The Inviscid Limit of the Navier-Stokes Equations with Kinematic and Navier Boundary Conditions
aa r X i v : . [ m a t h . A P ] D ec THE INVISCID LIMIT OF THE NAVIER-STOKES EQUATIONSWITH KINEMATIC AND NAVIER BOUNDARY CONDITIONS
GUI-QIANG G. CHEN, SIRAN LI, AND ZHONGMIN QIAN
Abstract.
We are concerned with the inviscid limit of the Navier-Stokes equations on boundedregular domains in R with the kinematic and Navier boundary conditions. We first es-tablish the existence and uniqueness of strong solutions in the class C ([0 , T ⋆ ); H r (Ω; R )) ∩ C ([0 , T ⋆ ); H r − (Ω; R )) with some T ⋆ > for the initial-boundary value problem with thekinematic and Navier boundary conditions on ∂ Ω and divergence-free initial data in the Sobolevspace H r (Ω; R ) for r ≥ . Then, for the strong solution with H r +1 –regularity in the spatialvariables, we establish the inviscid limit in H r (Ω; R ) uniformly on [0 , T ⋆ ) for r > . This showsthat the boundary layers do not develop up to the highest order Sobolev norm in H r (Ω; R ) inthe inviscid limit. Furthermore, we present an intrinsic geometric proof for the failure of thestrong inviscid limit under a non-Navier slip-type boundary condition. Introduction
We are interested in the analysis of strong solutions in the Sobolev spaces H r of theincompressible Navier-Stokes equations with positive viscosity coefficient ν > in a boundedregular domain Ω ⊂ R subject to the kinematic and Navier boundary conditions on ∂ Ω and thedivergence-free initial data at t = 0 , and their convergence to the corresponding strong solutionof the Euler equations in the inviscid limit as ν → . One of our main motivations for such ananalysis is to examine whether the boundary layers would develop in some high-order Sobolevnorm in the inviscid limit.We assume that the boundary, ∂ Ω , of domain Ω is an embedded oriented -dimensional(2-D) manifold, i.e. a regular surface. The incompressible Navier-Stokes equations in [0 , T ] × Ω take the following form: ∂ t u ν + ( u ν · ∇ ) u ν + ∇ p ν = ν ∆ u ν , ∇ · u ν = 0 . (1.1)In (1.1), the vector field u ν : Ω → R is the velocity of the fluid and the scalar field p ν : Ω → R isthe pressure, both of which depend on the viscosity constant ν > . The divergence-free conditionof u ν describes the incompressibility of the fluid. The existence, uniqueness, and regularity ofweak and strong solutions of the Navier-Stokes equations (1.1) are an important research topicin nonlinear PDEs and mathematical hydrodynamics; cf. [30, 31, 35, 47, 44] and the referencescited therein. In this paper, we focus on the Navier-Stokes equations (1.1) in a general boundedregular domain Ω , for which the geometry of Ω plays an important role in our analysis. Date : December 18, 2018.2010
Mathematics Subject Classification.
Primary: 35Q30, 35Q31, 35Q35, 76D03, 76D05, 76D09.
Key words and phrases.
Navier-Stokes equations, Euler equations, inviscid limit, vanishing viscosity limit, strongconvergence, higher-order, Navier boundary condition, kinematic boundary condition, weak solution, strong so-lution, Lie derivative, vorticity, boundary layers. onsider the initial condition: u ν | t =0 = u on Ω , (1.2)where u satisfies the compatibility condition: ∇ · u = 0 in Ω .The kinematic boundary condition is u ν · n = 0 on ∂ Ω × [0 , T ] , (1.3) i.e. the normal component of the velocity on the boundary vanishes.The Navier boundary condition is imposed as: u · τ = − ζ D u ( τ, n ) on ∂ Ω × [0 , T ] , (1.4)for any τ ∈ T ( ∂ Ω) , where the rate-of-strain tensor is the × matrix defined by D u := 12 Ä ∇ u + ( ∇ u ) ⊤ ä , (1.5) D u ( τ, n ) := τ ⊤ D u n , and constant ζ > is known as the slip length of the fluid.Traditionally, the Navier-Stokes equations (1.1) have been studied with the no-slip condi-tion, i.e. the Dirichlet boundary condition u = 0 on ∂ Ω . However, this does not always matchwith the experimental data; cf. [24, 42]. First proposed by Navier [39] in 1816, the Navier bound-ary condition (1.4) requires that the tangential component of the velocity field is proportional tothat of the normal vector field of the Cauchy stress tensor. The proportionality constant ζ > is known as the slip length . Physically, the Navier boundary condition (1.4) can be induced bythe effects of free capillary boundaries, perforated boundaries, or the exterior electric fields; cf. Achdou-Pironneau-Valentin [1], Bänsch [4], Beavers-Joseph [7], Einzel-Panzer-Liu [24], Maxwell[38], Jäger-Mikeli˘c [28, 29], and the references cited therein.To analyze the initial-boundary value problem (1.2)–(1.4) for the Navier-Stokes equations(1.1), we adopt an equivalent geometric formulation, as shown in Chen-Qian [15], for the bound-ary conditions on ∂ Ω × [0 , T ] : u ν · n = 0 ,ω ν · τ = − ζ ( R u ν ) · τ + 2 R ( S ( u ν )) · τ on ∂ Ω × [0 , T ] , (1.6)where ω ν := ∇ × u ν (1.7)is the vorticity of the fluid, τ ∈ T ( ∂ Ω) is an arbitrary tangential vector field on boundary ∂ Ω , S is the shape operator of surface ∂ Ω , and R is the operator corresponds to the left multiplicationby the matrix in the local coordinate on ∂ Ω : R = " −
11 0 , (1.8) i.e. the anti-clockwise rotation by π . In fact, R can be identified with the Hodge star operator ∗ defined for the differential forms on R : For a 2-D vector field V = ( V , V ) ⊤ , R V = ( ∗ ( V ♯ )) ♭ = ( − V , V ) ⊤ , (1.9) n which ♯ is the canonical isomorphism between vector fields and differential -forms, and ♭ isits inverse. The second equation in (1.6) ( i.e. the Navier boundary condition in the geomet-ric formulation) has the vorticity on the left-hand side, but it involves only the zero-th orderoperations on the velocity on the right-hand side.The problem of inviscid limits has been a central topic in mathematical hydrodynamics ( cf. Constantin [18]). In 1975, Swann [46] proved that, for
Ω = R , when the initial vorticity is in H δ , divergence–free, and vanishing at spatial infinity, and the right–hand side of the vorticityequation lies in C ([0 , T ); H ( R )) for some small T , then the initial-boundary value problem forthe Navier–Stokes equations with zero boundary condition has a unique strong solution, and thevanishing viscosity limit holds in L ∩ ˙ H . In 1986, Constantin [17] showed that, for Ω = R ,if the Cauchy problem for the Euler equations with initial data v ∈ H m +2 ( R ) for m ≥ hasa strong solution in X = C ([0 , T ]; H m ( R )) up to time T , then there exists ν ⋆ = ν ⋆ ( T, v ) suchthat the Cauchy problem for the corresponding Navier-Stokes equations for any ν ≤ ν ⋆ also hasa strong solution in X , and the vanishing viscosity limit holds in H m . In fact, for Ω = R d for d = 2 or , for any s > d + 1 and initial data v ∈ H s , the convergence can be obtained in the H s –norm; cf. Masmoudi [36]. Moreover, in Constantin-Wu [20], the vanishing viscosity limitswere also proved on
