Boundary layers for the Navier-Stokes equations linearized around a stationary Euler

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1 Boundary layers for te Navier-Stokes equations linearized around a stationary Euler flow Gung-Min Gie, James P. Kellier and Anna L. Mazzucato Abstract. We study te viscous boundary layer tat forms at small viscosity near a rigid wall for te solution to te Navier-Stokes equations linearized around a smoot and stationary Euler flow LNSE for sort) in a smoot bounded domain Ω R 3 under no-slip boundary conditions. LNSE is supplemented wit smoot initial data and smoot external forcing, assumed ill-prepared, tat is, not compatible wit te no-slip boundary condition. We construct an approximate solution to LNSE on te time interval [, T ], < T <, obtained via an asymptotic expansion in te viscosity parameter, suc tat te difference between te linearized Navier-Stokes solution and te proposed expansion vanises as te viscosity tends to zero in L Ω) uniformly in time, and remains bounded independently of viscosity in te space L [, T ]; H 1 Ω)). We make tis construction bot for a 3D cannel domain and a smoot domain wit a curved boundary. Te zero-viscosity limit for LNSE, tat is, te convergence of te LNSE solution to te solution of te linearized Euler equations around te same profile wen viscosity vanises, ten naturally follows from te validity of tis asymptotic expansion. Tis article generalizes and improves earlier works, suc as Temam and Wang [], Xin and Yanagisawa [3], and Gie [4]. Matematics Subject Classification 1). 35B5, 35C, 76D1, 35K5. Keywords. boundary layers, singular perturbations, vanising viscosity limit. 1. Introduction We study te boundary layer formed near a rigid wall by a low-viscosity incompressible fluid tat solves te Navier-Stokes equations NSEinearized

2 G.-M. Gie, J. Kellier and A. Mazzucato around a smoot and stationary Euler flow. In exterior domains, suc equations model te flow around an obstacle moving at constant velocity, te classical Oseen system, were te steady profile is also spatially omogeneous. In a smoot bounded domain, tey model te beavior of nearly inviscid flows in bodies wit cavities, as in simplified models of te eart s mantle. For simplicity, we assume tat te fluid occupies a bounded, connected region Ω in R 3 wit a C boundary Γ. We can ten write NSE as te system of PDEs, u ε t + uε u ε + p ε = f + ε u ε in Ω, div u ε = in Ω. Above, ε > is te given, constant viscosity coefficient, u ε is te velocity field, p ε is te pressure field, and f is a given time-dependent external force independent of ε). Togeter, u ε, p ε ) is te solution to NSE. We impose no-slip boundary conditions, u ε =, and we give an initial condition u on te velocity alone. Te no-slip condition is te most appropriate at rigid, smoot walls. As customary in fluid mecanics, we denote by H te function space H = {u L Ω) div u = and u n = on Γ}, 1.1) endowed wit te L norm, were n represents te unit outer normal to Γ. In te domain Ω, we consider a smoot vector field U of class H C Ω), wic is a solution to te stationary Euler equations EE), { U U + π = F in Ω, div U = in Ω, 1.) were F C Ω). We impose on U te no-penetration condition, U n = on Γ. One can always construct solutions to te equation above provided F is appropriately given. For example, wen F =, one can obtain steady Euler solutions, called Beltrami flows, from eigenfunctions of te curl operator. We linearize NSE about U in te usual manner. We let v ε = u ε U so tat u ε = v ε + U. Using 1.) gives an equation for v ε from NSE, in wic te pressure can be identified up to a constant) wit p ε π. LNSE is ten obtained by retaining all linear terms in v ε. Te pressure tat ensures te divergence-free condition on v ε can still be identified wit p ε π. Given U and te initial conditions on u ε, it follows tat v ε satisfies a non-omogeneous boundary condition on Γ, namely, v ε = U. However, by performing a lift of te boundary value tat is divergence free e.g. via an armonic vector potential), we can WLOG assume tat v ε =, provided te rigt-and-side of te equation is canged accordingly. Finally, wit abuse of notation, we

3 Boundary layers for te linearized NSE 3 relabel v ε by u ε and p ε π by p ε. Ten, te initial-boundary-value problem IBVP for sort) for LNSE is te system, u ε t ε uε + U u ε + u ε U + p ε = f F + ε U in Ω, T ), for any fixed T >. div u ε = in Ω, T ), u ε = on Γ, T ), u ε t= = u in Ω, 1.3) It sould be noted tat often in te literature, te Stokes equation, obtained by dropping te non-linear terms directly in NSE, is called te LNSE. Here, instead we refer to te LNSE as an Oseen-type equation, tat is NSE linearized around a non-trivial profile U. Te vanising viscosity limit and associated boundary layer for te Stokes system can be analyzed adapting tecniques used for te eat equation and do not require te use of correctors. Te goal of tis work is to analyze te boundary layer tat arise in te system 1.3) at small viscosity due to te mismatc in its boundary conditions and tose of te corresponding limit problem 1.6) below, wic is obtained by formally setting ε =. Our main task is to build an incompressible boundary layer corrector and ence an asymptotic expansion of u ε in ε) assuming sufficient regularity of te data. Since te minimal regularity requirement for te data is not our focus, we assume tat f C 1, T ; C Ω)), u H C Ω). 1.4) However, tis smoot data may not necessarily vanis on te boundary and, in tis sense, te initial data is ill-prepared; tat is, te boundary and initial conditions in 1.3) are not compatible. As is te case for te unsteady Stokes system see e.g [18, 19]), under te assumption 1.4), for any < T < tere exists a unique, strong solution v ε to te IBVP for LNSE 1.3) at fixed ε. Moreover, te solution u ε satisfies uniformly in ε, u ε L, T ; V ) C[, T ]; H), u ε t L, T ; H), 1.5) by standard energy estimates. Te limit problem satisfies tese same estimates.)

