Regularity Criteria for Navier-Stokes Equations with Slip Boundary Conditions on Non-flat Boundaries via Two Velocity Components

H.-O. Bae and H.J. Choe, in a 1997 paper, established a regularity criteria for the incompressible Navier-Stokes equations in the whole space $\R^3$ based on two velocity components. Recently, one of the present authors extended this result to the half-space case $\R^3_+\,.$ Further, this author in collaboration with J. Bemelmans and J. Brand extended the result to cylindrical domains under physical slip boundary conditions. In this note we obtain 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 the boundary. Otherwise, a reciprocal reduction between the two results looks not obvious, as shown in the last section below.


Introduction
The starting point of the present paper is the well known Prodi-Serrin (P-S) su cient condition for regularity of the solutions to the incompressible Navier-Stokes equations ∂ t u + u · ∇u − ∆u + ∇p = , in Ω × ( , T] , in Ω × ( , T] . (1.1) where u = (u , u , u ) denotes the unknown velocity of the uid and p the pressure. To immediately set limits to the circle of our interests, assume for now on that Ω ⊂ R is a bounded, 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 [1] and E. Hopf [2] classical references. See also A.A. Kiselev and O.A. Ladyzhenskaya [3], and J.L. Lions [4]. Below, solutions of (1.1) are intended in this sense.
A main classical open mathematical problem is to prove, or disprove, that weak solutions are necessarily strong under reasonable but general assumptions, where strong means that u ∈ L ∞ ( , T; H (Ω)) ∩ L ( , T; H (Ω)) . (1.2) In this context, a remarkable and classical su cient condition for uniqueness and regularity is the so-called Prodi-Serrin condition, P-S in the sequel, namely u ∈ L q ( , T; L p (Ω)) , q + p = , p > . (1.3) Concerning condition (1.3), we transcribe from [5], Section 1, the following considerations: Assumption (1.3) was rstly considered by G. Prodi in his paper [6] of 1959. He proved uniqueness under this last assumption. See also C. Foias, [7]. Furthermore, J. Serrin, see [8,9], particularly proved interior spatial regularity under the stronger (non-strict) assumption Concerning the above problems, see also O.A. Ladyzhenskaya's contributions [10,11]. 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 by H. Sohr in [12], W. von Wahl in [13], and Y. Giga in [14]. A simpli ed version of the proof was given in reference [15], to which we refer also for bibliography. For a quite complete overview on the main points, and references, on the initialboundary value problem for Navier-Stokes equations we strongly recommend Galdi's contribution [16]. Further, we refer to [9,17], as sources for information on the historical context of the P-S condition by the initiators themselves.
A signi cant improvement of the P-S condition was obtained by H.-O. Bae and H.J. Choe [19], see also [20]. They proved, in the whole space case, that it is su cient for regularity of solutions that two components of the velocity satisfy the above condition (1.3). For convenience we call here this situation as being the restricted P-S condition. In 2017, one of the authors, see [21], extended this result to the half-space R + under slip boundary conditions. In this case, the truncated 2-dimensional vector eldū cannot be chosen arbitrarily. The omitted component has to be the normal to the boundary.
Very recently, in reference [5], the result was extended to a cylindrical type three-dimensional domain, 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 boundary condition u · n = , [D(u)n] · τ = , on ∂Ω , (1.5) where D(u) = ∇u+(∇u) T is the shear stress. The above exclusion of an interior cylinder was done to avoid the radial coordinate singularities on the symmetry axis, which consideration is out of interest in our context. Below we obtain a similar result, extended to domains with general non-at boundaries, but under the slip boundary condition (2.1). The two boundary conditions coincide just on at portions of the boundary. Otherwise, a reciprocal reduction between the two results looks not obvious. This claim is shown in the last section.
Before going on we want to motivate the particular choice of the domain made below. It takes into account that the real signi cance of the result has essentially a local character. First of all, a global regular (i.e., without singularities) system of coordinates, two of them parallel and the third orthogonal to the boundary, does not exist in general, even in an arbitrarily thin neighbourhood of the full boundary, as in the case of a sphere and even in the case of a spherical corona. In fact, singularities typically appear, like on the above two cases, and even in full cylinders (due to the symmetry axis). The cylindrical case considered in reference [5] 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 su cient small neighbourhoods of any regular boundary point. Hence, to illustrate the full signi cance of our thesis in a simple, but still convincing way, it looks su cient to prove it near any "small" piece of smooth boundary with an arbitrary geometrical shape. This is our aim below. The restrictions on the domain Ω below are made in accordance with these lines, a choice which covers the very basic situation, in the simplest way.