Ω = R for the initial vorticity in L ( R ) ∩ L ∞ c ( R ) .On the other hand, in the case that Ω is a bounded domain with boundary, and the Navier-Stokes equations are equipped with the Dirichlet boundary condition, the vanishing viscosity limitfails in general: This is due to the formation of boundary layers , in which the Prandtl equationsserve as a candidate for matching the Navier-Stokes and Euler equations; see e.g. , Alexandre-Wang-Xu-Yang [3], Gérard-Varet-Dormy [25], and the references cited therein. In contrast, whenthe Navier and kinematic boundary conditions are imposed to the Navier-Stokes equations, thevanishing viscosity limit can be established in the affirmative. In 2007, Xiao-Xin [49] provedthat, for the initial data in H on a 3-D flat domain, there exists T ⋆ > such that the vanishingviscosity limit holds in C ([0 , T ]; H ) ∩ L p (0 , T ; H ) , for all ≤ p < ∞ . Various convergenceresults of this kind for a non-Navier “slip-type boundary condition” (first proposed by Bardos[5], which agrees with the Navier condition if and only if the domain is a part of the flat half–space) have been established, in W k,p , H s , or L p spaces and on 2-D or 3-D spatial domains; cf. Xiao-Xin [49], Beirão da Veiga-Crispo [8, 9], Bellout-Neustupa-Penel [12], Berselli-Spirito [13],Chen-Osborne-Qian [14], Clopeau-Mikeli˘c-Robert [16], Kelliher [32], Wang-Xin-Zang [48], Zhong[50], and the references cited therein.Furthermore, for the Navier boundary conditions, Chen-Qian [15] and Iftimie-Planas [26]obtained the vanishing viscosity limit in L ∞ t L x on smooth domains Ω ⊂ R and R d , d ≥ , pro-vided that strong solutions exist in H and H d/ ǫ , respectively; see also the related results byIfitimie-Raugel-Sell [27] on a 3-D thin domain and by Lopes Filho-Nussenzveig Lopes-Planas [34]on 2-D domains, and the recent results by Drivas-Nguyen [22]. In addition, by computations in lo-cal coordinates, Neustupa-Penel [40, 41] proved the convergence in L ∞ (0 , T ⋆ ; H ) ∩ L (0 , T ⋆ ; H ) ,provided that the initial data is in H , where T ⋆ > is a constant depending only on Ω andthe initial data. Moreover, using the geometric vector field approach, Masmoudi-Rousset [37]established the existence of strong solutions in L ∞ (0 , T ; E m (Ω; R )) ∩ L (0 , T ; H m +1 (Ω; R )) for m > and the inviscid limit in L ∞ t L x , where the anisotropic co-normal Sobolev space E m isgiven by E m := { u ∈ H m co : ∇ u ∈ H m − } , and u ∈ H m co whenever P ≤| l |≤ m k Z l u k L (Ω) < ∞ with { Z l } spanning the space of vector fields tangential to ∂ Ω . n this paper, by performing the higher-order energy estimates for the weak solutionsconstructed in [15], we first establish the existence and uniqueness of the strong solution of theNavier-Stokes equations in C (0 , T ⋆ ; H r (Ω; R )) ∩ C (0 , T ; H r − (Ω; R )) for some T ⋆ > and r ≥ , subject to the kinematic and Navier boundary conditions. We assume that domain Ω is regular, with the smooth second fundamental form II . In fact, in the estimates, we need k II k C r − ( ∂ Ω) < ∞ . Moreover, an explicit lower bound for T ⋆ is obtained. This is achieved byemploying more delicate energy estimates, which take into account the effects of the curvature(equivalently, the second fundamental form II ) of ∂ Ω and the Navier boundary conditions. Inaddition, we study the inviscid limit (also known as the vanishing viscosity limit) of the Navier-Stokes equations (1.1): We send ν → + and investigate whether the strong solutions u ν converge,in suitable norms, to the corresponding solution of the Euler equations describing the motion ofincompressible, inviscid fluids: ∂ t u + ( u · ∇ ) u + ∇ p = 0 in [0 , T ] × Ω , ∇ · u = 0 in [0 , T ] × Ω ,u | t =0 = u on Ω , (1.10)subject to the no-penetration boundary condition : u · n = 0 on [0 , T ] × ∂ Ω . (1.11)As discussed above, for the kinematic and Navier boundary conditions, the inviscid limitproblem was answered in the affirmative for strong solutions on domains with flat boundaries( e.g. the half-space) by Xiao-Xin [49] and Beirão da Veiga-Crispo [8, 9]. This is achieved byanalyzing the aforementioned simplified boundary condition in [5, 45], which agrees with theNavier boundary condition for flat boundaries. Similar affirmative results are also establishedfor several modified versions of the slip-type boundary conditions in [48, 50]. In addition, theinviscid limit for the strong solutions in L or H under the kinematic and Navier boundaryconditions are proved by Chen-Qian [15], Intimie-Planas [26], and Neustupa-Penel [40, 41] forbounded, regular, possibly non-flat domains in R .On the other hand, recently in [10, 11], Beirão da Veiga-Crispo proved that the inviscidlimits in strong topologies of W s,p for s > and p > fails for general non-flat domains, withthe Navier-Stokes equations equipped with the simplified boundary conditions as in [5, 45]. Incomparison, the inviscid limit in strong topologies always holds for regular domains in 2-D, whenthe Navier boundary condition is assumed. This is largely due to the fact that the vorticity istransported in 2-D; cf. [16, 34, 19].In view of the discussions above, it is important to understand whether the inviscid limitholds for strong solutions in the higher-order Sobolev norms in H r (Ω; R ) for r > in a bounded,regular, generally non-flat domain Ω ⊂ R , when the Navier-Stokes equations (1.1) are equippedwith the Navier boundary conditions ( i.e. Eq. (1.6)). To the best of our knowledge, this problemis still largely open. In Theorem 5.1, we answer this question in the affirmative: If the strongsolution exists in H r +1 (Ω; R ) for r > , we establish its strong convergence in H r (Ω; R ) asthe viscosity constant ν → . This implies that the boundary layers do not develop up to thehighest order Sobolev norm in H r (Ω; R ) for r > .The rest of the paper is organized as follows: In §2 we briefly sketch the derivation of theboundary conditions in terms of geometric quantities. In §3, we prove a lemma which expresses he H r –norm of a divergence-free vector field by the L –norm of the iterated curls, subject tothe kinematic and Navier boundary conditions. Next, in §4, we derive the a priori , higher-orderenergy estimates in H r (Ω; R ) for r ≥ for the Navier-Stokes equations with kinematic andNavier boundary conditions. We also deduce the existence of strong solutions from the energyestimates. Then, in §5, the inviscid limit is established. Finally, in §6, we discuss the inviscidlimit problem for other non-Navier slip-type boundary conditions.Before concluding this introduction, we present some notations that will be used from nowon in this paper. We denote H r (Ω; R ) = W ,r (Ω; R ) as the Sobolev space of vector fields φ : Ω → R with the norm in the multi-index notation: k φ k H r (Ω) := (cid:16) X ≤| α |≤ r Z Ω |∇ α φ | d x (cid:17) / < ∞ . (1.12)We write ∇ [ s ] to denote a generic differential operator ∇ i ∇ i · · · ∇ i s for any s ≥ . The Einsteinsummation convention is used. For the indices, we write i , i , . . . , j, k, l, . . . ∈ { , , } and α, β, γ, δ, . . . ∈ { , } . The angular bracket h· , ·i denotes the Euclidean inner product of twovectors in R . Furthermore, we write f . g if | f | ≤ C | g | for a generic constant C depends onlyon r , k II k C r − ( ∂ Ω) , and ζ ; and write f ≃ g whenever f . g and g . f . Denote H as the 2-DHausdorff measure. Finally, curl r := curl ◦ . . . ◦ curl means the composition of r curls.2. The Navier Boundary Condition
