On the Navier-Stokes equations on surfaces

We consider the motion of an incompressible viscous fluid that completely covers a smooth, compact and embedded hypersurface $\Sigma$ without boundary and flows along $\Sigma$. Local-in-time well-posedness is established in the framework of $L_p$-$L_q$-maximal regularity. We characterize the set of equilibria as the set of all Killing vector fields on $\Sigma$ and we show that each equilibrium on $\Sigma$ is stable. Moreover, it is shown that any solution starting close to an equilibrium exists globally and converges at an exponential rate to a (possibly different) equilibrium as time tends to infinity.


Introduction
Suppose Σ is a smooth, compact, connected, embedded (oriented) hypersurface in R d+1 without boundary. Then we consider the motion of an incompressible viscous fluid that completely covers Σ and flows along Σ.
Fluid equations on manifolds appear in the literature as mathematical models for various physical and biological processes, for instance in the modeling of emulsions and biological membranes. The reader may also think of an aquaplanet whose surface is completely covered by a fluid. The case of a planet with oceans and landmass will be considered in future work.
Fluid equations on manifolds have also been studied as mathematical problems in their own right, see for instance [1,4,5,7,8,11,24,25] and the references cited therein.
In this paper, we model the fluid by the 'surface Navier-Stokes equations' on Σ, using as constitutive law the Boussinesq-Scriven surface stress tensor where λ s is the surface shear viscosity, λ s the surface dilatational viscosity, u the velocity field, π the pressure, and the surface rate-of-strain tensor. Here, P Σ denotes the orthogonal projection onto the tangent bundle TΣ of Σ, div Σ the surface divergence, and ∇ Σ the surface gradient. We refer to Prüss and Simonett [13] and the Appendix for more background on these objects.
Boussinesq [3] first suggested to consider surface viscosity to account for intrinsic frictional forces within an interface. Several decades later, Scriven [22] generalized Boussinesq's approach to material surfaces having arbitrary curvature. The resulting tensor is nowadays called the Boussinesq-Scriven stress tensor.
It will be shown in the Appendix that (1.4) can we written in the form where ∆ Σ is the (negative) Bochner-Laplacian, κ Σ the d-fold mean curvature of Σ (the sum of the principal curvatures), and L Σ the Weingarten map. Here we use the convention that a sphere has negative mean curvature. We would like to emphasize that the tensor (κ Σ L Σ − L 2 Σ ) is an intrinsic quantity. In fact, we shall show in Proposition A.2 that where Ric Σ is the Ricci tensor and K Σ the Gaussian curvature of Σ (the product of the principal curvatures).
The formulation (1.4) coincides with [7, formula (3.2)]. In that paper, the equations for the motion of a viscous incompressible fluid on a surface were derived from fundamental continuum mechanical principles. The same equations were also derived in [8, formula (4.4)], based on global energy principles. We mention that the authors of [7,8] also consider material surfaces that may evolve in time.
Here we would like to point out that several formulations for the 'surface Navier-Stokes equations' have been used in the literature, see [4] for a comprehensive discussion, and also [7,Section 3.2]. It turns out that the model based on the Boussinesq-Scriven surface stress tensor leads to the same equations as, for instance, in [5, Note added to Proof] and [24]. Indeed, this follows from (1.5)-(1.6) and the relation where ∆ H denotes the Hodge Laplacian (acting on 1-forms).
The plan of this paper is as follows. In Section 2 we show that kinetic energy is dissipated by the fluid system (1.4), and we characterize all the equilibrium solutions of (1.4). It is shown that at equilibrium, the gradient of the pressure is completely determined by the velocity field. Moreover, it is shown that the equilibrium (that is, the stationary) velocity fields correspond exactly to the Killing fields of Σ. We finish Section 2 with some observations concerning the motion of fluid particles in the case of a stationary velocity field.
In Section 3, we prove that the linearization of (1.5) enjoys the property of L p -L q -maximal regularity. We rely on results contained in [13,Sections 6 and 7]. Moreover, we introduce the Helmholtz projection on Σ and we prove interpolation results for divergence-free vector fields on Σ. We then establish local well-posedness of (1.5) in the (weighted) class of L p -L q -maximal regularity, see Theorem 3.5.
In Section 4, we prove that all equilibria of (1.5) are stable. Moreover, we show that any solution starting close to an equilibrium exists globally and converges at an exponential rate to a (possibly different) equilibrium as time tends to infinity. In order to prove this result we show that each equilibrium is normally stable. Let us recall that the set of equilibria E coincides with the vector space of all Killing fields on Σ. It then becomes an interesting question to know how many Killing fields a given manifold can support. In Subsection 4.1, we include some remarks about the dimension of E and we discuss some examples.
In forthcoming work, we plan to use the techniques introduced in this manuscript to study the Navier-Stokes equations on manifolds with boundary.
We would like to briefly compare the results of this paper with previous results by other authors. Existence of solutions for the Navier-Stokes equations (1.5) has already been established in [24], see also [11] and the comprehensive list of references in [4]. The authors in [24,11] employ techniques of pseudo-differential operators and they make use of the property that the Hodge Laplacian commutes with the Helmholtz projection. Under the assumption that the spectrum of the linearization is contained in the negative real axis, stability of the zero solution is shown in [24]. The author remarks that this assumption implies that the isometry group of Σ is discrete. In contrast, our stability result in Theorem 4.3 applies to any equilibrium.
The Boussinesq-Scriven surface stress tensor has also been employed in the situation of two incompressible fluids which are separated by a free surface, where surface viscosity (accounting for internal friction within the interface) is included in the model, see [2].
Finally, we mention [7,12,18,19] and the references contained therein for interesting numerical investigations. These authors also observed that the equilibria velocities correspond to Killing fields.

