Regularity Criteria for Navier-Stokes Equations with Slip Boundary Conditions on Non-flat Boundaries via Two Velocity Components
aa r X i v : . [ m a t h . A P ] J a n Regularity Criteria for Navier-Stokes Equations with SlipBoundary Conditions on Non-flat Boundaries via Two VelocityComponents
Hugo Beir˜ao da Veiga , ∗ Jiaqi Yang , † Department of Mathematics, Pisa University, Pisa, Italy Key Laboratory for Mechanics in Fluid Solid Coupling Systems, Institute of Mechanics, ChineseAcademy of Sciences, Beijing 100190, China
Abstract
H.-O. Bae and H.J. Choe, in a 1997 paper, established a regularity criteria for theincompressible Navier-Stokes equations in the whole space R based on two velocity com-ponents. Recently, one of the present authors extended this result to the half-space case R . Further, this author in collaboration with J. Bemelmans and J. Brand extended theresult to cylindrical domains under physical slip boundary conditions. In this note weobtain a similar result in the case of smooth arbitrary boundaries, but under a distinct,apparently very similar, slip boundary condition. They coincide just on flat portions of theboundary. Otherwise, a reciprocal reduction between the two results looks not obvious,as shown in the last section below.
Mathematics Subject Classification:
Keywords:
Navier-Stokes equations; Slip boundary conditions; No flat boundaries; Twocomponents regularity criterium.
The starting point of the present paper is the well known Prodi-Serrin (P-S) sufficient condi-tion for regularity of the solutions to the incompressible Navier-Stokes equations ∂ t u + u · ∇ u − ∆ u + ∇ p = 0 , in Ω × (0 , T ] , ∇ · u = 0 , in Ω × (0 , T ] . (1.1) ∗ Partially supported by FCT (Portugal) under grant UID/MAT/04561/3013. † Hugo Beir˜ao da Veiga ( [email protected] ) and Jiaqi Yang ( [email protected] ) u = ( u , u , u ) denotes the unknown velocity of the fluid and p the pressure. Toimmediately set limits to the circle of our interests, assume for now on that Ω ⊂ R is abounded, smooth domain, even if many results quoted below hold for larger space dimensions.For the time being, assume that suitable boundary conditions are imposed to the velocity u .The global existence of the so called weak solutions to system (1.1) goes back to J. Leray[28] and E. Hopf [21] classical references. See also A.A. Kiselev and O.A. Ladyzhenskaya [22],and J.L. Lions [29]. Below, solutions of (1.1) are intended in this sense.A main classical open mathematical problem is to prove, or disprove, that weak solutionsare necessarily strong under reasonable but general assumptions, where strong means that u ∈ L ∞ (0 , T ; H (Ω)) ∩ L (0 , T ; H (Ω)) . (1.2)In this context, a remarkable and classical sufficient condition for uniqueness and regularityis the so-called Prodi-Serrin condition, P-S in the sequel, namely u ∈ L q (0 , T ; L p (Ω)) , q + 3 p = 1 , p > . (1.3)Concerning condition (1.3), we transcribe from [8], Section 1, the following considerations:Assumption (1.3) was firstly considered by G. Prodi in his paper [34] of 1959. He proveduniqueness under this last assumption. See also C. Foias, [15]. Furthermore, J. Serrin, see[36] and [37], particularly proved interior spatial regularity under the stronger (non-strict)assumption u ∈ L q (0 , T ; L p (Ω)) , q + 3 p < , p > . (1.4)Concerning the above problems, see also O.A. Ladyzhenskaya’s contributions [26] and [27].The above setup led to the nomenclature Prodi-Serrin condition.Complete proofs of the strict regularity result (i.e. under assumption (1.3)) were given byH. Sohr in [38], W. von Wahl in [40], and Y. Giga in [19]. A simplified version of the proof wasgiven in reference [18], to which we refer also for bibliography. For a quite complete overviewon the main points, and references, on the initial-boundary value problem for Navier-Stokesequations we strongly recommend Galdi’s contribution [17]. Further, we refer to [35] and[37], as sources for information on the historical context of the P-S condition by the initiatorsthemselves.Finally, we recall that L. Escauriaza, G. Seregin, and V. ˇSver´ak, see [14], extended theregularity result to the case ( q, p ) = ( ∞ , . A significant improvement of the P-S condition was obtained by H.