In this section, we briefly sketch the derivation of the boundary conditions in terms ofgeometric quantities.First of all, we justify that our geometric formulation of the Navier boundary condition( i.e. the second equation in (1.6), reproduced below): ω ν · τ = − ζ ( R u ν ) · τ + 2 R Ä S ( u ν ) ä · τ on ∂ Ω × [0 , T ] for any τ ∈ T ( ∂ Ω) is indeed equivalent to the one proposed by Navier in [39]. For simplicity, we drop superscript ν in this section.We start by remarking on the geometric notations. Recall that the boundary of the domainof fluid, ∂ Ω , is a regular surface embedded in R . We denote its second fundamental form by II : T ( ∂ Ω) × T ( ∂ Ω) → R , where T ( ∂ Ω) is the tangent bundle of ∂ Ω . Thus, writing n ∈ T ( ∂ Ω) ⊥ as the outward unit normal (viewed as the Gauss map n : ∂ Ω → S ), we have II = −∇ n . (2.1)In addition, take { e , e , e } to be an orthonormal frame such that e , e ∈ T ( ∂ Ω) and e = n .Then we have the local expression: II( u, v ) = X α =1 2 X β =1 II αβ u α v β . (2.2)The shape operator S : T ( ∂ Ω) → T ( ∂ Ω) is then defined as S ( u ) := −∇ u n , (2.3)where ∇ u means the directional derivative in the direction of u . ow, recall that the Navier boundary condition reads that, for any τ ∈ T ( ∂ Ω) , u · τ = − ζ D u ( τ , n ) on ∂ Ω × [0 , T ] , where, in local coordinates, the rate-of-strain tensor is given by ( D u ) ij = 12 Ä ∇ i u j + ∇ j u i ä , ≤ i, j ≤ . Suppose that { e , e , e } is an orthonormal moving frame adapted to ∂ Ω , with e = n . Then theNavier boundary condition is equivalent to the following: u = − ζ ( ∇ u + ∇ u ) , u = − ζ ( ∇ u + ∇ u ) on ∂ Ω × [0 , T ] . (2.4)The main issue of this paper is to derive the higher-order energy estimates of velocity u .As shown in §3 below, the H r –norm of u is estimated purely by the L –norm of the r -th iterated curls of u ( cf. Theorem 3.1). We now seek for the boundary condition with respect to thevorticity: ω = ∇ × u . For this purpose, note that ω = ∇ u − ∇ u ∇ u − ∇ u ∇ u − ∇ u in the local frame { e , e , e } . Then the Navier boundary condition (1.4) becomes ∇ k u + ∇ u k = − ζ u k for k ∈ { , } . (2.5)On the other hand, ∇ k u can be computed as ∇ k u = ∇ k ( u · n ) = ∂ ( u · n ) + X j =1 Γ kj u j , (2.6)where Γ kij = g kl ( ∂ i g jl + ∂ j g il − ∂ l g ij ) are the Christoffel symbols. Observe also that II jk = II( e j , e k ) · n = −∇ j e k · n = − X l =1 Γ ljk e l · n = − Γ jk . (2.7)Then, by collecting Eqs. (2.5)–(2.7), we have ∇ k u − ∇ u k = 2 ∂ k ( u · n ) − X j =1 II jk u j + 1 ζ u k . (2.8)Finally, in view of the kinematic boundary condition ( i.e. the first equation in (1.6)), u · n = 0 on ∂ Ω . Then, by taking k = 1 , , respectively, and recalling the definition of R , we immediatelyrecover the second equation in (1.6). Note that the term, R ( S ( u ν )) , reflects the geometry ofthe curvilinear fluid domain. It vanishes when the domain is flat, e.g. the half plane. In the restof the paper, this is referred to as the Navier boundary condition .3. A Div-Curl Estimate for Divergence-free Vector Fields
In this section, we show that the H r +1 –norm of a divergence-free vector field is equivalentto the sum of the L –norms of its iterated curls up to the ( r + 1) -th order. It is a variant of thewell-known div-curl estimate due to Caldéron-Zygmund for divergence-free vector fields. heorem 3.1. Let u ∈ H r +1 (Ω; R ) ∩ K (Ω) for r ≥ satisfy the kinematic and Navier boundaryconditions (1.6) , where K (Ω) := ¶ u ∈ L (Ω; R ) : ∇ · u = 0 © . (3.1) Then there exists a universal constant M = M ( r, Ω) > such that k∇ r +1 u k L (Ω) ≤ M r +1 X l =0 k curl l u k L (Ω) . (3.2)Here and in the sequel, the time variable t is always suppressed when only the spatialregularities are considered. The following Sobolev trace theorem is also frequently used:
Lemma 3.2 (Theorem 5.36 in [2]) . Let Ω be a domain in R n satisfying the uniform C m –regularitycondition. Assume that there exists a ( m, p ) –extension operator for Ω . Suppose that mp < n, p ≤ q ≤ p ∗ := ( n − pn − mp . (3.3) Then the continuous embedding W m,p (Ω) ֒ → L q ( ∂ Ω) holds. In particular, it implies that H (Ω) ֒ → L q ( ∂ Ω) for any q ∈ [2 , in the regular domain Ω ⊂ R . Proof of Theorem . We prove the theorem by induction on r . The arguments are divided intoseven steps. We first establish the base case r = 0 . Indeed, in view of the following identity (see Eq.(3.3) in Chen-Qian [15]): k∇ u k L (Ω) = k∇ × u k L (Ω) + k∇ · u k L (Ω) − Z ∂ Ω ( ∇ · u ) h u, n i d H + Z ∂ Ω h u · ∇ u, n i d H , for the incompressible velocity field satisfying the kinematic boundary condition, we have k∇ u k L (Ω) = k∇ × u k L (Ω) + Z ∂ Ω II( u, u ) d H , (3.4)where we have utilized the definition of the second fundamental form II := −∇ n . Since k II k L ∞ ( ∂ Ω) < ∞ , we bound (cid:12)(cid:12)(cid:12) Z ∂ Ω II( u, u ) d H (cid:12)(cid:12)(cid:12) ≤ k II k L ∞ ( ∂ Ω) k u k L ( ∂ Ω) ≤ ǫ k∇ u k L (Ω) + Cǫ k u k L (Ω) , (3.5)thanks to the Sobolev trace inequality and Young’s inequality. Thus, the case for r = 0 followsimmediately by choosing ǫ suitably small. We now assume the result for r ≥ and prove it for r + 1 . First of all, we applyintegration by parts twice to obtain k∇ r +1 u k L (Ω) = Z Ω Ä ∂ i · · · ∂ i r +1 u k äÄ ∂ i · · · ∂ i r +1 u k ä d x = Z Ω ∂ i n Ä ∂ i · · · ∂ i r +1 u k äÄ ∂ i · · · ∂ i r +1 u k ä o d x − Z Ω ∂ i n Ä ∂ i · · · ∂ i r +1 u k äÄ ∆ ∂ i · · · ∂ i r +1 u k ä o d x + Z Ω Ä ∆ ∂ i · · · ∂ i r +1 u k äÄ ∆ ∂ i · · · ∂ i r +1 u k ä o d x =: I + J + K. (3.6) sing the divergence theorem, the above three integrals are expressed as I = R ∂ Ω ∂ n |∇ r u | d H ,J = R ∂ Ω Ä ∂ i · · · ∂ i r +1 u k äÄ ∆ ∂ i · · · ∂ i r +1 u k ä h∇ i , n i d H ,K = R Ω |∇ r − ψ | d x, (3.7)where ψ = curl ω = − ∆ u is the stream function. Now we bound the surface integral I in (3.7). For this purpose, we introduce a localmoving frame { e , e , e } on surface ∂ Ω such that e , e ∈ T ( ∂ Ω) and e = n . Then I = Z ∂ Ω Ä ∇ i · · · ∇ i r u k äÄ ∇ i · · · ∇ i r ∇ u k ä d H + Z ∂ Ω Ä ∇ i · · · ∇ i r u k äÄ [ ∇ , ∇ i · · · ∇ i r ] u k ä