Prandtl boundary layer expansions of steady navier stokes flows over a moving plate (2)

arXiv:1411.6984v1 [math.AP] 25 Nov 2014Prandtl boundary layer expansions of steady Navier-Stokes flows over a moving plate December 1, 2014 Abstract This paper concerns the validity of t

arXiv:1411.6984v1 [math.AP] 25 Nov 2014

Prandtl boundary layer expansions of steady Navier-Stokes flows

over a moving plate

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 = 0.

We establish the validity of the Prandtl boundary layer expansion and its error estimates.

Contents

1.1 Boundary conditions 4

1.2 Main result and discussions 5

2 Construction of the approximate solutions 8 2.1 Zeroth-order Prandtl layers 9

2.2 ε1/2-order corrections 13

2.3 Euler correctors 14

2.3.1 Euler profiles 17

2.4 Prandtl correctors 18

2.4.1 Construction of Prandtl layers 19

2.4.2 Cut-off Prandtl layers 25

2.5 Proof of Proposition 2.1 27

3 Linear stability estimates 28 3.1 Energy estimates 29

3.2 Positivity estimates 31

3.3 Proof of Proposition 3.1 38

∗ 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.

We are interested in the problem when ε→ 0 The study of the inviscid limit and asymptoticboundary layer expansions of Navier-Stokes equations (1.1) in the presence of no-slip boundarycondition is one of the central problems in the mathematical analysis of fluid mechanics A for-mal limit ε → 0 should lead the Euler flow [U0, V0] inside Ω which satisfies only non-penetrationcondition at Y = 0 :

V0(X, 0) = 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 Generi-cally, there is a mismatch between the tangential velocities of the Euler flow ue≡ U0(X, 0)≡ u0

e(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 ofsize √

ε to connect different velocities ue and ub We shall work with the scaled boundary layer, orPrandtl’s, variables:

] = [Uε(x, y),√

εVε(x, y)]

in which we note that the scaled normal velocity Vε is √1

ε of the original velocity V Similarly,

P (X, Y ) = Pε(x, y) In these new variables, the Navier-Stokes equations (1.1) now read

UεUxε+ VεUyε+ Pxε= Uyyε + εUxxε

UεVxε+ VεVyε+P

ε y

Throughout the paper, we perform our analysis directly on these scaled equations together withthe same no-slip boundary conditions (1.2) Prandtl then hypothesized that the Navier-Stokes flowcan be approximately decomposed into two parts:

[ue+ ¯u]¯ux+ v¯uy+ ¯px= ¯uyy

¯

ux+ ¯vy = 0 (1.6)where ¯px = ¯px(x), since by the second equation in (1.4), ¯py = 0 Evaluating the first equation at

y = ∞, one gets ¯px = 0, which is precisely due to the assumption that the trace of Euler on theboundary 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

y→∞u(x, y) = 0.¯ (1.7)The Prandtl layer [¯u, ¯v] is subject to an “initial” condition at x = 0:

¯u(0, y) = ¯u0(y) (1.8)Regarded as one of the most important achievements of modern fluid mechanics, Prandtl’s theboundary 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 Inparticular, since the Prandtl layer solution [¯u, ¯v] satisfies an evolution equation in x, it is mucheasier to compute its solutions numerically than those of the original NS flows [Uε, Vε] whichsatisfy an elliptic boundary-value problem Many other shear layer phenomena in fluids, such aswake flows ([17, page 187]), plane jet flows ([17, page 190]), as well as shear layers between twoparallel 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 anoutstanding 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 scaledNavier-Stokes solutions [Uε, Vε, Pε] in the following form:

profile solutions also depend on ε, and the Euler flows are always evaluated at (x,√

εy), whereasthe Prandtl profiles are at (x, y)

Formally speaking, plugging the above ansatz into (1.4) and matching the order in ε, we ily get that [u0p, vp0, 0] solves the nonlinear Prandtl equation (1.6), whereas the next Euler profile[u1

(ue+ u0p)u1px+ u1pu0px+ (v0p+ v1e)u1py+ v1pu0px+ p1px− u1pyy = Ep,

u1px+ v1py= 0, (1.12)with p1p = p1p(x) Certainly, the remainder solutions [uε, vε, pε] solve the linearized Navier-Stokesequations around the approximate solutions, with a source that contains nonlinearity in [uε, vε];see Section 5 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 nonzeronormal velocity at infinity, and is one of the issues in controlling the remainders, since the profilesthen won’t even be integrable As a result, our Prandtl layers [u1p, v1p] in the expansion (1.9) arebeing cut-off for large y of the actual layers solving (1.12) In Section 2, we shall provide detailedconstruction of the approximate solutions and derive sufficient estimates for our analysis