-O. Bae and H.J. Choe[1], see also [2]. They proved, in the whole space case, that it is sufficient for regularity ofsolutions that two components of the velocity satisfy the above condition (1.3). For conve-nience we call here this situation as being the restricted P-S condition. In 2017, one of theauthors, see [7], extended this result to the half-space R under slip boundary conditions.2n this case, the truncated 2-dimensional vector field ¯ u cannot be chosen arbitrarily. Theomitted component has to be the normal to the boundary.Very recently, in reference [8], the result was extended to a cylindrical type three-dimensionaldomain, consisting on the complement set between two co-axial circular cylinders, with radius ρ and ρ , < ρ < ρ , periodic in the axial direction, under the physical slip boundarycondition u · n = 0 , [ D ( u ) n ] · τ = 0 , on ∂ Ω , (1.5)where D ( u ) = ∇ u +( ∇ u ) T is the shear stress. The above exclusion of an interior cylinder wasdone to avoid the radial coordinate singularities on the symmetry axis, which considerationis out of interest in our context. Below we obtain a similar result, extended to domains withgeneral non-flat boundaries, but under the slip boundary condition (2.1). The two boundaryconditions coincide just on flat portions of the boundary. Otherwise, a reciprocal reductionbetween the two results looks not obvious. This claim is shown in the last section.Again by following [8] we recall that after the contribution by H.-O. Bae and H.J. Choe,related papers appeared that particularly concerned assumptions on two components of ve-locity or vorticity, see [3], [5], [7], [10], and [13]. There are also many papers dedicated tosufficient conditions for regularity which depend merely on one component, see, for instance,[12], [20], [25], [33], [42], and [43].Before going on we want to motivate the particular choice of the domain made below. Ittakes into account that the real significance of the result has essentially a local character. Firstof all, a global regular (i.e., without singularities) system of coordinates, two of them paralleland the third orthogonal to the boundary, does not exist in general, even in an arbitrarilythin neighbourhood of the full boundary, as in the case of a sphere and even in the case of aspherical corona. In fact, singularities typically appear, like on the above two cases, and evenin full cylinders (due to the symmetry axis). The cylindrical case considered in reference [8]is an exception (see below) due to the removal of a neighbourhood of the symmetry axis.Luckily, the above type of coordinates’ system exists in sufficient small neighbourhoods ofany regular boundary point. Hence, to illustrate the full significance of our thesis in a simple,but still convincing way, it looks sufficient to prove it near any “small” piece of smoothboundary with an arbitrary geometrical shape. This is our aim below. The restrictions onthe domain Ω below are made in accordance with these lines, a choice which covers the verybasic situation, in the simplest way. In the sequel we assume the slip boundary condition u · n = 0 , ω × n = 0 on ∂ Ω , (2.1)3here ω = ∇ × u is the vorticity, n is the outward normal of ∂ Ω, and Ω ∈ R is a smoothdomain satisfying the following condition: Assumption 2.1.