4 4 G.-M. Gie, J. Kellier and A. Mazzucato Formally setting ε = in 1.3), we obtain te corresponding limit problem, u t + U u + u U + p = f F in Ω, T ), div u = in Ω, T ), u n = on Γ, T ), u t= = u in Ω. 1.6) By analogy wit 1.3), we call te above system te linearized Euler equations or LEE. Under te assumptions 1.4), for any < T < te system 1.6) possesses a unique strong solution u wit pressure p, unique up to an additive constant) suc tat u C 1, T ; H C Ω)). 1.7) We refer to te results in [13] for a proof.) In tis work, we systematically employ te metod of correctors as proposed by J. L. Lions [14] to analyse te boundary layer for LNSE. Te corrector tat we construct accounts for te difference between te solution to LNSE and LEE due to te discrepancies between te boundary values of te viscous and inviscid solutions, and it accounts for te rapid variation of te functions and teir normal derivatives in te boundary layer. We assume te same regular asymptotic expansions in powers of ε and scaling as in Prandtl teory [17]. In particular, te tickness of te layer were te effect of te corrector is not negligible is of order ε as in Prandtl teory. At te same time, as verified by rigorous analysis below, te corrector sares te major estimates and properties of te corrector introduced by Kato in [11] to study te vanising viscosity limit. Tis fact is not unexpected given tat te zero-viscosity limit olds in tis case. However, in Kato s work te effects of viscosity, in terms of viscous energy dissipation rate, must be controlled in a layer of order ε to pass to te zero-viscosity limit. Te main idea beind te corrector metod is to propose a form for an approximate solution to LNSE, wic is te given solution to te limit problem plus te corrector. Formal matced asymptotic analysis and pysical considerations are used to derive te form of te corrector and te effective equations it satisfies. Ten te validity of tis asymptotic expansions is establised by energy estimates on te difference of te viscous solution and te proposed expansion, performed on te wole domain Ω. To enforce te incompressibility condition on te corrector, we follow te original approac in [4], were te viscous boundary layer for te Stokes equations is investigated. In tis article, we generalize te analysis in [4] by studying te asymptotic beavior of te Navier-Stokes equations linearized around a stationary Euler flow. Te asymptotic expansion proposed in tis article provides complete structural information of te boundary layers for LNSE. Establising te zero-viscosity limit is nontrivial even in te absence of boundaries, due to te singular nature of te limit. Wen boundaries are

5 Boundary layers for te linearized NSE 5 present, te analysis of flows at small viscosity is significantly more callenging, given tat rigid walls generate vorticity. Wen no-slip boundary conditions are impose on te viscous solutions, te lack of control of growt normal to te boundary of te tangential velocity components keeps te problem still essentially open, unless strong conditions suc as analyticity or symmetry are imposed on te data or te solution we refer to [6, 16] and references terein for a survey of recent results) or te equations are linearized. In tis work, we consider anoter simplified situation were te vanising viscosity limit and te associated boundary layer can be rigorously studied. Te analysis of te boundary layer for LNSE can be te first step in elucidating te role of a potentially strong) tangential advection on te stability of te layer. Indeed, recent results on te ill-posedness of Prandtl equations ave as teir starting point linear instabilities around sear flows [3, 8, 9]. Wile Oseen-type equations were studied in te context of te vanising viscosity limit and Prandtl-type approximations in [15,, 1,, 3] in domains wit flat boundaries suc as a cannel, in tis article we consider te LNSE in a general smoot domain wit a curved boundary and ence extend tose earlier results. As a matter of fact, wen te boundary is curved, te expansion of te viscous solutions in powers of te viscosity, assumed small, tat is obtained on domains wit flat boundaries does not give a suitable approximation. As exemplified in Equation 1.8) and Teorem 1.1 below, an additional corrector for te pressure is required in order to account te lower-order error caused by curvature. In addition, prior works consider only te case of well-prepared or compatible initial data, tat is, data tat vanis on te boundary. Tis work, instead, is te first tat analyzes te boundary and initial layers for Oseentype equations or LNSE around a stationary Euler flow) wen te initial data is ill prepared. In te case of incompatible data, an initial layer forms in te viscous equations tat needs to be accounted for in te analysis. Indeed, wen te limit solution is steady, te contribution from te initial layer may persist in te limit of vanising viscosity. For a curved boundary, as in te case treated ere, it is necessary to introduce a pressure corrector at zero order. We terefore write te approximate expansion of te LNSE solution as u ε u + Θ, { p, for te case of a 3D cannel domain, p ε p + q, for te case of a 3D smoot domain. 1.8) To isolate and so clarify tese tecnical difficulties, we treat te case of LNSE in a cannel first in Section before tackling te te more tecnically involved case of a curved boundary in Section 3. Te corrector, Θ, is given explicity in.7) and.8) for te cannel and in 3.16) and 3.17) for a curved domain. Our main result is te following error estimate wit sarp rates of convergence in viscosity:

6 6 G.-M. Gie, J. Kellier and A. Mazzucato Teorem 1.1. Make te assumptions in 1.4) and let w ε := u ε u + Θ), te difference between te linearized Navier-Stokes solution and its asymptotic expansion, as given in 1.8). Ten w ε vanises wit te viscosity parameter in te sense tat w ε L,T ;L Ω)) + ε 1 w ε L,T ;L Ω)) κ T ε 1, 1.9) for a constant κ depending on te data, but independent of ε. Moreover, as ε tends to zero, u ε converges to te Euler solution u in te sense tat u ε u L,T ;L Ω)) κ T ε ) Tis paper is organized as follows. We introduce and study te velocity corrector Θ in Sections and 3 for a cannel geometry. We introduce te pressure corrector q in Section 3, were we discuss te case of a curved boundary and prove Teorem 1.1. Tere, we also introduce a suitable coordinate system in a collar neigborood of te boundary Γ, used in te analysis of te viscous layer. Trougout, we will use te fairly standard notation in wic a subscript on an equation number signifies te ordering of te equation in tat reference. For example,.3) 3 means te tird equation in system.3). Acknowledgments Te first and second autors were partially supported by NSF Grant DMS Te tird autor was partially supported by NSF grants DMS and DMS Te second and tird autor acknowledge te ospitality and support of te Institute for Computational and Experimental Researc in Matematics ICERM) during te Semester Program on Singularities and Waves In Incompressible Fluids, were part of tis work was discussed. ICERM receives major funding from te NSF and Brown University. Te autors tank Milton C. Lopes Filo and Helena J. Nussenzveig Lopes for many fruitful discussions.. Boundary layers for LNSE in a 3D cannel domain In tis section, we consider te problems 1.3) and 1.6) wen te domain is a 3D periodic cannel, identified wit Ω =:, L), ) R 3, under periodic conditions in te x 1 and x directions. Periodicity makes te domain bounded and ensures uniqueness of solutions to te fluid equations.

7 Boundary layers for te linearized NSE 7.1. Asymptotic expansion of solutions to LNSE We start wit te ansatz tat to te first-order, only te velocity needs to be corrected to obtain te LNSE solution from te LEE solution: u ε u + Θ, p ε p..1) Te ypotesis tat te approximate pressure is te Euler pressure is justified as in Prandtl teory, and will be rigorously verified. To obtain an equation for te corrector, we start by supposing tat.1) olds exactly, so tat Θ = u ε u and p ε p =. Subtracting 1.6) 1 from 1.3) gives t Θ ε u ε + U Θ + Θ U = ε U..) We refine.) by making an ansatz like tat of Prandtl: we assume tat.) olds exactly only outside of a boundary layer of widt proportional to ε. As in te Prandtl teory, tis gives tat in te boundary layer, / xi ε / x3 ) and Θ i ε 1/ Θ 3, i = 1,. We see tat only te x 3 derivatives contribute at leading order to ε u ε and tat ε U is of lower order in ε. Tis yields Θ i t ε u ε i x U j Θ i x j + Θ j U i x j =. Also, because ε x 3 u is of lower order in ε it can be added to te equation, allowing us to replace ε x 3 u ε i by ε x 3 Θ i. Supplemented wit initial and boundary conditions, we ave te following formal asymptotic expansion for te corrector: Θ i t Θ i ε x U j Θ i x j + Θ j U i x j = in Ω, T ), i = 1,, div Θ = in Ω, T ), Θ = u on Γ, T ), Θ t= = in Ω..3) Because tere are two boundary components, we will construct te corrector from two boundary layer functions, θ L and θ R, eac defined on a alf-space. We use te subscript L and R for left and rigt layer functions assuming te cannel is vertically oriented.) Te layer functions satisfy driftdiffusion equations on alf spaces wit boundary, respectively, given by te