Energy dissipation and equilibria
In the following, we set ̺ = 1. Let be the (kinetic) energy of the fluid system. We show that the energy is dissipated by the fluid system (1.4).
Proposition 2.1. Suppose (u, π) is a sufficiently smooth solution of (1.4) with initial value u 0 ∈ TΣ, defined on some interval (0, T ). Then Proof. By Remark 1.1 we know that u(t) ∈ TΣ for each t ∈ (0, T ). In order not to overburden the notation, we suppress the variables (t, p) ∈ (0, T ) × Σ in the following computation. It follows from (A.17) and Lemma A.1 that where (·|·) denotes the Euclidean inner product. Hence, the surface divergence theorem (A.14) and the relation div Σ u = 0 yield We now characterize the equilibria of (1.4). It will turn out that at equilibrium, the gradient of the pressure is completely determined by the velocity. Moreover, the equilibrium velocity fields correspond exactly to the Killing fields of Σ.
where c is an arbitrary constant.
(c) The C 1 -tangential fields satisfying the relation D Σ (u) = 0 correspond exactly to the Killing fields of Σ.
Proof. (a) Suppose (u, π) is an equilibrium of (1.4). By the second line in (1.4), div Σ u = 0. It follows from Proposition 2.1 that D Σ (u) = 0 and the first line in (1.4) then implies This, together with D Σ (u) = 0 and Lemma A.1(b),(c), yields Analogous arguments show that the inverse implication also holds true. Suppose that (u, π) is an equilibrium solution of (1.4). Then we point out the following interesting observation.
Using the assertion in Proposition 2.2(a), we obtain as D Σ (u) = 0. Hence, π is constant along stream lines of the flow. Furthermore, A short computation shows that for tangential vector fields u, hence (u|∇ Σ P Σ u) = (L Σ u|u)ν Σ . Therefore, we obtain the relationγ since ∇ Σ π = −P Σ (u · ∇ Σ u) in an equilibrium. Let us compare this ODE with the following second order system with constraints: Here, f : R d → R d and g : R d → R d−m are smooth with rank g ′ (x) = d− m for each x ∈ g −1 (0). In general, the ODEẍ = f (x,ẋ) does not leave Σ := g −1 (0) invariant. However, it can be shown that the movement of a particle under the constraint g(x) = 0 in the force field f is governed by the ODË In other words, the effective force field is the sum of the tangential part of f on Σ and the constraint force (ẋ|∇ Σ P Σ (x)ẋ), which results from the geodesic flow, see also [17,Section 13.5].

Existence of solutions
In this section, we show that there exists a unique solution ) of (1.4) resp. (1.5) for some suitable number a > 0. To this end, we first consider the principal linearization of (1.5) and show that the corresponding linear operator has L p -L q -maximal regularity in suitable function spaces. This will enable us to apply the contraction mapping principle to prove the existence and uniqueness of a strong solution to (1.5).
3.1. The principal linearization. We consider the following linear problem where ω > 0. Here and in the sequel, we assume without loss of generality that ̺ = 1. The main result of this section reads as follows.
Moreover, the solution (u, π) depends continuously on the given data (f, g, u 0 ) in the corresponding spaces.

Remark 3.2.
(a) In Theorem 3.1, we use the notationṡ

Pressure Regularity.
It is a remarkable fact that the pressure π has additional time-regularity in some special cases.
Proof. Let φ ∈ L q ′ (Σ), 1/q + 1/q ′ = 1 and solve the equation . This follows, for instance, from [13, Theorem 6.4.3 (i)] and the fact 0 is in the resolvent set of ∆ L Σ , acting on functions with zero average. We then obtain from the surface divergence theorem (A.14), (A.17), and the fact that (div Σ f, g) = 0 since ∂ α t and L Σ commute. This yields the claim. Without loss of generality, we may always assume that (div Σ f, g, u 0 ) = 0. To see this, let (u, π) be a solution of (3.1) and solve the parabolic problem by [13,Theorem 6.4.3 (ii)] to obtain a unique solution v ∈ E 1,µ (Σ). Next, we solve Of course, the converse is also true. If (ũ,π) solves (3.1) with (div Σ f, g, u 0 ) = 0, then one may construct a solution (u, π) of (3.1) with prescribed data (f, g, u 0 ) being subject to the conditions in Theorem 3.1, by reversing the above procedure.