There exists a curvilinear orthogonal system of coordinates q ( x ) = ( q ( x ) , q ( x ) , q ( x )) such that Ω can be transited into ˆΩ , { ( q , q , q ) : 0 ≤ q < , ≤ q < , < ρ ≤ q ≤ ρ } , where the axis q direct to the outward normal on the boundary ∂ ˆΩ := { ( q , q , q ) : q = ρ } (the inward normal on the boundary ∂ ˆΩ := { ( q , q , q ) : q = ρ } , respectively), and q , q are periodic.Remark . The above ”small” piece of a generical smooth boundary is here represented by q = ρ , and q = ρ . Remark . It is worth noting that the slip boundary condition (2.1) is equivalent to u · n = 0 , [ D ( u ) n ] · τ = − κ τ u · τ , (2.2)where τ stands for any arbitrary unit tangential vector on ∂ Ω , and κ τ is the principal curva-ture in the τ direction, positive if the center of curvature lies inside Ω . The above claim follows immediately by appealing to equation (5.2) in [9], namely[ D ( u ) n ] · τ = 12 ( ω × n ) · τ − κ τ u · τ . (2.3)For a mathematical treatment of some aspects related to slip boundary conditions imposedon smooth, but generic, boundaries see also [6], and the pioneering paper [39].Next we recall some facts on curvilinear coordinates. The Lam´e coefficients (scale factors)of the transition to the system of coordinates q are denoted by the letters H i H i ( q ) = X j =1 (cid:18) ∂x j ∂q i (cid:19) , i = 1 , , e i = H i ∂x∂q i , i = 1 , ,
3. Note that | ˆ e i | = 1 and ˆ e i · ∇ = H i ∂∂q i . One can write u ( x ) = ˆ u ( q ) = ˆ u ( q )ˆ e + ˆ u ( q )ˆ e + ˆ u ( q )ˆ e and ω ( x ) = ˆ ω ( q ) = ˆ ω ( q )ˆ e + ˆ ω ( q )ˆ e + ˆ ω ( q )ˆ e . It is well known (see for example [23] and [4]) that ∇ · u = 1 H H H (cid:18) ∂ (ˆ u H H ) ∂q + ∂ (ˆ u H H ) ∂q + ∂ (ˆ u H H ) ∂q (cid:19) (2.4)4nd ∇ × u = 1 H H (cid:18) ∂ (ˆ u H ) ∂q − ∂ (ˆ u H ) ∂q (cid:19) ˆ e + 1 H H (cid:18) ∂ (ˆ u H ) ∂q − ∂ (ˆ u H ) ∂q (cid:19) ˆ e + 1 H H (cid:18) ∂ (ˆ u H ) ∂q − ∂ (ˆ u H ) ∂q (cid:19) ˆ e . (2.5)We state our main result as follows. Theorem 2.2.
Let Ω satisfy Assumption 2.1, and suppose that there exist two positive con-stants c and C such that c ≤ H i ≤ C and (cid:12)(cid:12)(cid:12)(cid:12) ∂ x i ∂q i ∂q j (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) ∂ x i ∂q i ∂q j ∂q k (cid:12)(cid:12)(cid:12)(cid:12) ≤ C , (2.6) for any i , j , k = 1 , , . Let u be a weak solution of the system (1.1) under the boundarycondition (2.1) , and set ¯ u = ˆ u ˆ e + ˆ u ˆ e . If ¯ u satisfies ¯ u ∈ L q (0 , T ; L p (Ω)) , q + 3 p ≤ , p > , (2.7) then the solution u is strong, namely, u ∈ L ∞ (0 , T ; H (Ω)) ∩ L (0 , T ; H (Ω)) . Note that assumption (2.6) implies | ∂ i H j | , | ∂ ij H k | ≤ C .
It is worth noting that our proof applies to a more general set of geometrical situations.let’s just give some hint in this direction.
Remark . The above statement does not contain the result proved in reference [8], dueto the distinct boundary conditions, see Section 4. On the other hand, we may replace thetwo circular, vertical, cylinders by more general vertical cylinders where the external circle q = ρ is replaced by a smooth Jordan curve γ , and the internal circle q = ρ by aparallel Jordan curve γ , at a sufficient small distance δ > γ . The coordinate θ isnow an arc length coordinate on γ . All points in the same orthogonal segment to γ and γ enjoy the same θ coordinate. The coordinate r ∈ (0 , δ ) is given by the distance to γ .The “vertical” coordinate z preserves his periodic character. Clearly, the role played by theabove Jordan curve may be immediately extended to much more general situations.Another significant application is obtained by replacing the above two cylindrical bound-aries by two torus of revolution, generated by revolving two concentric circles γ and γ about an axis coplanar with the circles, which does not touch the circles (roughly, we obtainthe complement set between two closed tubes). Now z ∈ [0 , π ) is an angular periodiccoordinate, the toroidal coordinate. The result still applies by replacing the two circles bytwo parallel Jordan curves. 5et’s propose the following benchmark problem: Problem 2.3.
Consider two concentric spheres Ω R and Ω ρ , of radius respectively ρ and R , < ρ < R . Let u be a weak solution in Ω R × (0 , T ] of (1.1) under one of theabove slip boundary conditions. Further, assume that the restricted P-S condition holds in (Ω R − Ω ρ ) × (0 , T ] with respect to the tangential components of the velocity, and holds in Ω ρ × (0 , T ] with respect to two arbitrary components of the velocity. Problem: To prove that u is a strong solution in Ω R × (0 , T ] . Proof.