8 8 G.-M. Gie, J. Kellier and A. Mazzucato planes x 3 = and x 3 = : θ i, L ε θ i, L + U x3 θ i, L U 3 x3 θ i, L j + x t = 3 σ L x j x 3 = x 3 U i + θ j, L x j =,, L), ), T ), i = 1,, x3 = and θ i, L = u i, at x 3 =, θ i, L =, at t =, θ i, R ε θ i, R + U x3= θ i, R U 3 x3 θ i, R j x 3 σ R t x j x 3 = x 3 U i + θ j, R x j =,, L), ), T ), i = 1,, x3 = θ i, R = u i, at x 3 =, θ i, R =, at t =, were x 3 := x 3 and te cut-offs σ L and σ R are given by.4).5) σ L x 3 ) = { 1, x3 /4,, x 3 /, σ R x 3 ) = σ L x 3 )..6) Informally, te parabolic layer function θ i, L θ i, R ) represents te tangential component Θ i, i = 1,, of te corrector Θ near te boundary x 3 = x 3 = ). However, we want te corrector Θ to belong to te space H to avoid dealing directly wit te pressure in te convergence analysis of te error w ε. To tis end, we first introduce, as is customary in boundary layer analysis, appropriate cut-off functions σ L and σ R so tat te domain of te truncated layer functions is Ω and te approximate corrector satisfies te boundary conditions on Γ. Ten, to enforce te divergence-free condition, we define te tangential components of te corrector Θ i, i = 1,, as follows: Θ i x, t) = σ L θ i, L + σ L x3 x3 θ i, L dx 3 + σ R θ i, R + σ R θ i, R dx 3 }, i = 1, = x3 x3 {σ L θ i, L dx 3 + σ R θ i, R dx 3 x 3.7) Finally, we use te divergence-free condition on Θ to obtain te normal component of te corrector Θ 3 from its tangential components: Θ 3 x, t) = x3 {σ L θ i, L x i dx 3 + σ R x3 } θ i, R dx 3..8) x i

9 Boundary layers for te linearized NSE 9 Hence Θ belongs to te space H; div Θ = in Ω and Θ 3 = on Γ. We can write te corrector Θ as a sum of tree vector fields in te form, Θ = θ + φ + ψ,.9) were, for i = 1,, θ i = σ L θ i, L + σ R θ i, R, φ i = σ L ψ i = σ L x3 θ i, L dx 3 + σ R θ i, L dx 3 + σ R x3 θ i, R dx 3, θ i, R dx 3,.1) and θ 3 =, φ 3 = σ L ψ 3 = σ L x3 θ i, L x i θ i, L x i dx 3 σ R dx 3 σ R x3 θ i, R x i dx 3, θ i, R x i dx 3..11) As we will verify in te following subsection, te main part θ of Θ is a fast decaying boundary layer function, wic agrees wit te classical teory of boundary layers, wile te remaining parts φ and ψ are supplementary vector fields wic are small wen ε is small ) to maintain te corrector Θ in te space H... Estimates on te corrector In tis section, we will derive estimates on te correctors in various norms. Tese estimates are needed to establis error bounds on te approximate LNSE solution. We begin by estimating te layer functions θ i, L, θ i, R, i = 1,, in L p. Te main contribution in ε to te L p norm comes from te Laplacian, te zero-order term giving possibly an exponential growt in time only. Terefore, we utilize well-known estimates for solutions to te eat equation as an intermediate step. We let θ eat, i, i = 1,, be te solution of te following IBVP for te eat equation in a alf space: θ eat, i t ε θ eat, i x 3 θ eat, i = u i, at x 3 =, θ eat, i =, at t =. =, x 3 >,.1)

10 1 G.-M. Gie, J. Kellier and A. Mazzucato We recall te following L p estimates olds on θ eat, i see e.g. []; see [5, 4] for an application to viscous boundary layers): x3 p ε k+m θ eat, i x k j xm 3 L p,l), )) κ 1 + t 1 p ) 1 m ε p m, t >,.13) for p, k, l, 1 m 3, and i, j = 1,. Denoting θ i = θ i, L θ eat, i, one finds tat θ i, i = 1,, satisifies θ i t ε θ i + U x3 θ i U 3 θ i U i j + x = 3 σ L + θ j x j x 3 x3= x 3 x j x3= = Ẽi, in, L), ), θ i =, at x 3 = or t =, were Ẽ i = θ eat, i ε x j U i θ eat, j x j U x3= j x3=. θ eat, i U 3 θ eat, i x 3 σ L x j x 3 x3= x 3.14) A standard energy estimate gives bounds similar to.13) on θ i, ence θ i, L, and by symmetry θ i, R, satisfies te following estimates, wic we record in a lemma for convenience. Lemma.1. For i, j = 1, and k, l, and 1 p, we ave ) x3 l p k θ i, L + ε L,T ;L p,l), ))) x k j ε 1 p k θ i, L ε x3 x k j L,T ;L,L), ))) k+1 θ i, L x k j x + ε 1 k+1 θ i, L 3 L,T ;L,L), ))) x k j x 3 κ T ε 1 p,.15) L,T ;L,L), ))) κ T ε 1 4,.16) for a constant κ T depending on T and oter data, but independent of ε. Similar estimates old for θ i, R if θ i, L, x 3 / ε, and, L), ) are replaced by θ i, R, x 3 )/ ε, and, L), ) respectively. Even if te proof of tis lemma follows by standard arguments, we include a proof for te reader s sake. Proof. We prove.15) by induction on l.

