ≈ p¯(x, y), √ in which u0e ( εy) denotes the Euler flow as in (1.3) Putting the ansatz into the Navier-Stokes equations (1.4) and collecting the leading terms in ε, we obtain the Prandtl layer corrector [¯ u, v¯] which satisfies [ue + u ¯]¯ ux + v¯ uy + p¯x = u ¯yy (1.6) u ¯x + v¯y = where p¯x = p¯x (x), since by the second equation in (1.4), p¯y = Evaluating the first equation at y = ∞, one gets p¯x = 0, which is precisely due to the assumption that the trace of Euler on the boundary does not depend on x (a.k.a, Bernoulli’s law) Hence, the boundary layer satisfies u ¯(x, 0) = −ue + ub < 0, v¯(x, 0) = 0, lim u ¯(x, y) = y→∞ (1.7) The Prandtl layer [¯ u, v¯] is subject to an “initial” condition at x = 0: u ¯(0, y) = u ¯0 (y) (1.8) Regarded as one of the most important achievements of modern fluid mechanics, Prandtl’s the boundary layer expansion (1.5) connects the theory of ideal fluid (Euler flows) with the real fluid (Navier-Stokes flows) near the boundary, for a large Reynolds number (or equivalently, ε ≪ 1) Such a theory has led to tremendous applications and advances in science and engineering In particular, since the Prandtl layer solution [¯ u, v¯] satisfies an evolution equation in x, it is much easier to compute its solutions numerically than those of the original NS flows [U ε , V ε ] which satisfy an elliptic boundary-value problem Many other shear layer phenomena in fluids, such as wake flows ([17, page 187]), plane jet flows ([17, page 190]), as well as shear layers between two parallel flows, can also be described by the Prandtl layer theory (1.6) and (1.7) In spite of the huge success of Prandtl’s boundary layer theory in applications, it remains an outstanding open problem to rigorously justify the validity of expansion (1.5) in the inviscid limit The purpose of this paper is to provide an affirmative answer along this direction As it turns out, we will need higher order approximations, as compared to (1.5), in order to be able to control the remainders Precisely, we search for asymptotic expansions of the scaled Navier-Stokes solutions [U ε , V ε , P ε ] in the following form: √ √ √ √ U ε (x, y) = u0e ( εy) + u0p (x, y) + εu1e (x, εy) + εu1p (x, y) + εγ+ uε (x, y) √ √ V ε (x, y) = vp0 (x, y) + ve1 (x, εy) + εvp1 (x, y) + εγ+ v ε (x, y) √ √ √ P ε (x, y) = εp1e (x, εy) + εp1p (x, y) + εp2p (x, y) + εγ+ pε (x, y) (1.9) for some γ > 0, in which [uje , vej , pje ] and [ujp , vpj , pjp ], with j = 0, 1, denote the Euler and Prandtl profiles, respectively, and [uε , v ε , pε ] collects all the remainder solutions Here, we note that these √ profile solutions also depend on ε, and the Euler flows are always evaluated at (x, εy), whereas the Prandtl profiles are at (x, y) Formally speaking, plugging the above ansatz into (1.4) and matching the order in ε, we easily get that [u0p , vp0 , 0] solves the nonlinear Prandtl equation (1.6), whereas the next Euler profile [u1e , ve1 , p1e ] solves the linearized Euler equations around (u0e , 0): u0e u1ex + ve1 u0ey + p1ex = 0, u0e vex + p1ey = 0, u1ex + vey (1.10) = 0, or equivalently in the vorticity formulation, −u0e ∆ve1 + u0eyy ve1 = 0, (1.11) with [u1e , p1e ] being recovered from the last two equations in (1.10) The next Prandtl layer [u1p , vp1 , p1p ] solves the linearized Prandtl equations around [ue + u0p , vp0 + ve1 ], with a source term Ep : (ue + u0p )u1px + u1p u0px + (vp0 + ve1 )u1py + vp1 u0px + p1px − u1pyy = Ep , u1px + vpy = 0, (1.12) with p1p = p1p (x) Certainly, the remainder solutions [uε , v ε , pε ] solve the linearized Navier-Stokes equations around the approximate solutions, with a source that contains nonlinearity in [uε , v ε ]; see Section for details Note however that as we deal with functions in Sobolev spaces, all profile solutions are required to vanish at y = ∞ As it will be clear in the text, the actual Prandtl layers will introduce nonzero normal velocity at infinity, and is one of the issues in controlling the remainders, since the profiles then won’t even be integrable As a result, our Prandtl layers [u1p , vp1 ] in the expansion (1.9) are being cut-off for large y