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

u1p(x, 0) =−u1e(x, 0), vp1(x, 0) = 0,from the √

ε-order layers We also assume that

Next, we discuss boundary conditions at x = 0, L Since the Prandtl layers solve parabolic-typeequations, we require only “initial” conditions at x = 0:

u0p(0, y) = ¯u0(y), u1p(0, y) = ¯u1(y),whereas we prescribe boundary values for the Euler profiles at both x = 0, L:

u1e(0, Y ) = u1b(Y ), v1e(0, Y ) = Vb0(Y ), ve1(L, Y ) = VbL(Y )with compatibility conditions Vb0(0) = v0p(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, vε]:

[uε, vε]y=0 = 0 (no-slip), [uε, vε]x=0 = 0 (Dirichlet),

pε− 2εuεx = 0, uεy+ εvxε = 0 at x = L (Neumann or stress-free) (1.13)Certainly, one may wish to consider different boundary conditions for [uε, vε] at x = L However,

to avoid a possible formation of boundary layers with respect to x near the boundary x = L, theabove Neumann stress-free condition appears the most convenient candidate to impose

We are ready to state our main result:

Theorem 1.1 Let u0(Y ) be a given smooth Euler flow, and let u1

b(Y ), Vb0(Y ), VbL(Y ), with Y =

√

εy, and ¯u0(y) and ¯u1(y) be given smooth data and decay exponentially fast at infinity in theirarguments, and let ub be a positive constant We assume that |VbL(Y )− Vb0(Y )| L for small L,and

min

0≤y≤∞u0

e(√εy) + ¯u0(y) > 0 (1.14)Then, there exists a positive number L that depends only on the given data so that the boundarylayer expansions (1.9), with the profiles satisfying the boundary conditions in Section 1.1, hold for

γ ∈ (0,14) Precisely, [Uε, Vε, Pε] as defined in (1.9) is the unique solution to the Navier-Stokesequations (1.4), so that the remainder solutions [uε, vε] satisfy

k∇εuεkL 2 + ε1k∇εvεkL 2 + εγ2kuεkL ∞ + ε1+γ2kvεkL ∞ ≤ C0,for some constant C0 that depends only on the given data Here, ∇ε = (√

ε∂x, ∂y), and k · kL p

denotes the usual Lp norm over [0, L]× R+

As a direct corollary of our main theorem above, we obtain the inviscid limit of the steadyNavier-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 Theorem1.1, so that

sup

(x,y)∈Ω

sup

(x,y)∈Ω

as ε→ 0, for given Euler flow u0, the constructed Euler flows [u1, v1] and Prandtl layers [u0,√

εv0]

In particular, we have the convergence (U, V ) → (u0

e, 0) in the usual Lp norm, with a rate ofconvergence of order ε1/2p, 1≤ p < ∞, in the inviscid limit of ε → 0

Remark 1.3 We note that the Prandtl layers [u0p, vp0] are not exactly the layers [¯u, ¯v] solving(1.6)-(1.8) Indeed, whereas u0

p = ¯u, we have the normal velocity v0

p =R∞

y u¯x, but ¯v =−Ry

0 u¯x forthe true Prandtl layer The introduction of the Euler layer [u1e, v1e] was necessary to correct this.Let us give a few comments about the main result First, the nonzero condition (1.14) and

ub > 0 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 parallelflows 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 inviscidlimit problem is challenging due to its characteristic nature at the boundary (that is, v = 0 at

y = 0) and the instability of generic boundary layers ([6, 7, 8, 9]) Here, for steady flows, we areable 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 complicationoccurs due to the fact that we have to truncate the actual layers in order to fit in our functionalframework, and the lack of a priori estimates for linearized Prandtl equations The construction ofthe approximate solutions is done in Section 2

Next, once the approximate solutions are constructed, we need to derive stability estimates forthe remainder solutions Due to the limited regularity obtained for the Prandtl layers [u1

p, v1

p], weshall study the linearization around the following approximate solutions:

ε − ∆εvε = R2(uε, vε)

uεx+ vεy = 0,with ∆ε= ∂y2+ ε∂x2 Here, [us, vs] denotes the leading approximate solutions (see (1.15)), and theremainders R1,2(uε, vε) are defined as in (5.4) The standard energy estimate (Section 3.1) yieldsprecisely a control on k∇εuεkL 2 and √

εk∇εvεkL 2, but cannot close the analysis, due to the largeconvective term: R usyuεvε, for instance Indeed, this is a very common and central difficulty inthe stability theory of boundary layers

The most crucial ingredient (Section 3.2) in the proof is to give bound on k∇εvεkL 2 (in orderone, instead of order√