11 Boundary layers for te linearized NSE 11 Using te bounds on θ eat, i and te definition of Ẽi, i = 1,, we ave x3 ε k+m Ẽ i x k j xm 3 κ 1 + t 1 Lp,L), )) p m ) ε 1 p m, t >,.17) for p, k, l, 1 m 3, and i, j = 1,. In addition, Ẽi x3= L,T ),L) ) κ T..18) θ p 1 i To prove.15) for l =, we multiply.14) 1 by were p > 1 is a simple fraction q/r wit q an even integer. Integrating over, L), ) gives 1 d p dt θ i p L p,l), )) U x3= j +,L),L) + εp 1),L) θ i x j + x 3 σ L U 3 θ i θp i dx 3 dx 1 dx ) θ i x 3 x3= x 3 ) U i θ j x j + Ẽi θ p 1 i dx 3 dx 1 dx. x3= θ p 1 i dx 3 dx 1 dx.19) We bound eac term on te rigt-and side separately, starting wit te first: and,l) = 1 p = 1 p,l),l) U x3 θ i j =,L) x θp 1 i j U j x j x3= U 3 x3 θ i x 3 σ L x 3 = x 3 σ L ) U 3 x 3 x 3 x θp 1 i 3 x3 = dx 3 dx 1 dx θ p i dx 1dx dx 3 κ T p θ i p L p,l), )), dx 3 dx 1 dx.) θ p i dx 3dx 1 dx κ T p θ i p L p,l), ))..1)

12 1 G.-M. Gie, J. Kellier and A. Mazzucato For te second term on te rigt-and side of.19), we apply Hölder s and Young s inequalities wit 1/p and p 1)/p and write,l) κ T,L) ) U i θ j x j + Ẽi θ p 1 i dx 3 dx 1 dx x3 = ) θ j + Ẽi θ i p 1 dx 3 dx 1 dx κ T θ j Lp,L), ))) + Ẽi Lp,L), ))) ) θ i p 1 L p,l), ))) κ T p θ j p L p,l), ))) + κ T p Ẽi p L p,l), ))) +κ T θ i p L p,l), )))..) Now, it follows from.17) and.19).) tat ) d θ i p dt L p,l), )) + εpp 1),L) κ T ε 1 + κt p θ i p L p,l), )). θ i θp i.3) Ten, by applying Grönwall s inequality wit an integrating factor exp κ T p) and by using te continuity of L p norm in p, we deduce tat, i = 1,, θ i L,T ;L p,l), ))) + ε 1 p θi L,T ;L,L), ))) κ T ε 1 p,.4) for any 1 p. Again, we can replace θ i by θ i, L in.4), owing to estimate.13) for θ i, eat. Next, we observe tat any tangential derivative of θ i in x j, j = 1,, satisfies an equation similar to.14) up to lower-order derivatives in x j, in wic te source term is replaced by a tangential derivative of Ẽ i. Ten, tanks to.17), we can verify.15) wit l = for all k by an argument similar to te one above. Now we assume tat.15) olds true for l l 1 as our induction ypotesis, and establis.15) wit l = l. We multiply.14) 1 by x 3 / l ε) θp 1 i, were again p > 1 is a simple fraction q/r wit an even integer q. Integrating over, L), ) and

13 Boundary layers for te linearized NSE 13 integrating by parts gives 1 d x3 p dt ε + +κ p,l),l) x3 θi p + εp 1),L) U x3 θ i U 3 j + x = 3 σ L x j L p,l), )) p ε θi p x 3 x3= x 3 x3 ε θ i θp i ) θ i x3 ) U i θ j x3 x j + Ẽi θp 1 x3= ε i L p,l), )) + κ x3 ε p θi p ε θp 1 i L p,l), )).5) Tanks to te induction ypotesis and estimate.13), te same computations tat led to.4) give.15) for k =. Again, because any tangential derivative of θ i in x j, j = 1,, satisfies te equation similar to.14) up to lower order derivatives in x j, we deduce tat.15) olds for all k as well. To prove.16), we derive te IBVP for θ i / x 3. First, we differentiate.14) 1,3 in x 3 :. θ i ε θ i + t x 3 x 3 U 3 x3 θ i +x 3 x 3 = x + 3 θ i x 3 = at t =. U x3= θ i j + σ L + x 3 σ ) U 3 x3 θ i x j x 3 x 3 = x 3 θ j U i = Ẽi in, L), ), x 3 x j x 3 x3 =.6) Second, to obtain a boundary condition for θ i / x 3, given te regularity of te data, we simply restrict.14) to x 3 = : θ i x 3 = 1 ε Ẽi, at x 3 =,.7) wic is of order ε 1 by.18).

14 14 G.-M. Gie, J. Kellier and A. Mazzucato A standard energy estimate gives 1 d θ i dt x 3 ε +κ θ i x 3 L,L), )),L) {x 3 =} ) + ε θ i x 3 θ i θ i x dx 1 dx 3 x 3 + κ Ẽi x 3 L,L), )). L,L), )) L,L), )).8) To estimate te boundary integral in te rigt-and side, we use.7) and te trace teorem: ε,l) {x 3 =} θ i x 3 κ θ 1 i θ 1 i x 3 L,L), )) x 3 κ θ i L + κ θ 1 i x 3,L), )) x 3 θ i dx 1 dx x 3 κ θ i L x 3,L) {x 3 =}) H 1,L), )) L,L), )) κε 1 θ i + x 3 L,L), )) +κε 1 θ i L θ i L x 3,L), )) x 3,L), )) κε 1 so tat + θ i x 3 L,L), )) + 1 ε θ i x 3 θ 1 i x 3 L,L), )) L,L), )),.9) ) d θ i dt x 3 + ε θ i L,L), )) x 3 L,L), )) κ T 1 + t 1 1 )ε + κt 1 θ i. x 3 L,L), )).3) Finally, an application of Grönwall s inequality wit exp κt 1/ ) as integrating factor gives.16), employing again.13) to replace θ i wit θ i, L. Our goal in te rest of tis section is to derive te equations satisfied by te tree components of te corrector, and estimate te data in terms of ε. We begin wit θ, te main component of te corrector. Tanks to its

15 Boundary layers for te linearized NSE 15 definition, we can write, θ i t ε θ i + U x3= σ L θ i, L ) j + U x3= σ R θ i, R ) j x j x j U 3 +x 3 σ L σ L θ i, L ) U 3 x 3 x 3 ) σ R σ R θ i, R ) x3 = x 3 x 3 x3 = x 3 U i + σ L θ j, L U i x j + σ R θ j, R x3= x j = E temp,i θ i ), i = 1,, x3=.31) were { E temp,i θ i ) = ε σ L θ i, L + σ L x θ i, L σ θ } i, R R + σ R 3 x θ i, R 3 U 3 +x 3 σ L x 3 σ L θ U 3 j, L x 3 ) σ R x3= x 3 R x3=σ θ j, R..3) By te estimates in Lemma.1, E temp,i θ i ) L,T ;L Ω)) κ T ε ) Also, Taylor s teorem applied to U in x 3 at x 3 = gives, for i, j = 1,, ) σl U j U x3= θ i, L ) L σ L θ i, L ) L j κx 3 x j,t ;L Ω)) x j,t ;L Ω)) κ T ε 3 4,.34) ) U 3 U 3 x 3 σ L σl θ i, L ) L x 3 x3= x 3,T ;L Ω)) κx θ.35) i, L L 3 σ L x κ T ε 3 4, 3,T ;L Ω)) and Ui σ L θ j, L U i L L x j x j κσ L θ j, L x 3 x3 =),T ;L Ω)),T ;L Ω)) κ T ε ) Consequently, we can write te vectorial) equation for θ as θ ε θ + U θ + θ U = Eθ) in Ω, T ), t θ = u on Γ, T ),.37) θ t= = in Ω, were satisfies te estimate, Eθ) := E 1 θ 1 ), E θ ), ).38) E i θ i ) L,T ;L Ω)) κ T ε 3 4, i = 1,..39)