We start by reducing the system (1.1) under the boundary condition (2.1) into theclassical vorticity form ∂ t ω + u · ∇ ω − ω · ∇ u − ∆ ω = 0 , in Ω × (0 , T ] , ∇ · u = 0 , in Ω × (0 , T ] , u · n = 0 , ω × n = 0 , on ∂ Ω . Then we take the scalar product with ω , and integrate by parts. One easily gets12 ∂ t Z Ω | ω | dx + Z Ω |∇ ω | dx = Z ∂ Ω n · ∇ ω · ω dS + Z Ω ω · ∇ u · ω dx := I + I . (3.1)Next, we focus on the estimates of I and I . Control of I : First, it follows from (2.1) thatˆ u = 0 , ˆ ω = ˆ ω = 0 , as q = ρ , ρ . (3.2)Let ∂ Ω l = ∂ ˆΩ l := { ( q , q , q ) ∈ ˆΩ : q = ρ l } , where l = 0 ,
1. One can deduce from (3.2) that(2 l − Z ∂ Ω l n · ∇ ω · ω dS =(2 l − Z ∂ Ω l n · ∇ (cid:18) | ω | (cid:19) dS = Z Z (cid:20) ∂ q (cid:18) | ˆ ω | (cid:19) H H H − (cid:21) (cid:12)(cid:12)(cid:12) q = ρ l dq dq = Z Z (cid:2) ( ∂ q ˆ ω ) ˆ ω H H H − (cid:3) | q = ρ l dq dq = Z Z (cid:2) ∂ q ( H H ˆ ω ) H − ˆ ω (cid:3) | q = ρ l dq dq − Z Z (cid:2) ∂ q ( H H ) H − ˆ ω (cid:3) | q = ρ l dq dq . (3.3)Since ∇ · ω = 0, from (2.4) one gets ∂ (ˆ ω H H ) ∂q + ∂ (ˆ ω H H ) ∂q + ∂ (ˆ ω H H ) ∂q = 0 , Z Z (cid:2) ∂ q ( H H ˆ ω ) H − ˆ ω (cid:3) | q = ρ l dq dq = − Z Z (cid:2) ∂ q ( H H ˆ ω ) H − ˆ ω (cid:3) | q = ρ l dq dq − Z Z (cid:2) ∂ q ( H H ˆ ω ) H − ˆ ω (cid:3) | q = ρ l dq dq = 0 , since ˆ ω = ˆ ω = ∂ q ˆ ω = ∂ q ˆ ω = 0 on ∂ ˆΩ l . Hence, one obtains(2 l − Z ∂ Ω l n · ∇ ω · ω dS = − Z Z (cid:2) ∂ q ( H H ) H − ˆ ω (cid:3) | q = ρ l dq dq By appealing to (2.6) one shows that (cid:12)(cid:12)(cid:12)(cid:12)Z ∂ Ω n · ∇ ω · ω dS (cid:12)(cid:12)(cid:12)(cid:12) ≤ C Z ∂ Ω | ω | dS ≤ k | ω | k W , (Ω) , where we have used Gagliardo’s trace theorem, see [16]. See also [32], Theorem 4.2 (for anEnglish recent text see, for example, the Theorem III.2.21 in [11]). It follows that (cid:12)(cid:12)(cid:12)(cid:12)Z ∂ Ω n · ∇ ω · ω dS (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( ǫ ) k ω k L (Ω) + ǫ k∇ ω k L (Ω) , (3.4)for all 0 < ǫ < . Control of I : First, one has Z Ω ω · ∇ u · ω dx = X i,j,k Z ˆΩ ˆ ω i H − i ∂ q i (ˆ u j ˆ e j ) · (ˆ ω k ˆ e k ) H