d H =: I + I , (3.8)where [ · , · ] denotes the commutator. Since the commutator is of lower order, the second termin the integrand of I is schematically represented as ∇ [ r − u k . More precisely, by the Ricciidentity: ∇ i ∇ j V k − ∇ j ∇ i V k = X l C klij V l (3.9)for any vector field V ∈ T R and some constants C klij , each time we exchange ∇ with ∇ i j , azero-th order term is obtained. Then the Leibniz rule yields [ ∇ , ∇ i · · · ∇ i r ] u k ≃ ∇ [ r − u k . (3.10)Then the Cauchy-Schwarz inequality leads to | I | . k u k H r ( ∂ Ω) + k u k H r − ( ∂ Ω) . ǫ k∇ r +1 u k L (Ω) + (1 + 1 ǫ ) k u k H r (Ω) , (3.11)where the second line follows from the Sobolev trace embedding H r +1 (Ω) ֒ → H r ( ∂ Ω) for r ≥ ,together with the interpolation inequalities. To bound I , we make a crucial use of the kinematic and Navier boundary conditions(1.6). First, we rewrite it in the local frame { e , e , e } as u = 0 , ∇ u β = 2II αβ u α − ζ u β for β ∈ { , } , (3.12)where ∇ α u ≡ so that ω = −∇ u and ω = ∇ u . Moreover, from the incompressibilitycondition: ∇ · u = 0 , the following identities hold: ∇ u = −∇ α u α , ∇ ∇ u α = − ψ α − ∇ β ∇ β u α . (3.13)The key to Eqs. (3.12)–(3.13) is that the normal derivatives ∇ of the normal components can bereplaced by the tangential derivatives, and the normal derivatives of the tangential componentscan be replaced by the lower-order terms. We now estimate I . For simplicity, we introduce the short-hand notations: ∇ ( r − A · ∇ ( r − B := Ä ∇ i · · · ∇ i r − A ä · Ä ∇ i · · · ∇ i r − B ä , (3.14)for any sufficiently regular functions A and B . Then we split I into six terms: I := I , + I , + I , + I , + I , + I , , here I , = R ∂ Ω ( ∇ ( r − ∇ α ∇ β u ) · ( ∇ ( r − ∇ α ∇ β ∇ u ) d H ,I , = R ∂ Ω ( ∇ ( r − ∇ α ∇ β u γ ) · ( ∇ ( r − ∇ α ∇ β ∇ u γ ) d H ,I , = R ∂ Ω ( ∇ ( r − ∇ α ∇ u ) · ( ∇ ( r − ∇ α ∇ ∇ u ) d H ,I , = R ∂ Ω ( ∇ ( r − ∇ α ∇ u γ ) · ( ∇ ( r − ∇ α ∇ ∇ u γ ) d H ,I , = R ∂ Ω ( ∇ ( r − ∇ ∇ u ) · ( ∇ ( r − ∇ ∇ ∇ u ) d H ,I , = R ∂ Ω ( ∇ ( r − ∇ ∇ u γ ) · ( ∇ ( r − ∇ ∇ ∇ u γ ) d H . (3.15)In the sequel, we estimate these terms one by one.First of all, I , = 0 , since ∇ β u ≡ .To estimate I , , we first notice that (cid:12)(cid:12)(cid:12) ∇ ( r − ∇ α ∇ β ∇ u γ (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ∇ ( r − ∇ α ∇ β Ä γβ u β − ζ u γ ä (cid:12)(cid:12)(cid:12) = C |∇ [ r − u | + l . o . t ., (3.16)where C depends on k II k C r − ( ∂ Ω) and ζ − , and l . o . t . contains the derivatives of u of order lessthan or equal to r − . Next, considering the two cases: α = β and α = β separately, we deduce I , = Z ∂ Ω Ä ∇ ( r − ∆ u γ ä · Ä ∇ ( r − ∆ ∇ u γ ä d H + 2 Z ∂ Ω Ä ∇ ( r − ∇ ∇ u γ ä · Ä ∇ ( r − ∇ ∇ ∇ u γ ä d H . (3.17)For the first term, again by the Ricci identity, we write ∇ [ r − ∆ ∇ u γ = ∇ ( r − ∇ ∆ u γ + ∇ [ r − (∆ u γ ) = −∇ [ r − ∇ ψ γ − ∇ [ r − u γ , (3.18)and treat the second term as in Eq. (3.16) above. Then we obtain | I , | . (cid:12)(cid:12)(cid:12)(cid:12) Z ∂ Ω Ä ∇ [ r − ∇ ψ γ ä · Ä ∇ [ r − u γ ä d H (cid:12)(cid:12)(cid:12)(cid:12) + Z ∂ Ω |∇ [ r − u | d H . k u k H r − ( ∂ Ω) + k∇ [ r − ( ∇ × ψ ) k L ( ∂ Ω) (3.19)by the Cauchy-Schwarz inequality. By the trace and interpolation inequalities, we have k∇ [ r − ( ∇ × ψ ) k L ( ∂ Ω) . ǫ k∇ r +1 u k H r +1 (Ω) + 1 ǫ k u k H r (Ω) . Then | I , | . ǫ k∇ r +1 u k H r +1 (Ω) + 1 ǫ k u k H r (Ω) . (3.20)For I , , again by Eq. (3.13), the Ricci identity, the boundary condition (3.12), and thetrace and interpolation inequalities, we have | I , | = (cid:12)(cid:12)(cid:12)(cid:12) Z ∂ Ω Ä ∇ ( r − ∇ α ∇ β u β äÄ ∇ ( r − ∇ α ∇ ∇ γ u γ ä d H (cid:12)(cid:12)(cid:12)(cid:12) ≃ (cid:12)(cid:12)(cid:12)(cid:12) Z ∂ Ω Ä ∇ ( r − ∇ α ∇ β u β äÄ ∇ ( r − ∇ α ∇ γ ∇ u γ + ∇ [ r − u ä d H (cid:12)(cid:12)(cid:12)(cid:12) ≃ (cid:12)(cid:12)(cid:12)(cid:12) Z ∂ Ω (cid:16) ∇ ( r − ∇ α ∇ β u β (cid:17)(cid:16) ∇ ( r − ∇ α ∇ γ Ä δγ u δ − ζ u γ ä + ∇ [ r − u (cid:17) d H (cid:12)(cid:12)(cid:12)(cid:12) . k u k H r − ( ∂ Ω) . k u k H r (Ω) . (3.21) he treatment for I , is similar to the above for I , : | I , | = (cid:12)(cid:12)(cid:12)(cid:12) Z ∂ Ω (cid:16) ∇ ( r − ∇ α Ä βγ u β − ζ u γ ä (cid:17)(cid:16) ∇ ( r − ∇ α Ä − ψ α − ∇ δ ∇ δ u γ ä (cid:17) d H (cid:12)(cid:12)(cid:12)(cid:12) . (cid:12)(cid:12)(cid:12)(cid:12) Z ∂ Ω Ä ∇ [ r − u äÄ ∇ [ r ] u ä d H (cid:12)(cid:12)(cid:12)(cid:12) . k u k H r ( ∂ Ω) . ǫ k∇ r +1 u k L (Ω) + 1 ǫ k u k H r (Ω) . (3.22)For I , , we first substitute in Eq. (3.12) to derive I , = Z ∂ Ω Ä ∇ ( r − ∇ ∇ β u β äÄ ∇ ( r − ∇ ∇ ∇ α u α ä d H . Then, applying the Ricci identity once to the first term and twice to the second term in theintegrand, we have | I , | . (cid:12)(cid:12)(cid:12)(cid:12) Z ∂ Ω Ä ∇ ( r − ∇ β ∇ u β + ∇ [ r − u äÄ ∇ ( r − ∇ α ∇ ∇ u α + ∇ [ r − u ä d H (cid:12)(cid:12)(cid:12)(cid:12) ≃ (cid:12)(cid:12)(cid:12)(cid:12) Z ∂ Ω (cid:16) ∇ ( r − ∇ β Ä βγ u γ − ζ u γ ä + ∇ [ r − u (cid:17) × (cid:16) ∇ ( r − ∇ α Ä − ψ α − ∇ δ ∇ δ u α ä + ∇ [ r − u (cid:17) d H (cid:12)(cid:12)(cid:12)(cid:12) . (cid:12)(cid:12)(cid:12)(cid:12) Z ∂ Ω Ä ∇ [ r − u äÄ ∇ [ r ] u ä d H (cid:12)(cid:12)(cid:12)(cid:12) . ǫ k∇ r +1 u k L (Ω) + 1 ǫ k u k H r (Ω) , (3.23)where the last line follows analogously to the final inequality in Eq. (3.22).Finally, for I , , using Eqs. (3.12)–(3.13), we have | I , | = (cid:12)(cid:12)(cid:12)(cid:12) Z ∂ Ω (cid:16) ∇ ( r − ¶ − ψ α − ∇ β ∇ β u α © (cid:17)(cid:16) ∇ ( r − ∇ ∇ Ä αγ u γ − ζ u γ ä (cid:17) d H (cid:12)(cid:12)(cid:12)(cid:12) . k∇ [ r − u k L ( ∂ Ω) . k u k H r (Ω) . (3.24)Therefore, combining Eqs. (3.20)–(3.24) all together, I is estimated by | I | . ǫ k∇ r +1 u k L (Ω) + 1 ǫ k u k H r (Ω) . (3.25) Now we derive the estimates for the J term.In fact, J differs from I only by the lower-order terms so that the estimates follow imme-diately. More precisely, notice that J = Z ∂ Ω Ä ∇ i ∇ ( r − ∇ i r +1 u k äÄ ∆ ∇ ( r − ∇ i r +1 u k ä h∇ i , n i d H , (3.26)where we have relabelled ∇ ( r − = ∇ i · · · ∇ i r +1 as before. Then, invoking the Ricci identityagain, it follows that J . Z ∂ Ω Ä ∇ ( r − ∇ i ∇ i r +1 u k + ∇ [ r − u äÄ ∇ ( r − ∇ i r +1 ∆ u k + ∇ [ r − u ä d H = Z ∂ Ω Ä ∇ ( r − ∇ i ∇ i r +1 u k äÄ ∇ ( r − ∇ i r +1 ∆ u k ä d H + Z ∂ Ω Ä ∇ [ r − u äÄ ∇ [ r − u ä d H + Z ∂ Ω Ä ∇ i r +1 ∇ ( r − ∆ u k äÄ ∇ [ r − u ä d H + Z ∂ Ω Ä ∇ [ r − u äÄ ∇ [ r − u ä d H =: J + J + J + J . (3.27) y the trace, interpolation, and Young’s inequalities, again we have | J | + | J | . k∇ [ r − u k L ( ∂ Ω) . ǫ k∇ r +1 u k L (Ω) + 1 ǫ k u k H r (Ω) . (3.28)Also, J has the same decomposition as I into I , , . . . , I , so that, by Step 4, we conclude | J | . ǫ k∇ r +1 u k L (Ω) + 1 ǫ k u k H r (Ω) . (3.29)In the end, J is estimated via integration by parts again: Since ∂ Ω is a 2-D surface withoutboundary, the divergence theorem yields J = − Z ∂ Ω Ä ∇ ( r − ∆ u k äÄ ∇ i r +1 ∇ [ r − u ä d H ≃ Z ∂ Ω Ä ∇ [ r − u äÄ ∇ [ r − u ä d H . (3.30)Thus, this verifies the same estimate for J . Finally, putting together all the estimates for