16 16 G.-M. Gie, J. Kellier and A. Mazzucato We now turn to estimating te supplementary layer functions for te corrector Θ, starting wit φ. First, using.1),.11), and.15) we see tat k+m φ x k j xm 3 L,T ;L Ω)) κ k+1 θ i, L x k j x l i,l=1 + k+1 θ i, R x k j x l L,T ;L 1,L), ))) L,T ;L 1,L),))) κ T ε 1, j = 1,, k, m..4) Next, we can write φ i t = T i, 1 + T i,, i = 1,,.41) tanks to.4) and.1), were T i, 1 := εσ L = εσ L T i, := σ L σ L θ i, L x 3 +σ R σ R θ i, L x dx 3 + εσ R 3 εσ R x3= ε τ θ i, L θ i, R x 3 U x3 j = x U 3 x3 θ i, L 3σ L + x 3 = x 3 ε τ θ i, R x3=, U x3 j = θ i, R x 3 θ i, L x j dx 3 dx 3 U i θ j, L x j dx 3 x3= x U 3 x3 θ i, R 3σ R + x 3 = x 3 θ i, R dx 3 x j U i θ j, R x j dx 3, x3=.4) wit τ =, / x i and x 3 = x 3. By te standard trace teorem, T i, 1 L Ω) κ T ε θ i L x κ T ε θ 1 i θ 1 i, i = 1,. 3 Γ) x 3 L Ω) x 3 H 1 Ω).43) Ten, it follows again from te estimates in Lemma.1 tat T i, 1 L,T ;L Ω)) κ T ε 1, i = 1,..44) T i, L,T ;L Ω)) κ T ε 1, i = 1,..45) Combining.44) and.45) for φ i / t, i = 1,, and observing tat φ 3 / t enjoys te same estimates, we finally see tat φ κ T ε 1..46) t L,T ;L Ω))

17 Boundary layers for te linearized NSE 17 We deduce from te estimates above tat Eφ) := φ t ε φ + U φ + φ U L κ T ε 1..47),T ;L Ω)) In addition, φ Γ = φ t= =..48) We tackle ψ, wic is defined in.1). We temporarily set wic, by.4), satisfies ψ i, L := σ L θ i, L dx 3, i = 1,,.49) ψ i, L t + ε ψ i, L + U x3 j = U i ψ j, L x j = Êtemp,iθ i, L ), x3= ψ i, L U 3 x3 ψ i, L + x 3 σ L x j x 3 = x 3.5) were Ê temp,i θ i, L ) = εσ L x3 θ i, L dx 3 εσ Lθ i, L +x 3 σ L x3 Using Lemma.1 once again, we can estimate tis source term by x3 θ i, L dx 3, i = 1,..51) p Ê temp,i θ i, L ) κ T ε 1, l, 1 p,.5) ε L,T ;L p Ω)) noticing tat σ L and all its derivatives vanis for x 3 1/, so tat x3 σl ε x 3 θ i, L dx 1/ θi, 3 σ L x 3 L dx ε 3 κ x3 ) lθi, L ε L1,L), )),.53) for l and < t < T. An energy estimate ten gives p k ψ i, L + ε 1 ε L,T ;L p Ω)) x3 x k j p k ψ i, L ε x3 x k j L,T ;L Ω)) κ T ε 1 p,.54) for i, j = 1, and k, l, and 1 p. From.1) and.11), it follows tat ψ i ψ i, L ) ence ψ i ) enjoys te same estimate as in.54) wit x 3 replaced by x 3 = x 3. In addition, given its definition, te normal component ψ 3 satisfies te same type of estimates

18 18 G.-M. Gie, J. Kellier and A. Mazzucato as well. We conclude from.34),.35), and.36) tat ψ satisfies ψ t ε ψ + U ψ + ψ U = Eψ) in Ω, T ), ψ = on Γ, T ), ψ t= = in Ω,.55) were Eψ) L,T ;L Ω)) κ T ε 1, i = 1,..56) We are now in a position to prove our main result wen te geometry is tat of a periodized) cannel..3. Proof of Teorem 1.1: Te case of a 3D cannel domain Setting w ε := u ε u + Θ), π ε := p ε p, we employ te equations satisfied by te corrector Θ and by u ε and u equations 1.3), 1.6),.1),.37),.47),.48),.55)) along wit te divergence-free condition to write te IBVP for te error w ε, π ε ) as w ε t ε w ε + U w ε + w ε U + π ε = EΘ) in Ω, T ), div w ε = in Ω, T ), were w ε = on Γ, T ), w ε t= = in Ω,.57) EΘ) = Eθ) + Eφ) + Eψ)..58) A simple energy estimate gives 1.9), tanks to te bounds.39),.47), and.56). Finally, te vanising viscosity limit 1.1) follows from 1.9) and te smallness of te corrector in L, T ; L Ω)). 3. Te case of a 3D smoot domain We now turn to te study of te boundary layer of LNSE 1.3) in te more general and difficult case wen Ω is a bounded domain in R 3 wit curved boundary Γ. Following te analysis of te Stokes problem, we will utilize a curvilinear system adapted to te boundary.

19 Boundary layers for te linearized NSE Elements of differential geometry We assume tat a bounded domain Ω in R 3 as boundary Γ given by a compact, orientable D manifold of class C. We coose a small δ > and define a tubular collar) neigborood Ω 3δ of Ω as te set of all point in Ω witin distance 3δ of Γ. We will place coordinates on Ω 3δ following te procedure described in detail [7], wic we now summarize. Because Γ is compact, we can cover it wit a finite number of overlapping carts. We will develop te corrector in a single cart, te resulting estimates applying to te wole manifold because te number of carts is finite. Let us focus, ten, on a single cart on Γ, on wic ave a curvilinear coordinate system wic we label ξ = ξ 1, ξ ) ω, were ω is an open subset of R. Tis means tat tere exists a smoot function, x: ω Γ mapping points in ω to points on Γ. Tere is a condition on te transition maps between carts, but suc conditions do not concern us ere.) Letting g i ξ ) := x ξ i, i = 1, gives a covariant basis, g 1, g ), locally on Γ. We do not assume ortogonality of tis frame. We extend our coordinate system to a cart on Ω 3δ by setting ξ 3 to be te negative of te distance from a point in Ω 3δ to te boundary. We label te point, ξ = ξ, ξ 3 ) = ξ 1, ξ, ξ 3 ), and we ave a covariant basis, g 1, g, g 3 ), locally of Ω 3δ, were g i ξ) = x ξ i ξ) = g i ξ ) ξ 3 n ξ i ξ ), i = 1,, g 3 ξ) = nξ ). Let g ij := g i g j and g = detg ij ) 1 i,j 3. As sown in [7], we can write te metric tensor in covariant form as g ij ) = g g 1 g 1 g 11, 1 were g := detg ij ) 1 i,j 3 > locally in Ω 3δ. Te function, := g 1/ >, is te magnitude of te Jacobian determinant for te transformation from x to ξ. From te covariant basis, g i ), we introduce i := g i = i ξ), i = 1,, 3 = ) Defining te normalized covariant basis, {e 1, e, e 3 }, were e i := g i g i, we represent a vector-valued function, F, in te form 3 F = F i ξ)e i. 3.)