H H dq dq dq = X i,j,k Z ˆΩ ˆ u j ˆ ω i ˆ ω k ( ∂ q i ˆ e j · ˆ e k ) H − i H H H dq dq dq + X i,j Z ˆΩ ˆ ω i ( ∂ q i ˆ u j ) ˆ ω j H − i H H H dq dq dq := I + I . For I , from (2.6), one has | I | ≤ C Z Ω | u | | ω | dx ≤ C k u k L (Ω) k ω k L (Ω) ≤ C k u k L (Ω) k ω k L (Ω) ( k ω k L (Ω) + k∇ ω k L (Ω) ) ≤ C k u k L (Ω) k ω k L (Ω) + C k u k L (Ω) k ω k L (Ω) k∇ ω k L (Ω) ≤ C k u k L (Ω) k ω k L (Ω) + C ( ǫ ) k u k L (Ω) k ω k L (Ω) + ǫ k∇ ω k L (Ω) . (3.5)For I , we consider separately the three cases j = 3; j = 3 and i = 3; i = j = 3.7 ase I: j = 3. By integration by parts, one has Z ˆΩ ˆ ω i ( ∂ q i ˆ u j ) ˆ ω j H − i H H H dq dq dq = − Z ˆΩ ˆ u j ( ∂ q i ˆ ω i ) ˆ ω j H − i H H H dq dq dq − Z ˆΩ ˆ u j ˆ ω i ( ∂ q i ˆ ω j ) H − i H H H dq dq dq − Z ˆΩ ˆ u j ˆ ω i ˆ ω j ∂ q i (cid:0) H − i H H H (cid:1) dq dq dq . (3.6) Case II: j = 3 and i = 3. From (2.5) one hasˆ ω = 1 H H (cid:18) ∂ (ˆ u H ) ∂q − ∂ (ˆ u H ) ∂q (cid:19) . Hence, by integration by parts, it follows that Z ˆΩ ˆ ω i ( ∂ q i ˆ u ) ˆ ω H − i H H H dq dq dq = Z ˆΩ ˆ ω i ( ∂ q i ˆ u ) (cid:0) ∂ q ( H ˆ u ) − ∂ q ( H ˆ u ) (cid:1) H − i H dq dq dq = − Z ˆΩ ˆ u ∂ q (cid:0) ˆ ω i ( ∂ q i ˆ u ) H − i H (cid:1) H dq dq dq + Z ˆΩ ˆ u ∂ q (cid:0) ˆ ω i ( ∂ q i ˆ u ) H − i H (cid:1) H dq dq dq . (3.7) Case III: i = j = 3. Note that, due to ∇ · u = 0, it follows ∂ (ˆ u H H ) ∂q + ∂ (ˆ u H H ) ∂q + ∂ (ˆ u H H ) ∂q = 0 . (3.8)One has Z ˆΩ ˆ ω ( ∂ q ˆ u ) ˆ ω H H dq dq dq = Z ˆΩ ˆ ω ∂ q ( H H ˆ u ) ˆ ω dq dq dq − Z ˆΩ ˆ ω ˆ u ˆ ω ∂ q ( H H ) dq dq dq = − Z ˆΩ ˆ ω ∂ q ( H H ˆ u ) ˆ ω dq dq dq − Z ˆΩ ˆ ω ∂ q ( H H ˆ u ) ˆ ω dq dq dq − Z ˆΩ ˆ ω ˆ u ˆ ω ∂ q ( H H ) dq dq dq = Z ˆΩ ˆ u ∂ q (cid:0) ˆ ω (cid:1) H H dq dq dq + Z ˆΩ ˆ u ∂ q (cid:0) ˆ ω (cid:1) H H dq dq dq − Z ˆΩ ˆ u ˆ ω ∂ q ( H H ) dq dq dq , (3.9)where the first equality is an identity, the second is obtained by appealing to (3.8), and thethird one follows by integration by parts. From (3.6), (3.7), (3.9) and the assumption (2.6),one can obtain | I | ≤ C Z Ω | ¯ u ||∇ u ||∇ u | dx + C Z Ω | u ||∇ u | dx + C Z Ω | u | |∇ u | dx .