of the actual layers solving (1.12) In Section 2, we shall provide detailed construction of the approximate solutions and derive sufficient estimates for our analysis 1.1 Boundary conditions The zeroth Euler flow u0e is given Due to the no-slip boundary condition at y = 0, we require that ue + u0p (x, 0) = ub , ve1 (x, 0) + vp0 (x, 0) = 0, from the zeroth order in ε of the expansion (1.9), and from the √ u1p (x, 0) = −u1e (x, 0), vp1 (x, 0) = 0, ε-order layers We also assume that lim u0p (x, y) = 0, y→∞ lim u1p (x, y) = 0, y→∞ lim [uje , vej ](x, Y ) = Y →∞ The normal velocities vp0 (x, y), vp1 (x, y) in the boundary layers are constructed from u0p (x, y), u1p (x, y), respectively, through the divergence-free condition We note that in general limy→∞ vpj (x, y) = and hence, cut-off functions will be introduced to localize vpj Next, we discuss boundary conditions at x = 0, L Since the Prandtl layers solve parabolic-type equations, we require only “initial” conditions at x = 0: u0p (0, y) = u ¯0 (y), u1p (0, y) = u ¯1 (y), whereas we prescribe boundary values for the Euler profiles at both x = 0, L: u1e (0, Y ) = u1b (Y ), ve1 (0, Y ) = Vb0 (Y ), ve1 (L, Y ) = VbL (Y ) with compatibility conditions Vb0 (0) = vp0 (0, 0), VbL (0) = vp0 (L, 0) at the corners of the domain [0, L] × R+ Finally, we impose the following boundary conditions for the remainder solution [uε , v ε ]: [uε , v ε ]y=0 = (no-slip), pε − 2εuεx = 0, [uε , v ε ]x=0 = (Dirichlet), uεy + εvxε = at x = L (Neumann or stress-free) (1.13) Certainly, one may wish to consider different boundary conditions for [uε , v ε ] at x = L However, to avoid a possible formation of boundary layers with respect to x near the boundary x = L, the above Neumann stress-free condition appears the most convenient candidate to impose 1.2 Main result and discussions We are ready to state our main result: Theorem 1.1 Let u0e (Y ) be a given smooth Euler flow, and let u1b (Y ), Vb0 (Y ), VbL (Y ), with Y = √ εy, and u ¯0 (y) and u ¯1 (y) be given smooth data and decay exponentially fast at infinity in their arguments, and let ub be a positive constant We assume that |VbL (Y ) − Vb0 (Y )| L for small L, and √ ¯0 (y) > (1.14) u0e ( εy) + u 0≤y≤∞ Then, there exists a positive number L that depends only on the given data so that the boundary layer expansions (1.9), with the profiles satisfying the boundary conditions in Section 1.1, hold for γ ∈ (0, 14 ) Precisely, [U ε , V ε , P ε ] as defined in (1.9) is the unique solution to the Navier-Stokes equations (1.4), so that the remainder solutions [uε , v ε ] satisfy ∇ ε uε γ + ε ∇ε v ε + ε uε γ + ε + vε ≤ C0 , √ for some constant C0 that depends only on the given data Here, ∇ε = ( ε∂x , ∂y ), and denotes the usual Lp norm over [0, L] × R+ L2 L2 L∞ L∞ · Lp As a direct corollary of our main theorem above, we obtain the inviscid limit of the steady Navier-Stokes flows, with prescribed data up to the order of square root of viscosity Corollary 1.2 Under the same assumption as made in Theorem 1.1, there are exact solutions [U, V ] to the original Navier-Stokes equations (1.1) on Ω = [0, L] × R+ , with L being as in Theorem 1.1, so that y sup U (x, y) − u0e (y) − u0p (x, √ ) ε2 ε (x,y)∈Ω √ √ γ y ε 2+2 sup V (x, y) − εvp (x, √ ) − εve1 (x, y) ε (x,y)∈Ω √ as ε → 0, for given Euler flow u0e , the constructed Euler flows [u1e , ve1 ] and Prandtl layers [u0p , εvp0 ] In particular, we have the convergence (U, V ) → (u0e , 0) in the usual Lp norm, with a rate of convergence of order ε1/2p , ≤ p < ∞, in the inviscid limit of ε → u, v¯] solving Remark 1.3 We note that the Prandtl layers [u0p , vp0 ] are not exactly the layers [¯ ∞ y 0 ¯x , but v¯ = − u ¯x for ¯, we have the normal velocity vp = y u (1.6)-(1.8) Indeed, whereas up = u 1 the true Prandtl layer The introduction of the Euler layer [ue , ve ] was necessary to correct this Let us give a few comments about the main result First, the nonzero condition (1.14) and ub > are naturally related to the situation where boundary layers are near a moving plate: such as a wake flow of a moving body, a moving plane jet flow, and a shear layer