ε from the energy estimate) The key is to study the vorticity equation,

−us∆ǫvε+ vs∆ǫuε− uε∆ǫvs+ vε∆ǫus= ∆ǫωε+ R1y− εR2x.with a new multiplier uvε

s Here, the assumption (1.14) and ub > 0, together with the MaximumPrinciple for the Prandtl equations (see estimate (2.8)), assure that us is bounded away from zero.Formally, without worrying about boundary terms, the integral R ∆εωεvε vanishes Hence, the

leading term in the vorticity estimate lies in the convection: −us∆ǫvε+ vε∆ǫus, or to leading order

in the boundary layer analysis,−us∂2

yvε+ usyyvε Our key observation is then the positivity of thesecond-order operator:

−∂yy+usyy

us .Indeed, a direct calculation yields

∂y v

usus



∂y v

us



2

,

(1.17)

in which the last inequality used the estimate (3.8) on us

In addition, the Dirichlet boundary condition at x = 0 and the stress-free boundary condition

at x = L as imposed in (1.13) are carefully designed to ensure boundary contributions at x = 0and x = L are controllable

Our second ingredient is to derive L∞ estimate for the remainder solution [uε, vε] and to closethe nonlinear analysis; Sections 4 and 5 We have to overcome the issue of regularity of solutions tothe 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 thecorners We remark that in the case ub = 0, our analysis does not directly apply due to thepresence of zero points of the profile solutions us, and hence the function vuε

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) whichenables us to establish the error estimates and to close our nonlinear iteration Such constructionsare delicate (Section 2), due to the regularity requirement of v1pxx in the remainder R2(uε, vε) Inorder to control it, we need to create artificial new boundary layer at y = 0 in (2.30) to guaranteesufficient regularity for the first order Euler correction [u1

e, v1

e, p1

e] More importantly, to constructboth ve1 and v1p, the positivity estimate (1.16) once again plays the decisive role; see Sections 2.2and 2.4

We are not aware of any work in the literature that deals with the validity of the Prandtlboundary layer theory for the steady Navier-Stokes flows For unsteady flows, there are veryinteresting contributions [1, 15, 16] in the analyticity framework, [10] in the case where the initialvorticity is assumed to be away from the boundary, or [11] for special Navier-Stokes flows Ananalogous program for unsteady flows as done for the steady case in the precent paper appears notpossible, due to the fact that (unsteady) boundary layers are known to be very unstable; see, forinstance, [5, 6, 7, 8, 9]

Notation Throughout the paper, we shall use hyi = p1 + y2, and k · kL p or occasionally

k · kp to denote the usual Lp norms, p≥ 1, with integration taken over Ω = [0, L] × R+ We alsouse k · kL p (0,L) and k · kL p (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 givenEuler flow u0e and boundary data Occasionally, we simply write C or use the notation in theestimates By uniform estimates, we always mean those that are independent of smallness of εand L The smallness of L is determined depending only on the given data, whereas ε is takenarbitrarily small, once the given data and L are fixed In particular, ε≪ L

2 Construction of the approximate solutions

In order to construct the approximate solutions, we plug the Ansatz (1.9) into the scaled Stokes equations (1.4), and match the order in ε to determine the equations for the profiles Forour own convenience, let us introduce

We then calculate the error caused by the approximation:

Rappu := [uapp∂x+ vapp∂y]uapp+ ∂xpapp− ∆εuapp (2.2a)

Rappv := [uapp∂x+ vapp∂y]vapp+1

ε[u1e+ u1p]}+√

εvp1}+√1

ε∂y{p1e+ p1p+√

εp2p} − (∂y2+ ε∂2x){vp0+ v1e+√

εv1p},

in which we recall that the Euler profiles are always evaluated at (x, z) = (x,√

εy) We shallconstruct the approximate solutions so that Ru,vappare being small in ε In this long section, we shallprove that

Proposition 2.1 Under the same assumptions as in Theorem 1.1, there are approximate solutions[uapp, vapp, papp] so that

kRuappkL 2 +√

εkRvappkL 2 ≤ C(L, κ)ε3/4−κ,for arbitrarily small κ > 0 Furthermore, there hold various regularity estimates on the approximatesolutions which are summarized in Corollary 2.3 for the zeroth-order Prandtl layers [u0, v0], Section2.3.1 for the Euler profiles [u1e, v1e], and Section 2.4.2 for the Prandtl layers [u1p, v1p]

The (leading) zeroth order terms on the right-hand side of (2.2a) consist of

Ru,0:={u0e+ u0p}{u0e+ u0p}x+{v0p+ ve1}{u0e+ u0p}y− {u0e+ u0p}yy

in which we note that{u0

e}x = 0 Since the Euler flows are evaluated at (x, z) = (x,√

εy), we maywrite

εvez1 (√εy)yu0py+ E0

Z θ y

u0ezz(√

ετ )dτ dθ + εu0py

Z y 0

Z θ y

v1ezz(√

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:

εvez1 (√εy)yu0py− εu0ezz+ E0, (2.5)which will be put into the next order in ε

Lemma 2.2 Let u0p(0, y) := ¯u0(y) be an arbitrary smooth boundary data for the Prandtl layer at

x = 0 Assume that miny{ue+ ¯u0(y)} > 0 Then, there exists a positive number L so that the

problem (2.4) has the unique smooth solution u0(x, y) in [0, L]× R+ Furthermore, for all n≥ 0,

k≥ 0, there exists a constant C0(n, k, ¯u0) so that there holds the uniform bound:

sup

x∈[0,L]khyin/2∂xku0pkL 2 (R + )+khyin/2∂xk∂yu0pkL 2 (0,L;L 2 (R + ))≤ C0(n, k, ¯u0), (2.6)with hyi = p1 + |y|2 Here, the constant C(n, k, ¯u0) depends on n, k, and the hyin-weighted

H2k(R+) norm of the boundary value ¯u0(y)

Corollary 2.3 Let u0

p be the Prandtl layer constructed as in Lemma 2.2 Then, there holdssup

x∈[0,L]khyin/2∂xk∂yj[u0p, v0p]kL 2 (R + )≤ C0(n, k, j, ¯u0), (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 thePrandtl layer to bound ∂y2u0p 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:

η :=

Z y 0

w vanishes at both y = 0 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, hηinF (·) ∈ Wk,p(R+), for arbitrary n, k ≥ 0 and p ∈ [1, ∞] We shall solve this equationvia the standard contraction mapping

First, let us derive a priori weighted estimates We introduce the following weighted iterativenorm:

j

X

k=0

Z x 0

Zhηin|∂xkwη|2, j ≥ 0 (2.10)

By multiplying the equation (2.9) by hηinw, it follows the standard weighted energy estimate:

Z hhηin−1|w||wη| + hηine−η|wwη| + hηin|wFη|i,

for n≥ 0, which together with the Young’s inequality yields

12

ddx

Zhηin|w|2+

Zhηinw|wη|2≤ C

Zhηin|w|2+ C

Zhηin|Fη|2

The Gronwall inequality then yields

sup

0≤x≤L

Zhηin|w|2+

Z L 0

Zhηin|wη|2 ≤ C(L), (2.11)

which givesN0(L)≤ C(L), for some constant C(L) that depends only on large L and the give data

ddx

Zhηin|∂xjw|2+

Zhηinw|∂xjwη|2

≤ n

Zhηin−1w∂xjwη∂xjw + CX

α<j

Zhηin{∂xj−αw∂xαwη}∂xjwη (2.12)

+CX

α<j

Zhηin−1{∂xj−αw∂xαwη}∂xjw− [ub− ue]

Z[∂xjwe−η]ηhηin∂xjw

Let us treat each term on the right For arbitrary positive constant δ, we get

Zhηin−1w∂xjwη∂xjw ≤ δ

Z

whηin|∂xjwη|2+ Cδkwk∞

Zhηin|∂xjw|2Z

[∂xjwe−η]ηhηin∂xjw ≤ δ

Z

whηin|∂xjwη|2+ Cδ

Zhηin|∂xjw|2

By choosing δ sufficiently small, the first term in the above inequalities can be absorbed into theleft-hand side of the inequality (2.12) Next, for 0 < α < j, we have

hηin−1{∂xj−αw∂xαwη}∂jxw ≤ Ck∂xj−αwk∞kphηinw∂xαwηkL 2 (R + )khηin−22 ∂xjwkL 2 (R + )

≤ δ

Z

whηin|∂xαwη|2+ Cδk∂xj−αwk2∞

Zhηin|∂xjw|2Whereas in the case α = 0, we instead estimate

hηin−1{∂xjwwη}∂jxw ≤ kwηk∞kphηin∂jxwkL 2 (R + )khηin−22 ∂xjwkL 2 (R + )

It remains to give bounds on kwηk∞ and k∂xj−αwk∞, for 0 < α ≤ j We recall the definition

w= w + ue+ [ub− ue]e−η Using the Sobolev embedding, we get

k∂xj−αwk2∞ ≤ C + k∂xj−αwk2∞ ≤ C +k∂xj−αwkL 2 (R + )k∂xj−αwηkL 2 (R + )