8t is easy to get that Z Ω | u | |∇ u | dx ≤k u k L (Ω) k u k L (Ω) k∇ u k L (Ω) ≤k u k L (Ω) k u k L (Ω) + k∇ u k L (Ω) ≤k u k L (Ω) k u k L (Ω) + k∇ u k L (Ω) ( k∇ u k L (Ω) + k∇ u k L (Ω) ) ≤k u k L (Ω) k u k L (Ω) + C ( ǫ ) k∇ u k L (Ω) + ǫ k∇ u k L (Ω) , and similarly to the proof of (3.5) Z Ω | u ||∇ u | dx ≤ C k u k L (Ω) k∇ u k L (Ω) + C ( ǫ ) k u k L (Ω) k∇ u k L (Ω) + ǫ k∇ u k L (Ω) . Hence, one has | I | ≤ C Z Ω | ¯ u ||∇ u ||∇ u | dx + C k u k L (Ω) k u k L (Ω) + C ( ǫ ) k∇ u k L (Ω) + C k u k L (Ω) k∇ u k L (Ω) + C ( ǫ ) k u k L (Ω) k∇ u k L (Ω) + Cǫ k∇ u k L (Ω) . (3.10)By H¨older’s inequality, interpolation, and a Sobolev’s embedding theorem, one can easilyshow that Z Ω | ¯ u ||∇ u ||∇ u | dx ≤k| ¯ u |∇ u k L (Ω) k∇ u k L (Ω) ≤k ¯ u k L p (Ω) k∇ u k L pp − (Ω) k∇ u k L (Ω) ≤k ¯ u k L p (Ω) k∇ u k − p L (Ω) k∇ u k p L (Ω) k∇ u k L (Ω) ≤ C k ¯ u k L p (Ω) k∇ u k − p L (Ω) ( k∇ u k L (Ω) + k∇ u k L (Ω) ) p k∇ u k L (Ω) ≤ C (cid:18) k ¯ u k L p (Ω) k∇ u k L (Ω) k∇ u k L (Ω) + k ¯ u k L p (Ω) k∇ u k − p L (Ω) k∇ u k p L (Ω) (cid:19) ≤ C ( ǫ ) (cid:18) k ¯ u k pp − L p (Ω) + k ¯ u k L p (Ω) (cid:19) k∇ u k L (Ω) + ǫ k∇ u k L (Ω) . (3.11)Collecting (3.1) and the estimates (3.4), (3.5), (3.10) and (3.11), one obtains12 ∂ t Z Ω | ω | dx + Z Ω |∇ ω | dx ≤ C ( ǫ ) (cid:18) k u k L (Ω) + k u k L (Ω) + k ¯ u k pp − L p (Ω) + k ¯ u k L p (Ω) (cid:19) k∇ u k L (Ω) + C k u k L (Ω) k u k L (Ω) + Cǫ k∇ u k L (Ω) . On the other hand, the following well known estimates (see for instance Theorem IV.4.8and Theorem IV.4.9 in [11]), hold: k∇ u k L (Ω) ≤ C k ω k L (Ω) , k∇ u k L (Ω) ≤ C (cid:0) k u k L (Ω) + k ω k H (Ω) (cid:1) . (3.12)9herefore, from equation (3.12), by letting ǫ be sufficiently small, one has ∂ t Z Ω | ω | dx + Z Ω |∇ ω | dx ≤ C (cid:18) k u k L (Ω) + k u k L (Ω) + k ¯ u k pp − L p (Ω) + k ¯ u k L p (Ω) (cid:19) k ω k L (Ω) + C k u k L (Ω) k u k L (Ω) + C k u k L (Ω) . Finally (1.2) follows by taking into account equations (2.7) ( q ≥ pp − >
2) and (3.12), andby appealing to a well known argument, which is based on Gronwall’s inequality. Recall thatweak solutions verify k u k L (Ω) ∈ L ∞ (0 , T ) and k u k L (Ω) ∈ L (0 , T ). Hence we have provedthat u is a strong solution. In this section we present a first attempt to prove the statement of Theorem 2.2 with the slipboundary condition (2.1) replaced by the slip boundary condition (1.5) (assumed in reference[8]) by means of a simple modification of our proof. This attempt fails. Hence this significantproblem remains open to further investigation. This leads us to briefly show our calculations.Let’s start by explaining our guess. As still shown in Remark 2.2 condition (1.5) isequivalent to u · n = 0 , ( ω × n ) · τ = 2 κ τ u · τ , on ∂ Ω . (4.1)We may replace the arbitrary tangent vector τ simply by a couple of independent vectors like,for instance, the principal direction’s vectors τ and τ . In this case κ = κ τ and κ = κ τ are the maximum and the minimum principal curvatures.A more natural choice here is to consider the couple of tangent, orthogonal, vectors ˆ e and ˆ e . In this case κ and κ are the related curvatures. This second choice easily leads tothe couple of linear equations ˆ ω = 2 κ ˆ u , ˆ ω = − κ ˆ u . (4.2)Hence to replace the slip boundary condition (2.1) by [ D ( u ) n ] · τ = 0 means to replaceassumption (3.2) byˆ u = 0 , ˆ ω = − κ ˆ u , ˆ ω = 2 κ ˆ u , as q = ρ , ρ . (4.3)To prove our main statement with the boundary condition (2.1) replaced by the boundarycondition (4.1) we have to control some new boundary integrals, which no longer vanish sincenow ˆ ω and ˆ ω do not vanish. However, by (4.2), ˆ ω and ˆ ω can be expressed in terms of the(lower order) velocity components ˆ u and ˆ u . Well known inverse trace theorems allow us tocontrol boundary-norms of these two components by suitable internal norms. Since our P-Sassumption guarantees additional regularity just for these two velocity components, one could10xpect that the above internal norms could be estimated in a convenient way. Unfortunatelythis device seems not sufficient to prove our goal. So this interesting problem remains open.Next we pass to showing our calculations. Let’s turn back to equation (3.3), by takinginto account that now we can not apply to ˆ w = ˆ w = 0 . One has(2 l − Z ∂ Ω l n · ∇ ω · ω dS =(2 l − Z ∂ Ω l n · ∇ (cid:18) | ω | (cid:19) dS = Z Z (cid:20) ∂ q (cid:18) | ˆ ω | (cid:19) H H H − (cid:21) (cid:12)(cid:12)(cid:12) q = ρ l dq dq = Z Z X j ( ∂ q ˆ w j ) ˆ w j H H H − (cid:12)(cid:12)(cid:12) q = ρ l dq dq . We will drop terms which could be easily manipulated, called here ”lower order terms”.Dropping lower order terms and also cancelling non significant multiplication coefficients,lead us to introduce the symbols ” ≃ ” and ” (cid:22) ” , which have a clear meaning here.One has( ∂ q ˆ w j ) ˆ w j H H H − = ∂ q ( H H ˆ w j ) H − ˆ ω j − ∂ q ( H H ) H − ˆ ω j (cid:22) ∂ q ( H H ˆ w j ) H − ˆ ω j . (4.4)Since ∇ · ω = 0, from (2.4) one gets ∂ (ˆ ω H H ) ∂q + ∂ (ˆ ω H H ) ∂q + ∂ (ˆ ω H H ) ∂q = 0 , which gives, on ∂ ˆΩ ,∂ q ( H H ˆ ω ) H − ˆ ω = − ∂ q ( H H ˆ ω ) H − ˆ ω − ∂ q ( H H ˆ ω ) H − ˆ ω . Under the new boundary conditions we can not apply to ˆ ω = ˆ ω = ∂ q ˆ ω = ∂ q ˆ ω = 0 on ∂ ˆΩ l to claim the cancellation of the above right hand side. By noting that the two terms onthe right hand side are symmetric, with respect to the index 1 and 2, we may consider justthe first one.One has ∂ q ( H H ˆ ω ) H − ˆ ω = ( ∂ q ˆ ω )ˆ ω H − ˆ ω ˆ ω ∂ q ( H H ) H − ≃ ( ∂ q ˆ ω )ˆ ω . Note that the smooth coefficients H j , as their derivatives, are not significant on our estimatebelow. Further, since ∂ q is a tangential derivative, we may apply to the second equality (4.2)to assume that ∂ q ˆ ω ≃ − ∂ q ˆ u on ∂ ˆΩ . Hence Z Z (cid:2) ∂ q ( H H ˆ ω ) H − ˆ ω (cid:3) | q = ρ l dq dq ≃ Z Z ( ∂ q ˆ u ) ˆ ω | q = ρ l dq dq . (4.5)By appealing to Gagliardo’s theorem we show that the above right hand side is bounded by C ( ǫ ) k∇ u k + ǫ k∇ u k , which is sufficient to our purposes.11et’s now consider in equation (4.4) the terms ∂ q ( H H ˆ w j ) H − ˆ w j , for j = 1 , . Assume,for instance, j = 1 . One has ∂ q ( H H ˆ w ) H − ˆ w ≃ ( ∂ q ˆ w )ˆ u . Hence we need to controlthe integral Z Z ( ∂ q ˆ ω ) ˆ u | q = ρ l dq dq . (4.6)Roughly speaking the above integrand has the same order as that on the right hand side of(4.5). However in (4.6) the derivation symbol ∂ q appears now in the ”bad position”. Asuitable control of the above integral turns out to be not obvious. 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