20 G.-M. Gie, J. Kellier and A. Mazzucato We now ave te tools we need to represent covariant differential operators for smoot functions on Ω 3δ in an effective manner. Te divergence operator acting on F can ten be written in te ξ coordinates as div F = 1 F i) + 1 F 3 ), 3.3) ξ i i ξ 3 wile te Laplacian of F takes te form, 3 F = S i F + LF i + F i ) e i, 3.4) ξ 3 were ) linear combination of tangential derivatives S i F = of F j, 1 j 3, in ξ,, up to order LF i = 3.5) ) proportional to F i / ξ 3. Te coefficients of S i and L i, 1 i 3 in 3.5), are multiples of, 1/, i, 1/ i, i = 1,, g 1, g 1, and teir derivatives. Finally, we compute te covariant derivative F G, of G in te direction F for smoot vector fields F, G : Ω 3δ, ξ R 3 in te ξ coordinates: were F G = 3 P i F, G) = } {P i F, G) + F 3 Gi + Q i F, G) + R i F, G) e i, 3.6) ξ 3 linear combination of te products of te tangential component F 1 or F and te tangential derivative of G i in ξ j, j = 1,, 3.7) Q i F, G) = linear combination of F j G k, 1 j, k ), 3.8) R i F, G) = linear combination of F j G k, j = 3 or k = 3). 3.9) Te Q i F, G) and R i F, G) are related to te Cristoffel symbols of te second kind, wic comes from te twisting effects of te curvilinear system ξ. For te case of an ortogonal system, te explicit expressions are given in Appendix of [1]. Te formula above for te covariant derivative will be used to compute te convective term in te curvilinear coordinate system. 3.. Asymptotic expansion of solutions to LNSE As in te case of te 3D cannel, we postulate an expansion for te approximate LNSE in te form u ε u + Θ, p ε p + q, 3.1) were Θ is again te velocity corrector, and we ave now also a pressure corrector q. We construct te correctors using te coordinates ξ, tat is, in te collar neigborood Ω 3δ ). On tis collar neigborood, we implictly assume

21 Boundary layers for te linearized NSE 1 te representation 3.) for all vector fields. We remark tat δ is cosen independently of ε, and ence te collar neigborood contains te viscous boundary layer for all sufficiently small ε. We can again formally derive te equations for te correctors from 1.3) and 1.6), toug we no longer assume te pressures are identical. Tis gives Θ t ε Θ + U Θ + Θ U + pε p ) ε U + u ) in Ω, div Θ = in Ω, Θ = u on Γ, Θ t= = in Ω. 3.11) Te formal asymptotic expansion is performed along te same lines as tat for te cannel. Using te coordinate ξ 3 allows us to make scaling arguments similar to tose in te Prandtl teory, wic lead to a viscous boundary layer of tickness ε 1/, and to te assumption tat / ξ i ε / ξ 3 ), i = 1,. Ten, from 3.3) and 3.11), it follows tat Θ i ε 1/ Θ 3, i = 1,. Using tese observations as well as te differential geometric foumulas of Section 3.1, we write te equations 3.11) in te ξ variables, and collect te leading order terms in ε, yielding Θ i t Θ i ε ξ3 + P i U, Θ) + U 3 Θi + Q i U, Θ) + P i Θ, U) + Q i Θ, U) ξ 3 Q 3 Θ, U) + Q 3 U, Θ) + q ξ 3 = in Ω 3δ, T ) at least), i = 1,, in Ω 3δ, T ) at least), div Θ = in Ω, T ), Θ i = ũ i on Γ, T ), i = 1,, Θ 3 = on Γ, T ), Θ t= = in Ω. 3.1) Here and below, for any function f expressed in te ξ variables, we denote by f te restriction to te plane ξ 3 = in R 3 ξ. So, for instance, ũi := u e ) i ξ3 =. In contrast to te 3D cannel of Section, te curvature of te domain induces a small effect on te tangential components Θ i, i = 1,, in te normal direction, wic requires a pressure corrector q to cancel. Our task now is to build a corrector Θ as an approximating solution to tis system. We exploit te insigt gained from te construction of te incompressible corrector in Section.1, performing te matcing asymptotics in te equations 3.1) 1 and collecting te leading order terms wit respect to a small parameter ε. To construct te approximate solution to te corrector equations, we again solve a drift-diffusion equation in te alf space ξ 3 >

22 G.-M. Gie, J. Kellier and A. Mazzucato in R 3 ξ, and use te solution in te tangential components of Θ: θ i t ε ξθ i + P i Ũ, θ) + ξ 3 σ Ũ 3 ξ 3 θ i ξ 3 + Q i Ũ, θ) + P i θ, Ũ) + Q i θ, Ũ) =, in ω, ), T ), θ i = ũ i, at ξ 3 =, θ =, at t =, 3.13) were θ = θi e i and σ is a smoot cut-off function near te boundary, suc tat { 1, ξ3 δ, σξ 3 ) = 3.14), ξ 3 δ. Te P i Ũ, θ) is te value of Pi U, θ) wit U and te oter geometric functions, e.g., g and, evaluated at ξ 3 =. Te oter terms, Qi Ũ, θ), P i θ, Ũ), and Q i θ, Ũ), are defined in a similar way. In addition, for convenience, we ave set, wit a sligt abuse of notation, ξ v = 3 v ξ i for any scalar function v defined in ω, ), 3.15) wic is not te Laplacian expressed in te ξ variables.) Hence, equation 3.13) depends on ξ 3 only troug θ and te terms containing ξ 3. As we did for a cannel domain, we define te tangential components Θ i, i = 1,, of te corrector Θ to be Θ i ξ, t) = i ξ) ξ, ) { ξ3 } σξ 3 ) θ i ξ, η, t) dη, i = 1,. 3.16) i ξ 3 Ten, using 3.3), we define te normal component Θ 3 by enforcing te divergence-free constraint on Θ; tat is, Θ 3 ξ, t) = 1 ξ) σξ 3) ξ i { ξ3 } ξ, ) θ i ξ, η, t) dη. 3.17) i As a consequence of tis construction, Θ belongs to te space H, since div Θ = in Ω and Θ 3 = on Γ at ξ 3 =. To elucidate te structure of te corrector furter, we write Θ as a sum of tree vector fields in te form, Θ = θ + φ + ψ, 3.18)