≤ C + k∂xj−αwkL 2 (R + )

hk∂j−αx wη| x=0kL 2 (R + )+k∂xj−αwηk1/2L2 k∂xj−α+1wηk1/2L2

i

in which k · kL 2 denotes the usual L2 norm over [0, x]× R+ In term of the iterative normNj(x),this yields

k∂xj−αwk2∞ ≤ C + Nj−α1/2(x)hk∂xj−αwη| x=0kL 2 (R + )+Nj1/2(x)i, (2.13)

for 0 < α≤ j Next, to estimate kwηk∞, we use the embedding: kwηk2

∞≤ CkwηkL 2 (R + )kwηkH 1 (R + ).Using the equation (2.9) and the lower bound on w, we get

kwηηkL 2 (R + )≤ CkwxkL 2 (R + )+kwkH 1 (R + )+kw2ηkL 2 (R + )+kFηkL 2 (R + )



in which the Sobolev embedding for the supremum norm again yields

kwη2kL 2 (R + )≤ Ckwηk3/2L2 (R + )kwηk1/2H1 ≤ δkwηηkL 2 (R + )+ Cδkwηk3L2 (R + ).Choosing δ sufficiently small, we conclude that

kwηk∞ 1 +kwxkL 2 (R + )+kwkH 1 (R + )+kwηk2L2 (R + ) (2.14)Hence, integrating the above inequality over [0, x], recalling the definition of the iterative norm,and using the uniform bound onN0(L), we obtain

Z x

0 kwηk2∞.1 +

Z x 0

(1 +kwk∞+kwηk2∞)

Zhηin|∂xjw|2+k∂xj−αwk2∞X

α<j

Z Zhηinw|∂xαwη|2

to the original coordinates yields the lemma, upon noting that y∼ η thanks to the upper and lowerbound of w

Remark 2.4 It is possible to iterate our above scheme to obtain a global-in-x solution to thePrandtl equation Indeed, the L2 estimate (2.11) yields the global existence of a bounded weaksolution The standard Nash-Moser’s iteration applied to the parabolic equation (2.9) then yields

a uniform bound in C1 and hence H1 spaces By a view of the iterative estimate (2.16) which isonly nonlinear at the first step for N1, it follows a uniform bound for all Nk, for k ≥ 1, uniformly

in small L This yields the global smooth solution

2.2 ε1/2-order corrections

Next, we collect all terms with a factor √

ε from (2.2a), together with the new √

ε-order termsarising from Ru,0 (see (2.5)), to get

Ru,1:= [u1e+ u1p]∂x[u0e+ u0p] + [u0e+ u0p]∂x[u1e+ u1p] + vp1∂y[u0e+ u0p] + [v0p+ ve1]∂y[u1e+ u1p]+ p1ex+ p1px− ∂y2[u1e+ u1p] + [yu0px+ v0p+ v1e]u0ez+ yv1ezu0py

We construct the Euler and Prandtl layers so that Ru,1 is of order√

ε We rearrange terms withrespect to the interior variables (x,√

εy) and the boundary-layer variables (x, y), respectively Westress that when the partial derivative ∂y hits an interior term with scaling√

εy, that term can bemoved to the next order For instance, [v0p+ v1e]∂yu1e=√

ε[vp0+ ve1]u1ez(√

εy) Having this in mind,the leading interior terms consist of

u0eu1ex+ v1eu0ez+ p1ex= 0 (2.17)and the boundary-layer terms consisting of

py, whichleads to the fact that Prandtl’s pressure is independent of y:

p1p= p1p(x) (2.20)The next (zeroth) order in (2.2b) consists of

Rv,0 := [u0e+ u0p][vpx0 + vex1 ] + [vp0+ v1e]∂y[v0p+ ve1] + p1ez+ p2py− ∂y2[vp0+ v1e]

Again as above, we shall enforce Rv,0 = 0 (possibly, up to error of order √

ε) Note that vp1 hasnow been determined through the divergence-free condition and the construction of u1p We takethe interior layer [u1

e, v1

e, p1

e] to satisfy

u0evex1 + p1ez= 0 (2.21)

Whereas, the next layer pressure p2 is taken to be of the form

p2p(x, y) =

Z ∞ y

h[u0e+ u0p]v0px+ u0pvex1 + [v0p+ ve1][v0py+√

εvez1 ]− v0pyy− εv1ezz

i(x, θ) dθ (2.22)

With this choice of p2p and (2.21), the error term Rv,0 in this leading order is reduced to

Rv,0=√

ε[vp0+ ve1]vez1 − εv1ezz (2.23)The set of equations (2.17), (2.21), together with the divergence-free condition, constitutes theprofile equations for the Euler correction [u1e, ve1, p1e], whereas equations (2.18) and (2.20) are forthe divergence-free Prandtl layers [u1p, vp1, p1p]

We construct the Euler corrector [ve1, u1e, p1e] solving (2.17), (2.21) For sake of presentation, we dropthe superscript 1 Writing the equation for vorticity we = uez − vex and note that wex =−∆ve.This leads to the following elliptic problem for ve:

−u0e∆ve+ u0ezzve = 0, (2.24)with ∆ = ∂x2+ ∂z2, together with the boundary conditions as in Section 1.1, which we recall

ve(x, 0) =−vp0(x, 0), ve(0, z) = Vb0(z), ve(L, z) = VbL(z) (2.25)with the comparability assumption: Vb0(0) =−v0

p(0, 0) and VbL(0) =−v0

p(L, 0) We need to derivehigher regularity estimates for ve Due to the presence of corners in the domain for (x, z), singularitycould occur To avoid this, we instead consider the following elliptic problem:

−u0e∆ve+ u0ezzve = Eb, (2.26)with the same boundary conditions (2.25), in which Eb is introduced as a boundary layer corrector

To define Eb, let us introduce

for any n, k ≥ 0 and q ∈ [1, ∞], for some constant C that is independent of small L Let us thenintroduce the function w through

Lemma 2.5 Assume that Vb0 and VbL are sufficiently smooth, rapidly decaying at infinity, andsatisfy |∂k

z(VbL− Vb0)(z)| ≤ CL, for k ≥ 0, uniformly in z There exists a unique smooth solution

ve to the elliptic problem (2.25), (2.26), and (2.30), and there holds

kvek∞+khzinvekH 2 ≤ C0, khzinvekH 3 ≤ C0ε−1/2, khzinvekH 4 ≤ C0ε−3/2

for n ≥ 0 and for some constant C0 that depends on the given boundary data, and but does notdepend on L, when L is small In addition,

khzinvekW 2,q ≤ C(L), khzinvekW 3,q ≤ C(L)ε−1+1/q, khzinvekW 4,q ≤ C(L)ε−2+1/q,for n ≥ 0, q ∈ (1, ∞), and for some constant C(L) that could depend on small L Here, we notethat the integration in the above k · kW k,q norms is taken with respect to (x, z) in [0, L]× R+.Proof We write ve= B + w with the smooth B defined as in (2.27) It suffices to derive estimates

on w, solving (2.29) We first perform the basic L2 estimate Multiplying the equation by w/u0eand using the zero boundary conditions yield

Z Z 

− ∆ww +u

0 ezz

u0 |w|2=

Z Z 

|∇w|2+u

0 ezz

u0 |w|2=

Z Zw(Eb+ Fe)

u0 Thanks to the crucial positivity estimate (see (1.16) and (1.17) with u0e), we have

Z 

|wz|2+u

0 ezz

Putting the above estimates into the energy estimate, together with a use of the standard Young’sinequality, we get

Clearly, kGekL 2 ≤ C In addition, since Eb(x, 0) + Fe(x, 0) = 0, we have Ge = 0 and hence

w = wzz = 0 on the boundary z = 0 We thus have the following H2energy estimate by multiplyingthe elliptic equation by wz:

Z Z

|∇wz|2+

Z L 0

wzzwz(x, 0) +

Z ∞ 0

wzxwz

x=L x=0 =

Z Z(Ge)zwz =−

Z Z

Gewzz ≤ kGekL 2kwzzkL 2

Since wzz(x, 0) = 0 and wz(0, z) = wz(L, z) = 0, the boundary terms vanish Hence, togetherwith a use of the Young’s inequality, we have obtained the uniform boundkwzkH 1 ≤ C Using theequation to estimate wxx in term of the rest, we thus obtain the full H2 bound of the solution w,and hence of ve, uniformly in small L In addition, since w = 0 on the boundary, we have

|w(x, z)| ≤

Z x

0 |wx(θ, z)| dθ ≤

Z x 0

Z z

0 |wxwxz|(θ, η) dη1/2 dθ

≤√xkwxk1/2L2 kwxzk1/2L2 ≤ C0

√L,thanks to the H2 bound on w This proves the uniform boundedness of ve

As for the weighted estimates, we consider the elliptic problem for hzinw, with n ≥ 1, whichsolves

−∆(hzinw) = hzin

u0



Eb+ Fe− u0ezzw− w∂z2hzin− 2wz∂zhzin

the homogenous boundary conditions By induction, hzin−1w is uniformly bounded in H2, andhence the right-hand side of the above elliptic problem is uniformly bounded in H1 The sameproof given just above for the unweighted norm yieldskhzinwkH 2 ≤ C, for all n ≥ 1

Next, we derive higher regularity estimates for w We recall that from (2.28) and (2.31), thereholds

Here, we note that wzz = 0 on the boundary, precisely due to the layer corrector Eb and theequation (2.33) Next, note that the source term in the above equations has its L2 norm bounded

by Cε−1/2 and Cε−3/2, respectively Again, the above H2 energy estimates then give

khzin∂zkwkH 2 ≤ Cε−k+1/2, k = 1, 2, n≥ 0

We now estimate w in H3 and H4 norms Indeed, thanks to the H2 bound on wz, wzz, it remains

to estimate wxxx in L2 and H1, respectively Thanks to (2.33), we may write

Finally, the Wk,q estimates follow simply from the standard elliptic theory in [0, L]× R, when

we make the odd extension to z < 0 for (2.33) We note that the boundary layer construction(2.30) ensures that the odd extension of Ge ∈ W2,q([0, L]× R) This completes the proof of thelemma