I, J , and K in Steps 1–6, we conclude k∇ r +1 u k L (Ω) . ǫ k∇ r +1 u k L (Ω) + 1 ǫ k u k H r (Ω) + Z Ω |∇ r − ψ | d x. (3.31)Choose ǫ sufficiently small so that k∇ r +1 u k L (Ω) . k u k H r (Ω) + K, (3.32)where K := R Ω |∇ r − ψ | d x as before. The first term on the right-hand side, k u k H r (Ω) , is boundedby P rl =0 k curl l u k L (Ω) up to a multiplicative constant, thanks to the induction hypothesis.Now, it remains to show that K is bounded by the L –norm of the iterated curls: This isachieved by iterating the constructions in Step . Indeed, relabelling the indices yields K = Z Ω Ä ∆ ∂ i · · · ∂ i r − u k äÄ ∆ ∂ i · · · ∂ i r − u k ä d x. Then, as in Step 1, we integrate by parts twice to compute as K = Z ∂ Ω Ä ∆ ∂ i · · · ∂ i r − u k äÄ ∆ ∂ i · · · ∂ i r − u k ä h ∂ i , n i d H − Z ∂ Ω Ä ∆ ∂ i · · · ∂ i r − u k äÄ ∆∆ ∂ i · · · ∂ i r − u k ä h ∂ i , n i d H + Z Ω Ä ∆∆ ∂ i · · · ∂ i r − u k äÄ ∆∆ ∂ i · · · ∂ i r − u k ä d x = 12 Z ∂ Ω ∂ n |∇ r − ∆ u | d H − Z ∂ Ω Ä ∆ ∂ i · · · ∂ i r − u k äÄ ∆∆ ∂ i · · · ∂ i r − u k ä h ∂ i , n i d H + Z Ω | ∆∆ ∇ r − | u d x =: ˜ I + ˜ J + ˜ K. (3.33)It is crucial here that the flat gradient ∂ i l and flat Laplacian ∆ on R commute. We notice that ˜ I and ˜ J are obtained from I and J , respectively, by taking the trace over a pair of indices, sothat they satisfy the same estimates, which are given in Step above. Repeating this processfor finitely many times, we refine estimate (3.32) as k∇ r +1 u k L (Ω) . r X l =0 k curl l u k L (Ω) + R Ω | ∆ r +12 u | d x if r is odd , R Ω | ∆ r ∇ u | d x if r is even . (3.34)To conclude the proof, we notice that, for the divergence-free vector field u , ∆ u = − curl u .Thus, Eq. (3.34) gives the desired estimate for odd r . On the other hand, for even r , we apply q. (3.4) in Step 1 of the same proof to the divergence-free vector field ∆ r u to deduce Z Ω | ∆ r ∇ u | d x = Z Ω |∇ ∆ r u | d x = Z Ω | curl (∆ r u ) | d x + Z ∂ Ω II(∆ r u, ∆ r u ) d H , (3.35)where we need the commutativity of divergence, gradient, and curl. For the first term on theright-hand side, curl (∆ r u ) = ( − r curl r +1 u , while, for the second term, Z ∂ Ω II(∆ r u, ∆ r u ) d H . k ∆ r u k L ( ∂ Ω) ≃ k∇ r u k L ( ∂ Ω) . ǫ ′ k∇ r +1 k L (Ω) + 1 ǫ ′ k u k H r (Ω) , (3.36)again by the boundedness of the second fundamental form, as well as the trace and interpolationinequalities. The proof is then completed by choosing ǫ ′ sufficiently small. (cid:3) To conclude the section, we emphasize that Theorem 3.1 is independent of the Navier-Stokes equations (1.1). It is a general property of divergence-free vector fields satisfying thekinematic and Navier boundary conditions (1.6). For the Dirichlet boundary condition, Theorem3.1 also holds, which follows from the divergence-free condition.4.
Energy Estimates in H r for Strong Solutions In this section, we derive the higher-order energy estimates. We show that the solution isin the spatial Sobolev space H r for r ≥ , provided that the initial data lies in the same space.This allows us to prove the existence of strong solutions with spatial regularity H r .For this purpose, our starting point is the existence of weak solutions to the Navier-Stokesequations (1.1) under the kinematic and Navier boundary conditions. This can be established, e.g. via the Galerkin approximation scheme in [15]. We summarize it here for the subsequentdevelopments. In this section, we drop superscript ν in solution u ν of the Navier-Stokes equations(1.1), since we do not deal with the inviscid limits here.To begin with, consider the following vector space direct sum L (Ω; R ) = K (Ω) M G (Ω) (4.1)for the Hodge (or Helmholtz) decomposition, where K (Ω) is defined in (3.1). Next, for projection P ∞ onto the first factor, we introduce the Stokes operator : S := P ∞ ◦ ∆ , (4.2)where ∆ is the flat Laplacian on R . It is shown in §4 of [15] that S is densely defined on K (Ω) with a compact resolvent. Thus, it has a discrete spectrum λ ≥ λ ≥ λ ≥ . . . ↓ −∞ , and thecorresponding eigenfunctions { a n } form a complete orthonormal basis of K (Ω) . Now we lookat the graded chain of finite-D Hilbert spaces: K (Ω) ⊃ . . . ⊃ V N := N M j =1 R a j ⊃ V N − ⊃ . . . ⊃ V ⊃ V , (4.3)and denote by P N : K (Ω) → V N the canonical projection: ( P N u )( t, x ) := N X j =1 a j ( x ) Z Ω h a j ( y ) , u ( t, y ) i d y. (4.4)Thus, P ∞ is indeed the L -limit of P N , as N tends to ∞ .In §5 of [15], the weak formulation of Eq. (1.1) has been introduced. efinition 4.1. For
T > , we say that u ∈ L ([0 , T ]; H (Ω; R )) is a weak solution of the initialboundary problem (1.1) – (1.4) , provided that (i) u ( t, · ) ∈ K (Ω) for each t ∈ (0 , T ) ;(ii) For each φ ∈ C ∞ ([0 , T ] × Ω) with φ ( t, · ) ∈ K (Ω) , Z Ω h u ( T, · ) , φ ( T, · ) i d x = Z Ω h u ( x ) , φ (0 , x ) i d x + Z T Z Ω h u, ∂ t φ i d x d t − Z T Z Ω ¨ curl u, ( u × φ + ν curl φ ) ∂ d x d t − νζ Z T Z ∂ Ω h u, φ i d H d t + 2 ν Z T Z ∂ Ω II( u, φ ) d H d t ; (4.5)(iii) The energy inequality holds : k u ( T, · ) k L (Ω) + 2 ν Z T k∇ u ( t, · ) k L (Ω) d t + 2 ν Z T Z ∂ Ω Ä ζ | u | − II( u, u ) ä d H d t ≤ k u k L (Ω) . (4.6)Therefore, by solving the projected equations obtained via taking P N to Eq. (1.1) and de-riving the a priori estimates for the finite-D approximate solutions { u N } ⊂ L ([0 , T ]; H (Ω; R )) uniformly in N , we are able to deduce the existence of weak solutions via a compactness argu-ment. This method is known as the Galerkin approximation scheme, which relies crucially onthe spectral analysis of the Stokes operator S = P ∞ ◦ ∆ .More precisely, the following result is obtained: Lemma 4.2 (Theorem 5.1 in [15]) . For any u ∈ K (Ω) and T > , there exists a weak solution u ∈ L ([0 , T ]; K (Ω)) to the initial-boundary problem (1.1) – (1.4) . Such a solution u can beobtained as a weak subsequential limit of the family of finite-D approximate solutions { u N } . Now, taking the weak solution u of the Navier-Stokes equations (1.1) constructed by theGalerkin approximation scheme in Lemma 4.2 above, we derive the a priori estimate for thehigher-order energy of u in the Sobolev spaces H r with r ≥ . Indeed, the case, r = 2 , has beenproved in Theorem 5.3 of [15]. The higher-order energy estimate is proved by induction on r , forwhich purpose the reduction of order of differentiations in the boundary terms is essential. Thisis achieved by exploiting by the kinematic and Navier boundary conditions (1.6). In particular,we need to explore the role of the curl operator, the rotation matrix R , and the shape operator S (see §1).Our main theorem of this section is the following: Theorem 4.3.