23 Boundary layers for te linearized NSE 3 were, for i = 1,, and θ i = i φ i = i ψ i = i i σ θ i, σ θ i dη, i i σ ξ3 θ i dη, θ 3 =, φ 3 = 1 { σ ξ i i ψ 3 = 1 σ ξ i { i ξ3 } θ i dη, } θ i dη. 3.19) 3.) As was te cases for a cannel domain, θ, te main part of Θ, is a fast decaying boundary layer function wic agrees wit one in te classical teory of boundary layers, wile te remaining parts φ and ψ are small supplementary vector fields wit respect to a small ε) to ensure tat Θ belongs to te space H. We define te pressure corrector q in te form, q = σ ξ3 σ 1 Q3 θ, Ũ) + Q 3 Ũ, θ) ) dη. 3.1) Wit te coice made above for velocity corrector Θ, wic naturally follows from a Prandtl-type analysis, tere is small error of order ε 1/4 in L ) in 3.1) see 3.3) below as well).ten, q/ ξ 3 = Q 3 Θ, U) Q 3 U, Θ) up to a small error, as discussed in more detail later Estimates on te corrector As for te cannel, our goal in tis section is to derive estimates for te corrector, by deriving te equations tat its tree parts satisfy and estimating te data in terms of ε. First, we prove te estimates on θ i, i = 1,, below exactly in te same fasion as in te proof of Lemma.1. For i, j = 1, and k, l, and 1 p, we ave p k θ i ε ξ3 +ε 1 p ξ k j ξ3 L,T ;L p ω, ))) k θ i 3.) κ T ε 1 p, and ε L,T ;L ω, ))) ξ k j

24 4 G.-M. Gie, J. Kellier and A. Mazzucato k+1 θ i ξj k ξ + ε 1 k+1 θ i 3 L,T ;L ω, ))) ξj k ξ κ T ε 1 4, 3 L,T ;L ω, ))) 3.3) for a constant κ T depending on T and oter data, but independent of ε. We derive te equation for θ from its definition 3.18)-3.) and equation 3.13) for θ i : θ i t ε θ e i + P i 3 U Ũ, θ) + ξ θ i 3 σ + ξ 3 ξ Q i Ũ, θ) + P i θ, Ũ) + Q i θ, Ũ) 3 = E i tempθ), i = 1,, were E i tempθ) = ε θ e i ξ θ i ) ε ξ3 + σ i 1 j Ũ j ) i θ i ε ξ 3 σ i ) i θ i ξ 3 i ) σ θ i 3 U + ξ 3 σ ξ j i ξ 3 ξ 3 σ i ) i θ i. 3.4) 3.5) Using te differential geometric formulae for te differential operators as well as te estimates 3.), we notice tat E i tempθ) L,T ;L Ω)) κ T ε ) To estimate te convective terms, we apply Taylor s teorem at ξ 3 = to U and use te estimates in 3.) to obtain, P i U, θ) P i Ũ, θ) κ L ξ,t ;L Ω)) 3 σ θi L κ T ε 3 4, x j,t ;L Ω)) 3.7) U 3 ξ 3 σ Ũ 3 ) θ i L ξ ξ 3 x κ 3 σ θi L 3,T ;L Ω)) ξ κ T ε 3 4, 3,T ;L Ω)) 3.8) and Q i U, θ) Q i Ũ, θ) + L P i θ, U) P i θ, Ũ),T ;L Ω)) L,T ;L Ω)) + Q i θ, U) Q i θ, Ũ) κ L ξ,t ;L Ω)) 3 θ j L,T ;L Ω)) κ T ε ) From all te estimates above, we find tat te tangential part of te equation for θ can be written as, for i = 1,, θ i t ε θ e i + U θ) e i + θ U) e i = E i θ), 3.3)

25 Boundary layers for te linearized NSE 5 were E i θ) L,T ;L Ω)) κ T ε 3 4, i = 1,. 3.31) We proceed in a similar fasion for te normal component of te θ- equation. First, te differential geometric formulae 3.4) and 3.7)-3.9) give ) θ ε θ+u θ+θ U e 3 = εs 3 θ+q 3 U, θ)+p 3 θ, U)+Q 3 θ, U). t 3.3) Noticing tat te leading order term on te rigt-and side of 3.3) is Q 3 3 θ, Ũ) + Q Ũ, θ), we are led to define a pressure corrector q as in 3.1). Ten, θ ) 3 ε θ + U θ + θ U e 3 q = Eq), Eq) = E i q)e i, t 3.33) were E i q) = linear combination of te tangential derivatives of q), i = 1,, E 3 q) = RHS of 3.3)) ξ3 σ σ 1 ξ Q3 θ, Ũ) + Q ) ) 3 Ũ, θ) dη ) Tanks to 3.) and estimates similar to tose in 3.9), one can verify tat Eq) L,T ;L Ω)) κ T ε ) Combining 3.3) and 3.33), gives finally te system satisfied by θ, q): θ ε θ + U θ + θ U + q = Eθ) + Eq) in Ω, T ), t θ = u on Γ, T ), θ t= = in Ω. 3.36) We next turn to te supplementary part φ of te corrector Θ. Its definition in 3.18)-3.) gives We define te contribution of φ to te error as Eφ) := φ t φ Γ = φ t= =. 3.37) ε φ + U φ + φ U. To bound Eφ), we first observe tat, by 3.18)-3.), and 3.), for j = 1,, and k, m, k+m φ ξj k κ ξm 3 L,T ;L Ω)) k+1 θ i ξj k ξ κ T ε 1. l L,T ;L 1 ω, ))) i,l=1 3.38)