2.3.1 Euler profiles

We now construct the Euler corrector [u1

e, v1

e, p1

e] that is used in the boundary-layer expansion (1.9)

We take v1e = ve, where ve solves the modified elliptic problem (2.26), with an extra source Eb By

a view of (2.21) and the divergence-free condition, we take

u1e := u1b(z)−

Z x 0

vez1 (ξ, z)dξ,

p1e := pb−

Z x 0

[−u0evez1 + ve1u0ez](ξ, 0)dξ−

Z z 0

Similarly, by definition and the estimates from Lemma 2.5, we have

εku1ezz(√

ε·)kL 2 ≤ ε3/4ku1bzkL 2 + Cε3/4khzinv1ezzzkL 2 ≤ Cε1/4,and by the estimate (2.31),

Z ∞

√εyEb(x, θ) dθ

L 2 ≤ Cε−1/4khzinEb(·)kL 2 ≤ Cε1/4.Hence, we obtain the uniform error estimate:

kRu,1kL 2 ≤ Cε1/4 (2.38)Similarly, we give an estimate on Rv,0 defined as in (2.23) Thanks to the boundedness of v0

p

and v1e, and the estimates on ve1 (Lemma 2.5), we get

kRv,0kL 2 ≤√εkv0p+ ve1kL ∞kvez1 kL 2 + εkv1ezzkL 2 ≤ Cε1/4 (2.39)Finally, we estimate E0defined as in (2.3) Using the fact that u0p is rapidly decaying at infinityand v1

ε-order Prandtl corrector [up, up, pp], afterdropping the superscript 1, becomes

u0upx+ upu0x+ vpu0y+ [vp0+ v1e]upy− upyy

=−u0ez[yu0px+ v0p]− yvez1 u0py− u1eu0px− u0pu1ex− ppx

=: Fp

(2.41)

in which we note that the source term Fp includes the unknown pressure pp Thanks to (2.20),

pp = pp(x) and hence, by evaluating the equation (2.41) at y =∞, we get ppx= 0 We shall solve(2.41) together with the divergence-free condition

upx+ vpy= 0and the boundary conditions:

up(0, y) = ¯u1(y), up(x, 0) =−u1e(x, 0), vp(x, 0) = 0, lim

y→∞up(x, y) = 0

2.4.1 Construction of Prandtl layers

There is a natural energy estimate associated with the linearized Prandtl equation (2.41), yieldingbound on upyin term of vp The difficulty is in controlling the unknown vp Our construction startswith the crucial positivity estimate (1.16) Indeed, we introduce the inner product

u0 vwidy

Note that by (1.16) the quantity [[v, v]] yields a bound on kvyk2L2 (R + ) We therefore shall re-writethe equation in term of vp, putting up in the source term Precisely, taking y-derivative of (2.41)and using the divergence-free condition yield

−u0vpyy+ vpu0yy+ upu0xy+ [vp0+ v1e]upyy +√

εvez1 upy− upyyy = Fpy,

or equivalently, in a view of the inner product,

−vpyy+u

0 yy

vp(x, 0) = 0, vpy(x, 0) = u1ex(x, 0) (2.44)

We prove the following:

Lemma 2.6 There exists a unique smooth divergence-free solution [up, vp] solving (2.41) withinitial condition up(0, y) = ¯u1(y) and the boundary conditions (2.44) Furthermore, there hold

The proof consists of several steps First, we express the boundary conditions of vp in term ofthe given data up(0, y) = ¯u1(z)

Lemma 2.7 For smooth solutions [up, vp] solving (2.41) with up(0, y) = ¯u1(y) Then, there hold

Proof Define the stream function ψ =Ry

0 up dy Then, up = ψy and vp =−ψx We introduce thequantity w = u0ψy− u0

yψ Using (2.41), we get

wx =−u0xyψ− [vp0+ v1e]upy+ upyy+ Fp, (2.47)for Fp defined as in (2.41) Hence, the boundary values of w and wxcan be computed directly fromthe given boundary data ¯u1(y), [u0, v0], and Fp(0, y) Precisely, for k≥ 0, we get

k∂ykw(0,·)kL 2 ≤ C(u0, v0p)k¯u1kH k (R + ), k∂ykwx(0,·)kL 2 ≤ C(u0, vp0)k¯u1kH k+2 (R + )+k∂ykFp(0,·)kL 2 (R + )

for some constant C(u0, v0p) that depends on the high regularity norms of u0, v0p; see the estimates

w(x, θ){u0}2 dθand so we have

khyin∂yk+1vp(0,·)kL 2 (R + )=khyin∂yk+1ψx(0,·)kL 2 (R + )≤ C(kw(0, ·)kH k (R + )+kwx(0,·)kH k (R + )).This proves the claimed estimate for vp(0,·) in ˙Hk+1(R+) Next, as for vxestimate, we differentiate(2.47) with respect to x and get

kwxx(0,·)kH k (R + ) ≤ C(u0, v0p)khyi−2ψx(0,·)kH k (R + )+kvp(0,·)kH k+3 (R + )