Let u ∈ H r (Ω; R ) ∩ K (Ω) for some r ≥ . Then there exists some T ⋆ > suchthat the weak solution u ∈ L ([0 , T ⋆ ); K (Ω)) of the initial-boundary problem (1.1) – (1.4) satisfies sup ≤ t ≤ T ⋆ (cid:16) k u ( t, · ) k H r (Ω) + k ∂ t u ( t, · ) k H r − (Ω) (cid:17) ≤ C, (4.7) where constant C > depends only on ζ , ν , k II k C r − ( ∂ Ω) , and k u k H r (Ω) . As a consequence,there exists a unique strong solution u ∈ C ([0 , T ⋆ ); H r (Ω; R )) ∩ C ([0 , T ⋆ ); H r − (Ω; R )) .Proof. We divide the arguments in six steps. In Step 1, we set up the equations for the energyestimate. Then, in Steps 2–5, we control k u ( t, · ) k H r (Ω) and specify the lifespan, T ⋆ . Finally, inStep 6, we derive the energy estimate for ∂ t u . . We first deduce the evolution equation for the iterated curls of the velocity field u . Forthis purpose, we apply the divergence-free projection P ∞ : L (Ω; R ) → K (Ω) to the Navier-Stokes equations (1.1) to obtain ∂ t u − ν ∆ u + P ∞ ( u · ∇ u ) = 0 . (4.8)On the other hand, we have the following vectorial identity in 3-D: u · ∇ u = 12 ∇ ( | u | ) − u × ω, so that the projected Navier-Stokes equations (4.8) are equivalent to ∂ t u − ν ∆ u + P ∞ ( u × ω ) = 0 . (4.9)Here and in the sequel, we view P ∞ as extended to the bounded projection operator from H r (Ω; R ) to H r (Ω; R ) ∩ K (Ω) . This follows from the generalized Hodge decomposition theoryon the manifolds with boundaries subject to the kinematic boundary condition; see Theorem2.4.2 in Schwarz [43]. Then the Stokes’ operator: S := P ∞ ◦ ∆ gives rise to a densely defined, closable, self-adjoint bilinear form on H r (Ω; R ) ∩ K (Ω) : E r ( u, w ) := − Z Ω X ≤| α |≤ r ¨ ∇ α Su, ∇ α w ∂ d x. (4.10)In particular, the spectral analysis in Sections 4.1–4.2 in [15] also carries through in our settingto H r (Ω; R ) .For simplicity of presentation, we use the following abbreviation: Ψ := P ∞ ( u × ω ) . (4.11)Then, taking the iterated curls to Eq. (4.9), we obtain the evolution equation: ∂ t q r − ν ∆ q r + curl r Ψ = 0 , (4.12)where and in the sequel, we denote q r := curl r u. (4.13)To derive the energy estimate, we multiply q r to Eq. (4.12) and integrate over Ω to obtain ddt Z Ω | q r | d x − ν Z Ω h q r , ∆ q r i d x + Z Ω h q r , curl r Ψ i d x. (4.14)We integrate the last two terms by parts. For the second term, we have Z Ω h q r , ∆ q r i d x = Z ∂ Ω h ( q r · ∇ ) q r , n i d H − Z Ω |∇ q r | d x. For the final term, notice that, for any -D vector fields V and W , Z Ω h V, curl W i d x = Z Ω V k ǫ ijk ∂ i W j d x = Z ∂ Ω ǫ ijk V k W j h ∂ i , n i d H − Z Ω ǫ ijk W j ( ∂ i V k ) d x = Z ∂ Ω h W × V, n i d H + Z Ω h curl V, W i d x. (4.15) s a result, Z Ω h q r , curl ◦ curl r − Ψ i d x = Z ∂ Ω h curl r − Ψ × q r , n i d H + Z Ω h curl q r , curl r − Ψ i d x, so that Eq. (4.14) can be written as ddt Z Ω | q r | d x + ν Z Ω |∇ q r | d x = ν Z ∂ Ω h ( q r · ∇ ) q r , n i d H − Z ∂ Ω h curl r − Ψ × q r , n i d H − Z Ω h curl q r , curl r − Ψ i d x = I + J + K. (4.16)Our task is to estimate each of terms I, J , and K . Since the case, r = 2 , has beenestablished in Theorem 5.3 of [15], in the sequel, we assume the result for r − and prove it for r by induction, with r ≥ . For I in Eq. (4.16), observe that I = ν Z ∂ Ω ∂ n | q r | d H , (4.17)which has been treated in the proof of Theorem 3.1. Indeed, it coincides with I in Eq. (3.7) upto a constant ν . Utilizing the estimates in Steps 2–4 of the proof therein, we have | I | . ǫ k∇ q r k L (Ω) + 1 ǫ k u k H r (Ω) . (4.18) To prove for term K in Eq. (4.16), we first notice that, by the Cauchy-Schwarz inequalityand Young’s inequality, | K | ≤ k curl q r k L (Ω) k curl r − Ψ k L (Ω) . ǫ k∇ q r k L (Ω) + 1 ǫ k curl r − ( P ∞ ( u × ω )) k L (Ω) . (4.19)Since H s (Ω) for s > is a Banach algebra on R and P ∞ is a bounded linear operator, we have k curl r − ( P ∞ ( u × ω )) k L (Ω) . k u k H r (Ω) . (4.20)This gives us the estimate for K . Now it remains to control the boundary term J in Eq. (4.16): J := Z ∂ Ω h curl r − ( P ∞ ( u × ω )) × q r , n i d H . (4.21)It is crucial to reduce the order of differentiation by using the boundary conditions (1.6). Forthis purpose, we establish the following identity on ∂ Ω : π ¶ curl k ( P ∞ ( u × ω )) © = − ζ R ◦ π ¶ curl k − ( P ∞ ( u × ω )) © + 2 R ◦ π ¶ curl k − S ( π ◦ P ∞ ( u × ω )) © . (4.22)We recall that R is the orthogonal matrix rotating in the ( x, y ) –plane anti-clockwise by degrees, S is the shape operator corresponding to the second fundamental form II , and operator π denotes the projection onto the tangential components of a vector field, π ( V ) := V − h V, n i (4.23) iewed either as a 2-D vector or a 3-D vector with zero x –component. Here it suffices to considerthe tangential components, since h n × q r , n i ≡ .The above identity is proved by induction. The base step k = 1 is shown in the compu-tations preceding Eq. (5.21) in [15]. Now we assume the result for k . Then, by the inductionhypothesis, π ¶ curl k +1 ( P ∞ ( u × ω )) © = π ¶ curl ◦ curl k ( P ∞ ( u × ω )) © = π ◦ curl ¶ − ζ R ◦ π ◦ curl k − ( P ∞ ( u × ω )) + 2 R ◦ π ◦ curl k − S ( π ◦ P ∞ ( u × ω )) © =: π ◦ curl ¶ − ζ R ◦ π ◦ curl k − Ψ + 2
R ◦ π ◦ curl k − ( S ◦ π (Ψ)) © , (4.24)where we recall the short-hand notation Ψ in Eq. (4.11). Here and throughout, for a 2-D vectorfield W = ( W , W ) ⊤ ( e.g. W = S ◦ π (Ψ) ), we define its curl as curl W := curl ( W , W , ⊤ .It suffices to check that π ◦ curl ◦ R ◦ π ( V ) = R ◦ π ◦ curl V for any V ∈ T ( ∂ Ω) . (4.25)From here, Eq. (4.24) implies π ¶ curl k +1 ( P ∞ ( u × ω )) © = − ζ R ◦ π ◦ curl k Ψ + 2
R ◦ π ◦ curl k ( S ◦ π (Ψ)) . (4.26)Indeed, we observe π ◦ curl ◦ R ◦ π V V V = π ◦ curl − V V = π −∇ V −∇ V = " −∇ V −∇ V , and R ◦ π ◦ curl V V V = R ◦ π ∇ V − ∇ V ∇ V − ∇ V ∇ V − ∇ V = R " −∇ V ∇ V = " −∇ V −∇ V , since V = 0 . Therefore, Eq. (4.25) is proved, and the identity in Eq. (4.22) follows by induction.Now, in view of the above identity, J can be expressed as J = Z ∂ Ω ¨ n − ζ R ◦ π ( curl r − ( P ∞ ( u × ω )))+ 2 R ◦ π ◦ curl r − Ä S ◦ π ◦ P ∞ ( u × ω ) ä o × q r , n ∂ d H . (4.27)The crucial observation is that only the derivatives up to the ( r − -th order of u are involved.This is because S has a bounded norm in C r − owing to the assumption of bounded extrinsicgeometry, and P ∞ , π , and R are all smooth operators with the operator norm bounded by a niversal constant. Then we arrive at the following estimates: | J | . k curl r − ( u × ω ) k L ( ∂ Ω) k q r k L ( ∂ Ω) . k u k H r − ( ∂ Ω) k u k H r ( ∂ Ω) . (cid:16) ǫ k u k H r (Ω) + k u k H r − (Ω) (cid:17)(cid:16) ǫ k u k H r +1 (Ω) + k u k H r (Ω) (cid:17) ≃ ǫ k u k H r (Ω) k u k H r +1 (Ω) + ǫ k u k H r (Ω) + ǫ k u k H r − (Ω) k u k H r +1 (Ω) + k u k H r − k u k H r (Ω) . ( ǫ + ǫ ) k u k H r +1 (Ω) + k u k H r (Ω) . (4.28)In the above, the first line follows from the Cauchy-Schwarz inequality, the second line followsfrom the argument as for Eq. (4.20), the third line holds by the Sobolev trace inequality, andthe final line follows by