between two parallel flows It may also be related to the well-known fact in engineering that injection of moving fluids at the surface prevents the boundary layer separation It is widely known that the mathematical study of Prandtl boundary layers and the inviscid limit problem is challenging due to its characteristic nature at the boundary (that is, v = at y = 0) and the instability of generic boundary layers ([6, 7, 8, 9]) Here, for steady flows, we are able to justify the Prandtl boundary layer theory There are several issues to overcome The first is to carefully construct Euler and Prandtl solutions and derive sufficient estimates The complication occurs due to the fact that we have to truncate the actual layers in order to fit in our functional framework, and the lack of a priori estimates for linearized Prandtl equations The construction of the approximate solutions is done in Section Next, once the approximate solutions are constructed, we need to derive stability estimates for the remainder solutions Due to the limited regularity obtained for the Prandtl layers [u1p , vp1 ], we shall study the linearization around the following approximate solutions: √ √ √ √ vs (x, y) := vp0 (x, y) + ve1 (x, ε) (1.15) us (x, y) := u0e ( εy) + u0p (x, y) + εu1e (x, εy), A straightforward calculation (Section 5) yields the equations for the remainder solutions [uε , v ε , pε ] in (1.9): us uεx + uε usx + vs uεy + v ε usy + pεx − ∆ε uε = R1 (uε , v ε ) pεy − ∆ε v ε = R2 (uε , v ε ) us vxε + uε vsx + vs vyε + v ε vsy + ε uεx + vyε = 0, with ∆ε = ∂y2 + ε∂x2 Here, [us , vs ] denotes the leading approximate solutions (see (1.15)), and the remainders R1,2 (uε , v ε ) are defined as in (5.4) The standard energy estimate (Section 3.1) yields √ precisely a control on ∇ε uε L2 and ε ∇ε v ε L2 , but cannot close the analysis, due to the large convective term: usy uε v ε , for instance Indeed, this is a very common and central difficulty in the stability theory of boundary layers The most crucial ingredient (Section 3.2) in the proof is to give bound on ∇ε v ε L2 (in order √ one, instead of order ε from the energy estimate) The key is to study the vorticity equation, −us ∆ǫ v ε + vs ∆ǫ uε − uε ∆ǫ vs + v ε ∆ǫ us = ∆ǫ ω ε + R1y − εR2x ε with a new multiplier uv s Here, the assumption (1.14) and ub > 0, together with the Maximum Principle for the Prandtl equations (see estimate (2.8)), assure that us is bounded away from zero Formally, without worrying about boundary terms, the integral ∆ε ω ε v ε vanishes Hence, the leading term in the vorticity estimate lies in the convection: −us ∆ǫ v ε + v ε ∆ǫ us , or to leading order in the boundary layer analysis, −us ∂y2 v ε + usyy v ε Our key observation is then the positivity of the second-order operator: usyy −∂yy + us Indeed, a direct calculation yields − |vy | = vyy v = v us us ∂y u2s ∂y v us = = u2s ∂y = u2s ∂y v us v us 2 u2sy v us + u2sy v us + − usyy v us +2 ∂y − v us v us v us us usy [us usy ]y which gives the positivity estimate: −∂yy + usyy us |∂y v|2 + vv = usyy v = us u2s ∂y v us > (1.16) The desired bound on vyε , and in fact, ∇ε v ε is derived from this positivity estimate and the weighted estimates from the vorticity equation Precisely, vy2 = ≤2 ≤2 ∂y us u2s ∂y u2s ∂y u2s ∂y v us v us v us v us u2s ≤2 +2 ∂y v us v us × sup [1 + u2s x v us +2 {∂y us }2 × sup y{∂y us }2 dy y x (1.17) y{∂y us }2 dy] , in which the last inequality used the estimate (3.8) on us In addition, the Dirichlet boundary condition at x = and the stress-free boundary condition at x = L as imposed in (1.13) are carefully designed to ensure boundary contributions at x = and x = L are controllable Our second ingredient is to derive L∞ estimate for the remainder solution [uε , v ε ] and to close the nonlinear analysis; Sections and We have to overcome the issue of regularity of solutions to the elliptic problem in domains with corners In particular, it is a subtlety to justify the integrability of all terms in integration by parts, given the limited regularity provided for the solution near the corners We remark that in the case ub = 0, our analysis