+kFpx(0,·)kH k (R + ).Again by definition of Fp in (2.41), we have

kFpx(0, y)kH k (R + )≤ C(u0, v0p, [u1e, ve1](0,·))1 +khyi−mu1exx(0,·)kH k (R + )



in which C(u0, v0p, v1e(0,·)) depends on high regularity norms of u0, v0p, and v1e(0,·) Similarly,

vpyy(0, y) can be written in term of up(0, y), vp(0, y), Fpy(0, y), according to (2.42) This gives

kvp(0,·)kH k+3 (R + )≤ C(u0, vp0, [u1e, ve1](0,·))1 +kvp(0,·)kH˙ k+1 (R + )



This yields estimates on vpx on the boundary as claimed

Lemma 2.8 There exists a positive number L > 0 so that for each N large, the fourth orderelliptic equation:

− vyyx+u

0 yy

 (2.49)

for j = 0, 1, for any n≥ 0, as long as the right-hand side is finite

Proof Let us choose an orthogonal basis{ei(y)}∞

i=1 in H2([0, N ]) with [ei, eiy] = 0 at y = 0, N , forall i≥ 1 The orthogonality is obtained with respect to the [[·]] inner product:

[[ei, ej]] = δij.Such an orthogonal basis exists, since [[·]] is equivalent to the usual inner product in H1([0, N ]).Then, we introduce the weak formulation of (2.48):

for all ei(y), i≥ 1 Next, for each fixed k, we construct an approximate solution in Span{ei(y)}k

u0 =

Z(−feiy + gei) dy (2.50)

which by orthogonality yields a system of ODE equations:

Since f, g∈ L2(R+), the ODE system has the unique smooth solution ak and hence, vk is defineduniquely and smooth Multiplying (2.50) by ai

x and taking the sum over i, we get[[vxk, vxk]] +

Z vkyyvkyyx

u0 =

Z(−fvkxy+ gvkx) dywhich is equivalent to

[[vxk, vkx]] + 1

2

ddx

Z {vk

yy}2

u0 =

Z(−fvxyk + gvxk−1

By the positivity estimate (1.16) and (1.17) with u0, we note that [[vk, vk]]≥ θ0kvk

xyk2

L 2 (0,N ) Thestandard Gronwall inequality then yields

Recall now that vi

xx =Pk

i=1ai

xxei, which is a smooth function, since ai, ei are both smooth Using

vixx as a test function, we then get

[[vkxx, vxxk ]] + 1

2

ddx

The Gronwall inequality, together with (2.51), yields the claim (2.49) for the unweighted estimates,upon integrating by parts in y the third term on the right Almost identically, we may now insertthe weight function w(y) =hyi2n and take inner products against w(y)vk

0 yy

u0 yχu1exx(x, 0) +(yχ)yy

in which Gp is defined as in (2.42) Explicitly, we have defined

u0



xvp− (yχ)yyu1exx(x, 0) +u

0 yy

u0 yχu1exx(x, 0) +(yχ)yy

u0



yyu1ex(x, 0)with

Fp =−u0ez[yu0px+ v0p]− yv1ezu0py− u1eu0px− u0pu1ex.Here, we note that the divergence-free condition is imposed: upx= −vpy =−¯vy+ (yχ)yu1

ex(x, 0)

We construct the unique solution ¯v to the above problem, and hence the solution vp to (2.43) via

a contraction mapping theorem We shall work with the norm:

The uniform bound onk¯vyyk2

Z Zhyi−n|¯vy|2 ≤ C sup

x kvyyk2L 2 (R + )

Z Zhyi−n+1≤ CL|||¯v|||,for some large n Such a spatial decay hyi−n is produced by the rapid decay property of u0p.Similarly, we have

Z Z

hyi−n|up|2 ≤ C

Z Zhyi−nh|¯u1|2+ L

Z L

0 |upx|2i≤ CLk¯u1k2L2 (R + )+ku1ex(x, 0)k2L2 (0,L)+|||¯v|||,using the fact that upx = −vpy = −¯vy + (yχ)yu1

ex(x, 0) As for upy, we use |upy| ≤ |¯u1y| +

√

LkupxykL 2 (0,L) Next, we bound hyi−nupyy, which by observation the decaying factor hyi−n is

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. .. plays the decisive role; see Sections 2.2and 2.4

We are not aware of any work in the literature that deals with the validity of the Prandtlboundary layer theory for the steady Navier- Stokes