Main Results
In the sequel we assume the slip boundary condition where ω = ∇ × u is the vorticity, n is the outward normal of ∂Ω, and Ω ∈ R is a smooth domain satisfying the following condition: There exists a curvilinear orthogonal system of coordinates such that Ω can be transited intô where the axis q direct to the outward normal on the boundary ∂Ω := {(q , q , q ) : q = ρ } (the inward normal on the boundary ∂Ω := {(q , q , q ) : q = ρ }, respectively), and q , q are periodic.
Remark 2.1. The above "small" piece of a generical smooth boundary is here represented by q = ρ , and q = ρ .
Remark 2.2. It is worth noting that the slip boundary condition (2.1) is equivalent to where τ stands for any arbitrary unit tangential vector on ∂Ω , and κτ is the principal curvature 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 [32], namely For a mathematical treatment of some aspects related to slip boundary conditions imposed on smooth, but generic, boundaries see also [33], and the pioneering paper [34].
Next we recall some facts on curvilinear coordinates. The Lamé coe cients (scale factors) of the transition to the system of coordinates q are denoted by the letters H i and ω(x) =ω(q) =ω (q)ê +ω (q)ê +ω (q)ê .
It is well known (see for example [35,36]) that and (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 constants c and C such that for any i , j , k = , , . Let u be a weak solution of the system (1.1) under the boundary condition (2.1), and set u =û ê +û ê . Ifū satis esū then the solution u is strong, namely, 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 2.3. The above statement does not contain the result proved in reference [5], due to the distinct boundary conditions, see Section 4. On the other hand, we may replace the two 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 a parallel Jordan curve γ , at a su cient small distance δ > from γ . The coordinate θ is now 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 ∈ ( , δ) is given by the distance to γ . The "vertical" coordinate z preserves his periodic character. Clearly, the role played by the above Jordan curve may be immediately extended to much more general situations.
Another signi cant application is obtained by replacing the above two cylindrical boundaries 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 obtain the complement set between two closed tubes). Now z ∈ [ , π) is an angular periodic coordinate, the toroidal coordinate. The result still applies by replacing the two circles by two parallel Jordan curves.
Let's propose the following benchmark problem:

H H H dq dq dq
:=I + I . For I , we consider separately the three cases j ≠ ; j = and i ≠ ; i = j = . where the rst equality is an identity, the second is obtained by appealing to (3.8), and the third one follows by integration by parts. From (3.6), (3.7), (3.9) and the assumption (2.6), one can obtain It is easy to get that Hence, one has (3.10) By Hölder's inequality, interpolation, and a Sobolev's embedding theorem, one can easily show that Finally (1.2) follows by taking into account equations (2.7) (q ≥ p p− > ) and (3.12), and by appealing to a well known argument, which is based on Gronwall's inequality. Recall that weak solutions verify u L (Ω) ∈ L ∞ ( , T) and u L (Ω) ∈ L ( , T). Hence we have proved that u is a strong solution.

On related slip boundary conditions.
In this section we present a rst attempt to prove the statement of Theorem 2.2 with the slip boundary condition (2.1) replaced by the slip boundary condition (1.5) (assumed in reference [5]) by means of a simple modication of our proof. This attempt fails. Hence this signi cant problem remains open to further investigation. This leads us to brie y show our calculations.
Let's start by explaining our guess. As still shown in Remark 2.2 condition (1.5) is equivalent to u · n = , (ω × n) · τ = κτ 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 = κτ and κ = κτ are the maximum and the minimum principal curvatures. A more natural choice here is to consider the couple of tangent, orthogonal, vectorsê andê . In this case κ and κ are the related curvatures. This second choice easily leads to the couple of linear equations ω = κ û , Hence to replace the slip boundary condition (2.1) by [D(u)n] · τ = means to replace assumption (3.2) bŷ To prove our main statement with the boundary condition (2.1) replaced by the boundary condition (4.1) we have to control some new boundary integrals, which no longer vanish since nowω andω do not vanish. However, by (4.2),ω andω can be expressed in terms of the (lower order) velocity componentsû andû . Well known inverse trace theorems allow us to control boundary-norms of these two components by suitable internal norms. Since our P-S assumption guarantees additional regularity just for these two velocity components, one could expect that the above internal norms could be estimated in a convenient way. Unfortunately this device seems not su cient 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 taking into account that now we can not apply toω =ω = . One has We will drop terms which could be easily manipulated, called here "lower order terms". Dropping lower order terms and also cancelling non signi cant multiplication coe cients, lead us to introduce the symbols " " and " " , which have a clear meaning here. Under the new boundary conditions we can not apply toω =ω = ∂q ω = ∂q ω = on ∂Ω l to claim the cancellation of the above right hand side. By noting that the two terms on the right hand side are symmetric, with respect to the index and , we may consider just the rst one. One has ∂q H H ω H − ω = (∂q ω )ω H −ω ω ∂q (H H )H − (∂q ω )ω .
Note that the smooth coe cients H j , as their derivatives, are not signi cant on our estimate below. Further, since ∂q is a tangential derivative, we may apply to the second equality (4.2) to assume that ∂q ω −∂q û on ∂Ω . Hence ∂q H H ω H − ω |q =ρ l dq dq (∂q û )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(ϵ) ∇ u + ϵ ∇ u , which is su cient to our purposes.
Let's now consider in equation (4.4) the terms ∂q (H H ω j ) H − ω j , for j = , . Assume, for instance, j = . One has ∂q (H H ω )H − ω (∂q ω )û . Hence we need to control the integral (∂q ω )û |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". A suitable control of the above integral turns out to be not obvious.