the interpolation and Young’s inequalities. Now, combining the estimates in Steps 2–4 for I , J , and K (especially Eqs. (4.18)–(4.20)and (4.28)), Eq. (4.16) becomes ddt Z Ω | q r | d x + ν Z Ω |∇ q r | d x . ǫ k∇ r +1 u k L (Ω) + 1 ǫ k u k H r (Ω) + ǫ k∇ q r k L (Ω) + ( ǫ + ǫ ) k u k H r +1 (Ω) + (1 + 1 ǫ ) k u k H r (Ω) , (4.29)where, in light of Theorem 3.1, k∇ r +1 u k L (Ω) ≃ k∇ q r k L (Ω) ≃ k curl r +1 u k L (Ω) , and similarly k q r k L (Ω) . k u k H r (Ω) . Then, choosing ǫ suitably small in comparison with ν and consideringthe energy at the r -th order: E r := k q r k L (Ω) , (4.30)we obtain the following differential inequality: E ′ r ( t ) ≤ E ′ r ( t ) + νE r +1 ≤ M E r ( t ) + M E r ( t ) , (4.31)where M depends on ν , ζ , and k II k C r − (Ω) .To proceed, consider the auxiliary Cauchy problem for ODE: A ′ ( t ) = M Ä A ( t ) + A ( t ) ä ,A (0) = E r (0) + η (4.32)for arbitrary η > . It is solved explicitly by A ( t ) = Ä η + E r (0) ä e Mt − Ä η + E r (0) äÄ e Mt − ä , so that, for any t > before the blowup time: T ⋆ = 1 M log (cid:16) η + E r (0) (cid:17) > , (4.33)we see that A ( t ) < ∞ . Comparing the differential inequality (4.31) with the ODE in (4.32), wefind that E r ( t ) ≤ A ( t ) for all ≤ t < T ⋆ . In particular, since η > is arbitrary, the upper boundfor A (hence for E r ) is controlled by E r (0) := k u k H r (Ω) and M . This implies sup ≤ t
Let u ∈ C ∞ (Ω; R ) be a divergence-free vector field, and let domain Ω ⊂ R be of smooth second fundamental form. Then there exists T ⋆ > such that the initial-boundary value problem (1.2) – (1.4) for the Navier-Stokes equations (1.1) has a smooth solution u ∈ C ([0 , T ⋆ ); C ∞ (Ω; R )) satisfying the kinematic and Navier boundary conditions (1.6) . Inviscid Limit
In this section, we establish the inviscid limit from the Navier-Stokes equations under thekinematic and Navier boundary conditions (1.6) to the Euler equations under the no-penetrationcondition (1.11). The existence and uniqueness of u , the strong solution of the Euler equations(1.10) satisfying the no-penetration boundary condition (1.11), have been known ( cf . Ebin-Marsden [23]). We obtain the convergence in the Sobolev spaces H r , r > , via strong compact-ness arguments.To this end, a priori estimates of the evolution equations for u ν − u , i.e. the differencebetween the Navier-Stokes solution and the Euler solution, are required. In particular, techni-calities are involved in the estimates for the higher-order Sobolev norms of the nonlinear terms, or instance, the iterated curls of ( u ν − u ) · ∇ ( u ν − u ) . To deal with the nonlinearities, we need tomake full use of the incompressibility condition of u ν and u , as well as the kinematic and Navierboundary conditions (1.6).The main theorem of this section is stated as follows: Theorem 5.1 (Inviscid Limit) . Let u ∈ C ([0 , T ⋆ ); H r +1 (Ω; R ) ∩ K (Ω)) be the unique strongsolution of the incompressible Euler equations (1.10) subject to the no-penetration boundary con-dition (1.11) . Then there exists some ν ⋆ = ν ⋆ ( T ⋆ , R T ⋆ k u ( t, · ) k H r +1 (Ω) ) such that, whenever < ν ≤ ν ⋆ , the strong solution of the Navier-Stokes equations (1.1) with the kinematic andNavier boundary conditions (1.6) exists in L ∞ ([0 , T ⋆ ); H r (Ω; R )) for r > . Moreover, if r > ,then there exists a constant C depending only on T ⋆ , k u k L ([0 ,T ⋆ ); H r +1 (Ω)) , and k II k C r ( ∂ Ω) suchthat sup ≤ t
Remark 5.2.
A key point of Theorem is that the strong solutions u ν to the Navier-Stokesequations do not blow up before T ⋆ in H r , where T ⋆ is the lifespan of the corresponding Eulerequations in H r +1 for r > . The arguments ( Step of the proof ) are adapted from § inConstantin [17] , in which the case of periodic boundary conditions are treated. This does notdirectly follow from our proof of Theorem . In fact, constant M is proportional to ν − in Eq. (4.31) , so that the lifespan for the Navier-Stokes equations ( in H r +1 in space ) is proportional toviscosity ν , which goes to zero in the vanishing viscosity limit. Remark 5.3.
In Theorem , the rate of convergence in the inviscid limit is O ( ν / ) . It can beimproved to O ( √ ν ) , provided that {∇ u ν − ∇ u } is uniformly bounded in space-time. Moreover,in this case, the H r +1 –norm of u ν is also close to that of u in the average in time. roposition 5.4. Let u ν , u, ν , T ⋆ , and r be as in Theorem . In addition, suppose that {∇ u ν − ∇ u } is uniformly bounded in L ∞ ([0 , T ⋆ ) × Ω; R ) . Then there exists a constant C ,depending only on T ⋆ , k u k L ([0 ,T ⋆ ); H r +1 (Ω)) , and k II k C r ( ∂ Ω) , such that sup ≤ t The proof follows essentially from the arguments in Theorem 5.1 above, i.e. by consideringthe evolution equation for v ν := u ν − u . We only emphasize the differences.Indeed, starting from Eq. (5.7), we estimate the terms, V , V , V , and V , as in Steps 1–2and 4– 5 in the proof of Theorem 5.1. The only difference occurs in Step 3. Recall Eq. (5.14)therein: |V | . k v ν k H r (Ω) k∇ u k L ∞ (Ω) + k v ν k H r (Ω) k u k H r (Ω) k∇ v ν k L ∞ + k v ν k H r (Ω) k∇ v ν k L ∞ . Under the additional assumption, k∇ v ν k L ∞ ≤ C so that |V | . Ä k u k H r (Ω) ä k v ν k H r (Ω) . (5.38)As a consequence, by choosing ǫ suitable small, estimate (5.28) in Step 6 can be improved to ddt Z Ω | V νr | d x + ν k curl r +1 v ν k L (Ω) ≤ C (cid:16) k u k H r +1 (Ω) (cid:17) k v ν k H r (Ω) + C ν k u k H r +1 (Ω) , (5.39)which does not contain the cubic terms in k v ν k H r .From here, the usual Gronwall inequality yields k v ν ( t, · ) k H r (Ω) ≤ k v ν (0 , · ) k H r (Ω) exp n C t + Z t k u ( s, · ) k H r +1 (Ω) d s o + C ν Z t n exp (cid:16) C ( t − s ) + Z ts k u ( τ, · ) k H r +1 (Ω) d τ (cid:17) k u ( s, · ) k H r +1 (Ω) o d s = C ν Z t n exp (cid:16) C ( t − s ) + Z ts k u ( τ, · ) k H r +1 (Ω) d τ (cid:17) k u ( s, · ) k H r +1 (Ω) o d s (5.40)for any t ∈ [0 , T ⋆ ) . This is because v ν (0 , · ) = 0 , since the Navier-Stokes and the Euler solutionshave the same initial data. Thus, for some constant C = C ( ν , k II k C r (Ω) , T ⋆ , k u k L (0 ,T ⋆ ; H r +1 (Ω)) ) , sup t ∈ [0 ,T ⋆ ) k v ν ( t, · ) k H r (Ω) ≤ C √ ν −→ as ν → + . (5.41)Finally, integrate Eq. (5.39) with Eq. (5.40) substituted into the right-hand side. In thisway, we find a constant C with the same dependence as C such that sup t ∈ [0 ,T ⋆ ) Z t k curl r +1 v ν ( s, · ) k L (Ω) d s ≤ C . (5.42)Therefore, in view of Theorem 3.1 and Eq. (5.41), we have sup t ∈ [0 ,T ⋆ ) Z t k v ν ( s, · ) k H r +1 (Ω) d s ≤ C = C ( ν , k II k C r (Ω) , T ⋆ , k u k L (0 ,T ⋆ ; H r +1 (Ω)) ) , (5.43)which completes the proof. (cid:3) emark 5.5. Combining the results in § –§ together, we have established the existence ofstrong solutions in H r +1 for r > of the Navier-Stokes equations, while the inviscid limit hasbeen proved in H r . Therefore, it remains an open question whether the inviscid limit holds orfails ( e.g., due to the development of boundary layers ) in H r +1 , i.e. the highest order the