does not directly apply due to the ε presence of zero points of the profile solutions us , and hence the function uv s can no longer be used as a multiplier Our positivity estimate is lost in this limiting, but classical, case Finally, the third ingredient is the construction of profiles (or approximate solutions) which enables us to establish the error estimates and to close our nonlinear iteration Such constructions in the remainder R2 (uε , v ε ) In are delicate (Section 2), due to the regularity requirement of vpxx order to control it, we need to create artificial new boundary layer at y = in (2.30) to guarantee sufficient regularity for the first order Euler correction [u1e , ve1 , p1e ] More importantly, to construct both ve1 and vp1 , the positivity estimate (1.16) once again plays the decisive role; see Sections 2.2 and 2.4 We are not aware of any work in the literature that deals with the validity of the Prandtl boundary layer theory for the steady Navier-Stokes flows For unsteady flows, there are very interesting contributions [1, 15, 16] in the analyticity framework, [10] in the case where the initial vorticity is assumed to be away from the boundary, or [11] for special Navier-Stokes flows An analogous program for unsteady flows as done for the steady case in the precent paper appears not possible, due to the fact that (unsteady) boundary layers are known to be very unstable; see, for instance, [5, 6, 7, 8, 9] Notation Throughout the paper, we shall use y = + y , and · Lp or occasionally · p to denote the usual Lp norms, p ≥ 1, with integration taken over Ω = [0, L] × R+ We also use · Lp (0,L) and · Lp (R+ ) to denote the Lp norms with integration taken over [0, L] and R+ , respectively We shall denote by C(us , vs ) a universal constant that depends only on the given Euler flow u0e and boundary data Occasionally, we simply write C or use the notation in the estimates By uniform estimates, we always mean those that are independent of smallness of ε and L The smallness of L is determined depending only on the given data, whereas ε is taken arbitrarily small, once the given data and L are fixed In particular, ε ≪ L Construction of the approximate solutions In order to construct the approximate solutions, we plug the Ansatz (1.9) into the scaled NavierStokes equations (1.4), and match the order in ε to determine the equations for the profiles For our own convenience, let us introduce √ √ √ √ uapp (x, y) = u0e ( εy) + u0p (x, y) + εu1e (x, εy) + εu1p (x, y) √ √ (2.1) vapp (x, y) = vp0 (x, y) + ve1 (x, εy) + εvp1 (x, y) √ √ √ papp (x, y) = εpe (x, εy) + εpp (x, y) + εpp (x, y) We then calculate the error caused by the approximation: u Rapp := [uapp ∂x + vapp ∂y ]uapp + ∂x papp − ∆ε uapp v Rapp := [uapp ∂x + vapp ∂y ]vapp + ∂y papp − ∆ε vapp , ε or explicitly, √ √ √ u Rapp = {u0e + u0p + ε[u1e + u1p ]}∂x + {vp0 + ve1 + εvp1 }∂y {u0e + u0p + ε[u1e + u1p ]} √ √ √ + ε∂x {p1e + p1p + εp2p } − (∂y2 + ε∂x2 ){u0e + u0p + ε[u1e + u1p ]} √ √ √ v Rapp = {u0e + u0p + ε[u1e + u1p ]}∂x + {vp0 + ve1 + εvp1 }∂y {vp0 + ve1 + εvp1 } √ √ + √ ∂y {p1e + p1p + εp2p } − (∂y2 + ε∂x2 ){vp0 + ve1 + εvp1 }, ε (2.2a) (2.2b) √ in which we recall that the Euler profiles are always evaluated at (x, z) = (x, εy) We shall u,v construct the approximate solutions so that Rapp are being small in ε In this long section, we shall prove that Proposition 2.1 Under the same assumptions as in Theorem 1.1, there are approximate solutions [uapp , vapp , papp ] so that √ u v 3/4−κ Rapp , L2 + ε Rapp L2 ≤ C(L, κ)ε for arbitrarily small κ > Furthermore, there hold various regularity estimates on the approximate solutions which are summarized in Corollary 2.3 for the zeroth-order Prandtl layers [u0p , vp0 ], Section 2.3.1 for the Euler profiles [u1e , ve1 ], and Section 2.4.2 for the Prandtl layers [u1p , vp1 ] 2.1 Zeroth-order Prandtl layers The (leading) zeroth order terms on the right-hand side of (2.2a) consist of Ru,0 := {u0e + u0p }{u0e + u0p }x + {vp0 + ve1 }{u0e + u0p }y − {u0e + u0p }yy √ in which we note that {u0e }x = Since the Euler flows are evaluated at (x, z) = (x, εy), we may