26 6 G.-M. Gie, J. Kellier and A. Mazzucato From calculations similar to tose in.41)-.45), we also ave φ κ T ε ) t L,T ;L Ω)) Tese estimates imply tat Eφ) κ T ε ) L,T ;L Ω)) Finally, we derive te equation for ψ defined also in 3.18)-3.). We use 3.13) and write te equation of ψ in te ξ i direction, i = 1,, as were ψ i t ε ψ e i + P i 3 U Ũ, ψ) + ξ ψ i 3 σ + ξ 3 ξ Q i Ũ, ψ) + P i ψ, Ũ) 3 + Q i ψ, Ũ) = Êi tempψ), i = 1,, E i tempψ) = ε ψ e i ξ ψ i ) ε ξ j ε ξ3 + i σ i ) ) ξ3 σ i i ξ3 θ i dη ε θ i dη ε ξ 3 1 Ũ j i ) ξ3 σ θ i dη j ξ j i ξ3 3.41) i ) ξ3 σ θ i dη ξ j i ξ j σ i ) i θ i 3 U +ξ 3 σ σ i ) θ i dη, i = 1,. ξ 3 ξ 3 i 3.4) Since σ and all its derivatives vanis for ξ 3 δ, we use 3.) and find tat ξ3 p Ê i ε tempψ) κ T ε 1, L,T ;L p Ω)) l, 1 p. 3.43) By te energy estimate for 3.41) wic is identical to one in.19)), we ten obtain ξ3 p k ψ i + ε 3 ) 4 ξ3 l k ψ i κ T ε 1, ε L,T ;L p Ω)) ε L,T ;L Ω)) ξ k j 3.44) for i, j = 1, and k, l, and 1 p. Te normal component ψ 3 satisfies similar bounds given tat te two expressions for ψ i, i = 1,, and for ψ 3 are similar. Using tese estimates as well as 3.7), 3.8), and 3.9), we conclude tat te second supplementary ξ k j

27 Boundary layers for te linearized NSE 7 part ψ of te corrector satisfies ψ ε ψ + U ψ + ψ U = Eψ) in Ω, T ), t ψ = on Γ, T ), ψ t= = in Ω, were 3.45) Eψ) L,T ;L Ω)) κ T ε 1, i = 1,. 3.46) 3.4. Proof of Teorem 1.1: Te case of a 3D smoot domain Recall tat te error is given by w ε := u ε u +Θ), π ε = p ε p +q). Ten, tanks to te equations satisfied by u ε, u, and te corrector Θ, along wit te divergence-free condition on Θ, te equation for w ε, π ε ) can be written as w ε ε w ε + U w ε + w ε U + π ε = EΘ) + Eq) in Ω, T ), t div w ε = in Ω, T ), were w ε = on Γ, T ), w ε t= = in Ω. 3.47) EΘ) = Eθ) + Eφ) + Eψ). 3.48) for w ε, π ε ) Using te bounds derived in te previous sections specifically 3.31), 3.35), 3.4), and 3.46)), te error estimate in 1.9) follows from a simple energy estimate. Te vanising viscosity limit 1.1) is a consequence of 1.9) and te smallness of te corrector in L, T ; L Ω)). Te proof of Teorem 1.1 is complete. References [1] G. K. Batcelor. An introduction to fluid dynamics. Cambridge Matematical Library. Cambridge University Press, Cambridge, paperback edition, [] J. R. Cannon. Te one-dimensional eat equation, volume 3 of Encyclopedia of Matematics and its Applications. Addison-Wesley Publising Company, Advanced Book Program, Reading, MA, Wit a foreword by Felix E. Browder. 1 [3] D. Gérard-Varet and E. Dormy. On te ill-posedness of te Prandtl equation. J. Amer. Mat. Soc., 3):591 69, 1. 5 [4] G.-M. Gie. Asymptotic expansion of te Stokes solutions at small viscosity: te case of non-compatible initial data. Commun. Mat. Sci., 1):383 4, 14. 1, 4, 1

28 8 G.-M. Gie, J. Kellier and A. Mazzucato [5] G.-M. Gie, M. Hamouda, and R. Temam. Boundary layers in smoot curvilinear domains: parabolic problems. Discrete Contin. Dyn. Syst., 64):113 14, 1. 1 [6] G.-M. Gie, C.-Y. Jung, and R. Temam. Recent progresses in boundary layer teory. Discrete Contin. Dyn. Syst., 365):51 583, [7] G.-M. Gie and J. P. Kellier. Boundary layer analysis of te Navier-Stokes equations wit generalized Navier boundary conditions. J. Differential Equations, 536): , [8] E. Grenier, Y. Guo, and T. T. Nguyen. Spectral stability of Prandtl boundary layers: an overview. Analysis Berlin), 354): , [9] Y. Guo and T. Nguyen. A note on Prandtl boundary layers. Comm. Pure Appl. Mat., 641): , [1] D. s. Iftimie and F. Sueur. Viscous boundary layers for te Navier-Stokes equations wit te Navier slip conditions. Arc. Ration. Mec. Anal., 1991): , 11. [11] T. Kato. Remarks on zero viscosity limit for nonstationary Navier-Stokes flows wit boundary. In Seminar on nonlinear partial differential equations Berkeley, Calif., 1983), volume of Mat. Sci. Res. Inst. Publ., pages Springer, New York, [1] W. Klingenberg. A course in differential geometry. Springer-Verlag, New York, Translated from te German by David Hoffman, Graduate Texts in Matematics, Vol. 51. [13] H. Koc. Transport and instability for perfect fluids. Mat. Ann., 333):491 53,. 4 [14] J.-L. Lions. Perturbations singulières dans les problèmes aux limites et en contrôle optimal. Lecture Notes in Matematics, Vol. 33. Springer-Verlag, Berlin-New York, [15] M. C. Lombardo and M. Sammartino. Zero viscosity limit of te Oseen equations in a cannel. SIAM J. Mat. Anal., 33):39 41, 1. 5 [16] Y. Maekawa and A. Mazzucato. Te Inviscid Limit and Boundary Layers for Navier-Stokes Flows, pages Springer International Publising, Cam, [17] L. Prandtl. Verber flüssigkeiten bei ser kleiner reibung. Verk. III Intem. Mat. Kongr. Heidelberg, pages , 195, Teuber, Leibzig. 4 [18] H. Sor. Te Navier-Stokes equations. Modern Birkäuser Classics. Birkäuser/Springer Basel AG, Basel, 1. An elementary functional analytic approac, [13 reprint of te 1 original] [MR198881]. 3 [19] R. Temam. Navier-Stokes equations. AMS Celsea Publising, Providence, RI, 1. Teory and numerical analysis, Reprint of te 1984 edition. 3 [] R. Temam and X. Wang. Asymptotic analysis of Oseen type equations in a cannel at small viscosity. Indiana Univ. Mat. J., 453): , , 5 [1] R. Temam and X. Wang. Boundary layers for Oseen s type equation in space dimension tree. Russian J. Mat. Pys., 5): ), [] R. Temam and X. M. Wang. Asymptotic analysis of te linearized Navier- Stokes equations in a cannel. Differential Integral Equations, 87): ,

29 Boundary layers for te linearized NSE 9 [3] Z. Xin and T. Yanagisawa. Zero-viscosity limit of te linearized Navier-Stokes equations for a compressible viscous fluid in te alf-plane. Comm. Pure Appl. Mat., 54): , , 5 Gung-Min Gie Department of Matematics University of Louisville Louisville, KY 49 gungmin.gie@louisville.edu James P. Kellier Department of Matematics University of California, Riverside 9 University Ave. Riverside, CA kellier@mat.ucr.edu Anna L. Mazzucato Department of Matematics Penn State University University Park, PA alm4@psu.edu