spatialregularity of the strong solutions. Remarks on the Non-Navier Slip-type Boundary Condition In the introduction (§1), a modified version of the Navier boundary condition, which isoriginally introduced by Bardos [5] and Solonnikov-˘S˘cadilov [45], has been briefly discussed.Physically, it describes the phenomenon that the tangential part of the normal vector field of theCauchy stress tensor is uniformly vanishing, and it agrees with the Navier boundary conditionif and only if boundary ∂ Ω is flat. Together with the kinematic boundary condition, we have u ν · n = 0 , ω ν × n = 0 on ∂ Ω . (6.1)The second line is referred to as the non-Navier slip-type boundary condition.In Beirão da Veiga-Crispo [10, 11], the inviscid limit problem is analyzed for the Navier-Stokes equations subject to the boundary conditions (6.1) and the Euler equations subject to theno-penetration boundary condition (1.11). In this section, we write K for the Gauss curvature of surface ∂ Ω . We first introduce the following notions (see also Definitions 2.1 and 2.3 in [11]): Definition 6.1. For the non-Navier slip-type boundary conditions (6.1) , we say that (i) u ∈ C ∞ (Ω; R ) is an admissible initial data if ∇ · u = 0 in the closure Ω , as well as u · n = 0 and ω × n = 0 on ∂ Ω ;(ii) The inviscid limit u ν → u holds “strongly” in L p ([0 , T ]; W s,q (Ω; R )) for some T > , p, q ≥ , and s > , if the convergence holds with respect to the strong topology on L p ([0 , T ]; W s,q (Ω; R )) . In particular, we notice that, if u ν → u strongly, ω ν × n = 0 on [0 , T ] × ∂ Ω implies ω × n = 0 on [0 , T ] × ∂ Ω . This is termed as the “ persistence property ” in [10, 11]. In the presence of such aproperty, the following non-convergence result is established by Beirão da Veiga-Crispo, first byconsidering a special example on S in [10] and then proved in full generality via computationsof the principal curvatures on ∂ Ω in local coordinates in [11]: Theorem 6.2 (Beirão da Veiga-Crispo, [10, 11]) . Let Ω ⊂ R be a bounded regular domain. Letthe admissible initial data u be given for the initial-boundary value problem (1.1) and (6.1) andproblem (1.10) – (1.11) such that the following condition holds : ω ( x ) = 0 for some x ∈ ∂ Ω such that K ( x ) = 0 . (6.2) Then, for arbitrary δ > , p, q ≥ and s > , the “strong” inviscid limit fails : u ν u in L p ([0 , δ ]; W s,q (Ω; R )) . (6.3)This theorem says that, if the initial vorticity vanishes somewhere on the curved part ofthe boundary, i.e. the Gauss curvature is non-vanishing at this point, then the “strong” inviscidlimit fails in an arbitrarily short time interval, so that the Prandtl boundary layers must bedeveloped. Heuristically, this is due to the incompatibility of the vorticity directions of theslip-type boundary conditions in the limiting process ν → + . ow we give an alternative proof of Theorem 6.2, which avoids the computations in localcoordinates on ∂ Ω as in [10, 11]. This offers a new, global perspective for the above theorem,and the proof makes essential use of the properties of Lie derivatives in R . Proof. First of all, following the original proof in [10, 11], we consider the inviscid vorticityequation, which is obtained by taking the curl of the Euler equation (1.10): ∂ t ω + ( u · ∇ ) ω − ( ω · ∇ ) u = 0 . (6.4)Then taking the cross product with the outer unit normal n : ∂ Ω → S leads to ∂ t ( ω × n ) + ¶ ( u · ∇ ) ω − ( ω · ∇ ) u © × n = 0 . (6.5)It is observed by Xiao-Xin ( cf. Corollary 8.3 in [49]) that a necessary condition for the “strong”inviscid limit in the time interval (0 , δ ) is: ω × n ≡ on (0 , δ ) × ∂ Ω , (6.6)that is, the persistence property is verified ( cf. Definition 6.1). Thus, if the “strong” inviscidlimit were valid, then Eq. (6.5) implies that { ( u · ∇ ) ω − ( ω · ∇ ) u } × n = 0 for all time. Inparticular, sending t → + , the following condition must be fulfilled: ¶ ( u · ∇ ) ω − ( ω · ∇ ) u © × n = 0 on ∂ Ω . (6.7)Our crucial observation is that the expression in the bracket in (6.7) coincides with the Liebracket of the vector fields u , ω ∈ T R : ( u · ∇ ) ω − ( ω · ∇ ) u = [ u , ω ] . (6.8)To prove Eq. (6.8), let V = P i =1 V i ∂ i and W = P j =1 V j ∂ j be two smooth vector fieldsin T R , where { ∂ , ∂ , ∂ } denotes the canonical Euclidean frame. We follow the convention indifferential geometry to identify vector fields with first-order differential operators. Thus, theLie bracket of V and W can be computed as [ V, W ] = V W − W V = (cid:16) V i ∂ i ( W j ∂ j ) − W j ∂ j ( V i ∂ i ) (cid:17) =: ( V · ∇ ) W − ( W · ∇ ) V, which verifies the above identity. In the above, the Einstein summation convention is adopted.To proceed, we recall that the Lie bracket of two vector fields on a differentiable manifoldequals the Lie derivative (denoted by L ) of one vector field along the other: [ u , ω ] = L u ω . (6.9)This result is standard in the differentiable manifold theory, which can be found in the classicaltexts ( cf. do Carmo [21]). Furthermore, the Lie derivative at a point x ∈ ∂ Ω is given by L u ω ( x ) := lim s → + ( θ s ) ∗ { ω ( θ s ( x )) } − ω ( x ) s , (6.10)where { θ s ( x ) : s ≥ , x ∈ R } is the one-parameter subgroup defined by the following ODE: dds θ s ( x ) = u ( θ s ( x )) for all s ≥ , x ∈ R ,θ ( x ) = x for all x ∈ R . (6.11)In other words, trajectory { θ s ( x ) } s ≥ is the integral curve of the vector field u emanating frompoint x . Also, ( θ s ) ∗ denotes the pullback operation under map θ s . y the admissibility of the initial data, ω ∈ T ( ∂ Ω) ⊥ is orthogonal to the boundary since ω × n = 0 , and u ∈ T ( ∂ Ω) is tangential to the boundary because of the kinematic boundarycondition: u · n = 0 . The expression on the right-hand side of Eq. (6.9) is well-defined since ω is a vector field defined along the manifold ∂ Ω . In geometric terminologies, it means that ω ∈ ι ∗ T R , where ι : ∂ Ω ֒ → R is the embedding of Riemannian submanifold, and ι ∗ T R is thepullback vector bundle. This enables us to take the Lie derivative on ω along any vector field( e.g. u ) tangent to ∂ Ω .Now, by the assumptions, there is a point x ∈ ∂ Ω such that ω ( x ) = 0 and K ( x ) = 0 .Owing to the non-vanishing curvature, there exists some small neighbourhood U ⊂ ∂ Ω of x suchthat every smooth curve γ : ( − δ, δ ) → ∂ Ω satisfying γ (0) = x is not a straight line segment in R . In addition, as vorticity ω is non-vanishing at x , we have h u , ˙ γ i 6 = 0 on U. (6.12)On the other hand, using the definition of the Lie derivative in terms of the integral curve, i.e. (6.10), we have L ˙ γ ω = 0 on T ( ∂ Ω) . (6.13)This is because the parallel-transport of ω along γ cannot be obtained by a Euclidean translation,so that ( θ s ) ∗ { ω ( θ s ( x )) } − ω ( x ) = 0 in Eq. (6.10). Therefore, we conclude that L h u , ˙ γ i ˙ γ ω = 0 on T ( ∂ Ω) , from which it follows L u ω ( x ) × n ( x ) = 0 in R . (6.14)This contradicts Eq. (6.7). (cid:3) Acknowledgement . The research of Gui-Qiang G. 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