write √ {vp0 + ve1 }∂y u0e = ε{vp0 + ve1 }u0ez and, with ue = u0e (0), u0e u0px + ve1 u0py = ue u0px + ve1 (x, 0)u0py + √ √ √ √ εuez ( εy)yu0px + εvez ( εy)yu0py + E in which E satisfies E := = √ y εu0px εu0px y √ √ √ {u0ez ( εθ) − u0ez ( εy)}dθ + εu0py θ y √ u0ezz ( ετ )dτ dθ + εu0py y θ y y √ √ ( εy)}dθ {vez ( εθ) − vez √ vezz ( ετ )dτ dθ (2.3) In particular, E0 is in the high order in ε, as to be proved rigorously in the next section; see (2.40) To leading order, this yields the nonlinear Prandtl problem for u0p :  ∞   {ue + u0p }u0px + {vp0 + ve1 (x, 0)}u0py = u0pyy , u0px dy, vp0 (x, y) := (2.4) y   u0 (x, 0) = u − u , vp (x, 0) + ve (x, 0) = 0, up (0, y) = u ¯0 (y) e b p Having constructed the Prandtl layer [u0p , vp0 ], the zeroth order term Ru,0 is reduced to Ru,0 = √ ε{vp0 + ve1 }u0ez + √ √ √ √ εuez ( εy)yu0px + εvez ( εy)yu0py − εu0ezz + E , (2.5) which will be put into the next order in ε Lemma 2.2 Let u0p (0, y) := u ¯0 (y) be an arbitrary smooth boundary data for the Prandtl layer at x = Assume that miny {ue + u ¯0 (y)} > Then, there exists a positive number L so that the problem (2.4) has the unique smooth solution u0p (x, y) in [0, L] × R+ Furthermore, for all n ≥ 0, k ≥ 0, there exists a constant C0 (n, k, u ¯0 ) so that there holds the uniform bound: sup y x∈[0,L] n/2 k ∂x up L2 (R+ ) + y n/2 k ∂x ∂y u0p L2 (0,L;L2 (R+ )) ≤ C0 (n, k, u¯0 ), (2.6) with y = + |y|2 Here, the constant C(n, k, u ¯0 ) depends on n, k, and the y n -weighted H 2k (R+ ) norm of the boundary value u ¯0 (y) Corollary 2.3 Let u0p be the Prandtl layer constructed as in Lemma 2.2 Then, there holds sup n/2 k j 0 ∂x ∂y [up , vp ] L2 (R+ ) y x∈[0,L] ≤ C0 (n, k, j, u¯0 ), (2.7) for arbitrary n, k, j Proof Indeed, the proof follows directly from Lemma 2.2 and a use of equations (2.4) for the Prandtl layer to bound ∂y2 u0p by those of lower-order derivative terms Proof of Lemma 2.2 Let ue = u0e (0) Following Oleinik [12], we use the von Mises transformation: y η := (ue + u0p (x, θ))dθ, w(x, η) := ue + u0p (x, y(η)) The function w then solves wx = {wwη }η on [0, L] × R+ Note that by the standard Maximum Principle (to the equation for w2 ), we have w ≥ min{ub , ue + u0p |x=0 } ≥ c0 > 0, (2.8) y for some positive constant c0 Hence, the above is a non-degenerate parabolic equation Since w does not vanish on the boundary, we introduce w = w − ue − [ub − ue ]e−η Hence, it follows that w vanishes at both y = and y = ∞, and there holds wx = [wwη ]η − [ub − ue ][we−η ]η − Fη , F (η) := [ub − ue ][ue + [ue − ub ]e−η ]e−η (2.9) Clearly, η n F (·) ∈ W k,p (R+ ), for arbitrary n, k ≥ and p ∈ [1, ∞] We shall solve this equation via the standard contraction mapping First, let us derive a priori weighted estimates We introduce the following weighted iterative norm: j Nj (x) := j η k=0 n |∂xk w|2 x + k=0 η n |∂xk wη |2 , j ≥ (2.10) By multiplying the equation (2.9) by η n w, it follows the standard weighted energy estimate: d dx η n |w|2 + η n w|wη |2 ≤ η n−1 |w||wη | + η n e−η |wwη | + η n |wFη | , for n ≥ 0, which together with the Young’s inequality yields d dx η n |w|2 + η n w|wη |2 ≤ C 10 η n |w|2 + C η n |Fη |2 equation multiplied by the test function v us ∂y − = v us ∂y This yields {us ux + usx u + vs uy + v∂y us + px − ∆ε u} v us ∂x v us {ε[us vx + u∂x vs + vs vy + v∂y vs ] + py − ε∆ε v} f − ε∂x v us (3.10) g √ 1/2 y Again, we use the inequality |v| ≤ y vy2 , together with the estimates (3.6)-(3.8) on [us , vs ] to estimate the right-hand side of (3.10) We have v∂y us vy − us u2s supx | y ≤ { f L2 1/2 ∇ f −ε vx v∂x us − us u2s ε us |2 + sup us {min us }2 √ + ε g L2 } ∇ ε v L2 , ∞ { f L2 g + √ ε g L2 } ∇ε v L2 in which the Young inequality can be applied to absorb the L2 norm of ∇ε v to the left hand side of (3.9) Next, we treat each term on the left-hand side of (3.10) First, integrating by parts multiple times, we have ∂y = ∂y v us v us {us ux + v∂y us } − = − = − = − = − εus vx vx ∂x us v − us u2s {−us vy + v∂y us } − vy ∂y us v − us u2s = v us ∂x εus vx ε∂x us vvx u2s 2 {∂y us } v ε∂y us vvx ∂y us − −ε vx2 + vy2 + vvy us us u2s ∂y us ε∂y us vvx {∂y us }2 vy2 − ∂y −ε vx2 + v2 − v2 us us u2s ε∂y us vvx ∂yy us v −ε vx2 + vy2 − us u2s ε∂y us vvx v u2s |∂y , |2 − ε vx2 + us u2s vx2 + {−us vy + v∂y us } − ε in which the last equality is precisely due to the positivity estimate (1.16) From (1.17), we obtain a lower bound ∂y v us {us ux + v∂y us } − ∂x v us εus vx − |∇ε v|2 + ε∂y us vvx , u2s (3.11) which crucially yields a bound on the L2 norm of ∇ε v; or precisely, the L2 norm of ∇ε v appearing on the left-hand side of (3.9) 32 Next, we treat the pressure term Integrating by parts, with recalling that p = 2εux at x = L, we have ∂y v us px − ∂x v us v us ∂y py = x=L ∂y = 2ε x=L = −2ε x=L p v us ux = −2ε vy2 − 2ε us ∂y x=L v us ∂y x=L us vy vvy in which we can estimate ∂y ε x=L us vvy ≤ ε ≤ x=L ∂y us ∂y us vvy ≤ ε sup 3/2 us us {C(u0ey , u0py ) 1/2 +ε u1ez ∞ }Lε vy v√ us x=L vx L2 x=L vy2 us 1/2 Here, thanks to the bounds on the Euler flows, summarized in Section 2.3.1, we in particular have ε1/2 u1ez ∞ ≤ Cε1/2 u1e H Together with the Young inequality, we thus obtain v us ∂y px − ∂x v us py ε ≤− x=L vy2 + C(us , vs )L2 ∇ε v us L2 , (3.12) in which we stress that the boundary term is favorable Next, we shall treat terms involving the Laplacian Again, we recall from [13, 14] that the Stokes problem yields [u, v] ∈ H 3/2+ and p ∈ H 1/2+ , and so their traces [∇u, ∇v] ∈ L2+ (Γ) and p ∈ L2+ (Γ) on any smooth curve Γ Moreover, [u, v] ∈ H and p ∈ H away from the four corners of [0, L] × [0, N ] Hence, we can evaluate − ∂y I := − ∂y = v us v us ∆ε u + ∂x v us ε∆ε v [uyy + εuxx ] + ∂x v us ε[vyy + εvxx ] vy v∂y us vx v∂x us − [uyy + εuxx ] + − ε[vyy + εvxx ] us us us u2s uyy ux vx vxx uxx ux vx vyy +ε +ε + ε2 us us us us v∂y us v∂x us [uyy + εuxx ] − [εvyy + ε2 vxx ] us u2s − = = + Now taking integration by parts respectively in each integration above, with a special attention on 33 the boundary contributions, we get ε + I = − ε ε2 + − − +ε u2y 1 + ∂x us x=L us ux ε − ∂x us x=L us vy ε + ∂x us x=L us 2 vx ε vx − − u u x=L s x=0 s v∂y us ∂y uy − ε u2s v∂x us ∂y vy + ε2 u2s u2y − ∂y us uy ux (3.13) u2x vy2 − ε ε2 ∂x ∂y us us vx2 vy v x v∂y us v∂y us ux + ε ux us u2s x=L v∂y us v∂x us vx − ε2 vx ∂x u2s u2s x=L ∂x Let us first take care of boundary contributions Notice that there is only one boundary term at 2 x = 0, which is a favorable term: − ε2 x=0 uvxs Now as for boundary terms at x = L, one can use the fact that uy + εvx = and ux + vy = at x = L, and hence, the boundary contributions at x = L can be simplified as B L := − x=L +ε x=L = −ε x=L = −ε x=L u2y ε + us v∂y us u2s v∂y us u2s v∂y us u2s vy2 vx2 ε2 + x=L us x=L x=L us v∂y us ux − ε2 vx u2s x=L v∂y us vy − ε uy u2s x=L v∂y us ∂y vy + ε u, u2s x=L u2x ε − us which can be estimated by √ ∂y us B L ≤ ε L sup us x +εL sup x vx L2 x=L √ vy2 ∂y us + ε L sup us us x y|∂yy us | dy + sup u2s x y{∂y us }2 dy u3s ux ux L2 L2 x=L x=L vy2 us vy2 us √ We estimate norms on us in the same lines as done in (3.6)-(3.8) Recall that us = u0e + u0p + εu1e and us is bounded below away from zero thanks to the assumption (1.14) Now, similarly as done 34 for (3.8), we have y|usyy |dy ≤ sup sup x x y ε|u0ezz | + |u0pyy | + ε3/2 |u1ezz | dy ≤ C(u0p ) + sup z|u0ezz | + ε1/2 z|u1ezz | dz x ≤ C(u0e , u0p ) + ε1/2 L−1 z|u1ezz |dxdz + ≤ C(u0e , u0p ) + ε1/2 L−1 z n ve1 −1/2+1/q + C(L)ε (3.14) z|u1exzz |dxdz W 3,q , for any q > Here, Lemma 2.5 was used Taking q → in the above estimates so that −1/2+1/q > 0, the above is bounded uniformly by a constant C(us , vs ), which is independent of small ε, L The integral y|usy |2 dy is already estimated in (3.8) Also, we get √ (3.15) usy ∞ ≤ ε u0ez ∞ + u0py ∞ + ε u1ez ∞ ≤ C(u0e , u0p ) + ε ve1 (·) H ≤ C This together with the Young inequality yields vy2 + C(us , vs )L ∇ε v 22 , (3.16) u x=L s √ upon using the divergence-free condition ux = −vy Here, ∇ε = ( ε∂x , ∂y ) The boundary term in (3.16) can be absorbed into the good boundary term in (3.12) We now combine the untreated terms on the left-hand side of (3.10), all the interior terms in (3.13), and the last term in (3.11), altogether We shall use the standard embedding inequalities √ √ 1/2 1/2 y y u L2 ≤ L ux L2 , |v| ≤ y vy2 and |u| ≤ y u2y Respectively, we have BL ≤ R0 := − + ε 1 ∂x u2y − ∂y uy ux us us ε ε ε2 1 − ∂x u2x + ∂x vy2 − ε∂y vy vx − ∂x vx2 us us us us v∂y us v∂y us v∂x us v∂x us ∂y uy − ε∂x ux + ε∂y vy + ε2 ∂x vx u2s u2s u2s u2s ε∂y us vvx v v ∂y {∂x us u + vs uy } − ε∂x {u∂x vs + vs vy + v∂y vs } + us us u2s Let us give estimates on each term on the right We claim that R0 ∇ε u L2 + ∇ε u L2 ∇ε v L2 √ + C(L) ε ∇ε v L2 , (3.17) Proof of (3.17) First, we have ∂x us u2y − ∂y us uy ux ≤ C ∇us 35 ∞ uy L2 ( uy L2 + vy L2 ) in which the bounds (3.5) and (3.15) gives ∇us √ ∇ε = ( ε∂x , ∂y ), the Hă older inequality yields x us ≤ C ∇us us ε u2x + ∂x ∇ε u ∞ L2 v∂y us u2s + ∇ε u ∂y us u2s uy = ≤C usy us vy2 − ε∂y ∇ε v L2 which again gives the bound as claimed, since ∇us right of R0 We first have ∂y Next, upon recalling the definition ∞ ∂y ∞ ∞ + sup x + √ us vx2 L2 ε ∇ε v We now estimate the third line on the ∞ u y vy + L2 ε2 ∂x vy vx − ∂y us u2s uy v y(|usyy |2 + |usy |4 ) 1/2 uy L2 vy L2 , and ε∂x v∂y us u2s ∂y us u2s ux = −ε ≤C √ ε usy ∞ vy vx − ε ∂x ∞ + ε sup x ∂y us u2s vy v y(|usxy |2 + |usx usy |2 ) 1/2 ∇ε v L2 The last two terms on the third line on the right of R0 can be estimated very similarly We give bounds on the norms of [us , vs ] Similarly as done in (3.14), we get sup x y|usyy |2 dy ≤ sup x y ε2 |u0ezz |2 + |u0pyy |2 + ε3 |u1ezz |2 dy ≤ C(u0p ) + sup x εz|u0ezz |2 + ε2 z|u1ezz |2 dz ≤ C(u0e , u0p ) + ε2 L−1 z|u1ezz |2 dxdz + ≤ C(u0e , u0p ) + ε2 L−1 z n ve1 z|u1exzz |2 dxdz H3 + C(L)ε, and sup x y|usy |4 dy ≤ sup x y ε2 |u0ez |4 + |u0py |4 + ε4 |u1ez |4 dy ≤ C(u0e , u0p ) + ε3 L−1 z|u1ez |4 dxdz + ≤ C(u0e , u0p ) + ε3 L−1 z n ve1 z|u1exz |4 dxdz W 2,4 + C(L)ε3 , both of which are thus bounded, thanks to the estimates from Lemma 2.5 Same bounds can be given for the weighted integrals of ε|usxy |2 and ε|usx |4 , using the extra factor of ε in these integrals 36 In addition, we have εy |u0pxx | + ε1/2 |u1exx | dy εy|usxx |dy ≤ sup sup x x ≤ C(u0p ) + ε1/2 sup z|vexz | dz x ≤ C(u0p ) + ε1/2 L−1 z|vexz |dxdz + ≤ C(u0e , u0p ) + ε1/2 L−1 z n ve1 z|vexxz |dxdz W 3,q + C(L)ε−3/2+2/q , which are again bounded, thanks to the Euler bounds from Lemma 2.5, with q being arbitrarily close to We now give bounds on the last line on the right of R0 We have v us ∂y = {∂x us u + vs uy } ∂y ≤ C sup x us us {∂x us u + vs uy }v + ∞ {∂x us u + vs uy }vy 1/2 y(|usy usx | + |usy vs |2 + |usx |2 ) + vs uy ∞ L2 vy L2 in which we note that vs is uniformly bounded The estimate (3.8) gives the weighted bound on usy Next, we estimate sup x y{usx }2 dy ≤ sup x y |u0px |2 + ε|u1ex |2 dy ≤ C(u0p ) + sup z|vez | dz ≤ C(u0p ) + C ||x=0 dz + z|vez x | dxdz + z|vez |2 dxdz , z|vexz This gives the desired bound on the first term on the last line in R0 Next, we have ε ∂x =ε v us {u∂x vs + vs vy + v∂y vs } ∂x us ≤ Cε sup x ∞ us {u∂x vs + vs vy + v∂y vs }v + ε y(|usx vsx | + |usx vs |2 + |usx vsy |) + C(L)ε ||vsx ||∞ + vs ∞ + sup x ∞ y|vsy |2 1/2 uy 1/2 {u∂x vs + vs vy + v∂y vs }vx L2 ux vy L2 L2 vx L2 in which the last estimate used the inequality: u L2 ≤ L ux L2 and the divergence-free condition vy = −ux As estimated above, it remains to give a uniform estimate on ||vsx ||∞ ≤ ||vpx ||∞ + ||vex ||∞ 37 + ve1 W 2,q ≤ C(L) thanks to the estimates on ve1 , with q > Finally, we estimate ε ∂y us vvx ≤ Cε sup u2s x ∞ y|usy |2 1/2 vy L2 vx L2 in which the integral y|usy |2 dy is already estimated in (3.8) Putting all above estimates together, we have completed the proof for the claim (3.17) Finally, using the Young inequality and the smallness of ε, the L2 norm of ∇ε v can be absorbed into the left-hand side of (3.9) This completes the proof of the positivity estimate and the lemma 3.3 Proof of Proposition 3.1 The proof of Proposition 3.1 now follows straightforwardly from the energy estimate (Lemma 3.2) and the positivity estimate (Lemma 3.3), as a direct application of the Schaefer’s fixed point theorem; see [4, Theorem 4, p 504] Indeed, first combining these estimates together and choosing L sufficiently small, we get ∇ε u L2 + ∇ε v L2 ≤ C(us , vs ) f L2 + √ ε g L2 uniformly in N Taking N → ∞ yields the stability estimate (3.4) To apply the fixed point theorem of Schaefer ([4, Theorem 4, p 504]), we consider the following system u + px − ∆ε u = λ f + u − [us ux + uusx + vs uy + vusy ] v v py + − ∆ε v = λ g + − [us vx + uvsx + vs vy + vvsy ] ε ε ε ux + vy = 0, or in the operator form S[u, v] = λT [u, v], for parameter λ ∈ [0, 1] The existence of a solution to (3.1)-(3.2) is equivalent to the existence of a fixed point of S −1 T The compactness of the operator S −1 T follows directly from that of the Stokes operator To derive uniform bounds on the set of solutions [uλ , v λ ], we may rewrite the above system as (1 − λ)u + λ[us ux + uusx + vs uy + vusy ] + px − ∆ε u = λf py v − ∆ε v = λg (1 − λ) + λ[us vx + uvsx + vs vy + vvsy ] + ε ε together with the divergence-free condition and the same boundary conditions (3.3) The uniform estimates now follow almost identically from the above energy estimates and positivity estimates We omit to repeat the details This completes the proof of Proposition 3.1 L∞ estimates In order to perform nonlinear iteration, we shall need to derive bounds in L∞ for the solution We prove the following: 38 Lemma 4.1 Consider the scaled Stokes system py − ∆ε v = g ε px − ∆ε u = f, together with the divergence-free condition ux + vy = and the (same) boundary conditions [u, v]y=0 = (no-slip), p − 2εux = 0, [u, v]x=0 = (Dirichlet), uy + εvx = (4.1) at x = L (Neumann or stress-free) Then, there holds γ ε4 u γ ∞ + ε +2 v Cγ,L ∞ u H1 + √ ε v H1 + f L2 + √ ε g L2 , for some constant Cγ,L Proof Since our rectangle domain can be covered by two C 0,1 charts, we may apply the standard extension theorem (see, for instance, [3, Theorem 5.4]) such that there exists u ¯ ∈ H 1+β (R2 ) and 1+β v¯ ∈ H (R ), for β ∈ (0, 1), such that [¯ u, v¯] = [u, v] in [0, L] × R+ , and u ¯ H 1+β ≤ Cβ,L u H 1+β , v¯ H 1+β ≤ Cβ,L v H 1+β , for some constant Cβ,L that depends only on β and L By the Sobolev’s imbedding in R2 and an interpolation inequality for u ¯ and v¯, we have for any < τ < α, u ∞ ≤ u ¯ ∞ ≤ u ¯ H 1+τ α−τ H1 α ≤ Cτ,L [ u ¯ ≤ Cτ,α,L [ u ] H1 ] α−τ α and similarly, ε1/2 v ∞ ≤ Cτ,α,L [ε1/2 v [ u ¯ H1 ] [ u α−τ α τ H 1+α ] α τ H 1+α ] α , [ε1/2 v τ H 1+α ] α for some constant Cτ,α,L Here, we have used the standard interpolation between Sobolev spaces H , H 1+τ , and H 1+α , with < τ < α We note that thanks to our uniform estimates for u H √ and ε1/2 v H , it suffices to give estimates on the H 1+α norm of [u, εv] In what follows, we fix α ∼ 1/2 and take τ so that τ such that there holds √ √ √ u H 1+α + ε v H 1+α ε−mα { f L2 + ε g L2 + u H + ε v H } (4.2) Given the claim, we then have u ∞ + √ ε v ∞ ≤ Cτ,α,L ε− mα τ α f L2 + √ ε g L2 + u H1 + √ ε v H1 The lemma would then be proved at once by choosing τ ≪ α so that mαα τ ≤ γ4 We shall now prove the claim for α = 1/2 and mα = 7/4 To so, let us introduce the (original) scaling: y uε (x, y) ≡ u(Lx, L √ ), ε vε (x, y) ≡ √ y εv(Lx, L √ ), ε 39 pε (x, y) ≡ L y p(Lx, L √ ) ε ε (4.3) Clearly, direct calculations yield uεx + vεy = and y y L2 uyy (Lx, L √ ), ∂xx uε (x, y) = L2 uxx (Lx, L √ ), ∂yy uε (x, y) = ε ε ε √ y L y ∂xx vε (x, y) = L2 εvxx (Lx, L √ ), ∂yy vε (x, y) = 1/2 vyy (Lx, L √ ) ε ε ε y y L2 L2 px (Lx, L √ ), ∂x pε (x, y) = ∂y pε (x, y) = 3/2 py (Lx, L √ ) ε ε ε ε Plugging these in the Stokes problem, we yield a normalized Stokes system: y pεx − ∆uε = ε−1 L2 f (Lx, L √ ), ε y pεy − ∆vε = ε− L2 g(Lx, L √ ) ε (4.4) in a fixed domain ≤ x ≤ and ≤ y ≤ ∞ with boundary conditions [uε , vε ] = on both boundaries: {y = 0} and {x = 0} pε + 2uεx = 0, vεx + uεy = at x = We now invoke the standard elliptic estimate for the Stoke problem in such a fixed domain Recall that there holds the Poincare’s inequality: uε L2 ≤ uεx L2 , vε L2 ≤ vεx L2 Next, the standard energy estimates yield ∇uε L2 + ∇vε L2 Ly Ly ≤ C{ε−1 L2 f (Lx, √ ) L2 + ε− L2 g(Lx, √ ) ε ε √ ≤ Cε−3/4 L f L2 + ε g L2 L2 } (4.5) Next, let us give an L2 estimate on the pressure pε First, for h ∈ L2 , we show that there is a h L2 Invector-valued function φ ∈ H such that φ(0, y) ≡ 0, φ(x, 0) ≡ 0, ∇ · φ = h and φ H ∞ deed, we can decompose h = n=0 1n≤y