Localization.
In this subsection we prove the existence and uniqueness of a solution to (3.1). We start with the proof of uniqueness. To this end, let (u, π) be a solution of (3.1) with (div Σ f, g, u 0 ) = 0.
By compactness of Σ, there exists a family of charts Note that without loss of generality, we may assume that ) and g (k)ij is defined in a very similar way, see also the Appendix. Then, withū = u • ϕ −1 k ,π = π • ϕ −1 k and so on, the system (3.1) with respect to the local charts (U k , ϕ k ), k ∈ {1, . . . , N }, reads as follows.
. . , d}, B (k) is a linear differential operator of order one and Here, upon translation and rotation,ḡ ij (k) (0) = δ i j and the coefficients have been extended in such a way that |ḡ ij where η > 0 can be made as small as we wish, by decreasing the radius r > 0 of the ball B R d (0, r).
In order to handle system (3.3), we define vectors in R d as follows: For each k ∈ {1, . . . , N }, we define operators L k,ω by the first two lines on the left side of (3.5). Then each operator is invertible and bounded. This can be seen by first freezing the coefficients at x = 0, leading to full-space Stokes problems, which enjoy the property of L p -L q -maximal regularity by [13, Theorem 7.1.1]. Secondly, a Neumann series argument yields the claim, since G (k) is a perturbation of the identity in R d×d . With the operator L k,ω at hand, we may rewrite (3.5) in the more condensed form see also [13, (7.41)]. Definẽ . The couple (ũ (k) ,π (k) ) then solves the equation We note on the go that div(F (k) (ū,π) − G (k) ∇Φ k ) = 0 and that the pressureπ (k) enjoys additional time regularity. This can be seen exactly as in the proof of [13, Proposition 7.3.5 (ii)] with an obvious modification concerning the matrix G (k) . In particular, there exists a constant C > 0, such that Let us now introduce a norm for the solution, taking the parameter ω into account.
For each k ∈ {1, . . . , N } there exists ω 0 > 0 such that the operator L k,ω has the property of L p -L q -maximal regularity, provided ω > ω 0 . In particular, there exists a constant C > 0 such that (ũ (k) ,π (k) ) ω ≤ Cω −γ (u, π) ω , where for the last inequality, we made use of Proposition 3.3, implying the estimate for some constant γ > 0, by interpolation between L p,µ (R + ; L q (Σ)) and L p,µ (R + ; H 2 q (Σ)). In the same way, making also use of (3.6) and the definition of Φ k , we obtain by (3.7), Proposition 3.3 forπ and the Poincaré inequality for π 0 (k) and Φ k , since we may assume without loss of generality, that π 0 (k) as well as Φ k have mean value zero on B(0, R) ⊂ R d . By interpolation with (3.6) this yields In conclusion, we obtain the estimate Since χ k = 1 on supp(ψ k ), this finally yields the estimate valid for all ω > ω 0 . Choosing ω 0 > 0 sufficiently large, we conclude This in turn implies uniqueness of a solution to (3.1). It remains to prove the existence of a solution to (3.1). To this end, we may assume that (div Σ f, g, u 0 ) = 0. Solve the parabolic problem by [13,Theorem 6.4.3 (ii)] to obtain a unique solution v ∈ 0 E 1,µ (Σ). Next, we solve ∆ L Σ φ = div Σ v by [13, Theorem 6.4.3 (i)] to obtain a solution φ such that ∇ Σ φ is unique with regularity Making use of local coordinates, one can show that where A Σ is a second order operator. Settingû =ũ,π =π − µ s ∆ L Σ φ and Sf := (û,π), we obtain L ω Sf = L ω (û,π) = f + Rf, where the constant C > 0 does not depend on ω. We note that (∆ L Σ ) −1 , acting on functions with average zero, is well defined. A Neumann series argument implies that (I + R) is invertible provided ω > 0 is sufficiently large. Hence the operator S(I + R) −1 is a right inverse for L ω , which means that L ω is surjective. This completes the proof of Theorem 3.1. u ∈ H 1 p,µ (R + ; L q (Σ, TΣ)) ∩ L p,µ (R + ; H 2 q (Σ, TΣ)), π ∈ L p,µ (R + ;Ḣ 1 q (Σ)), if and only if the data (f, g, u 0 ) are subject to the conditions (1)-(4) in Theorem 3.1. Moreover, the solution (u, π) depends continuously on the given data (f, g, u 0 ) in the corresponding spaces.

The surface Stokes operator. By Proposition
Proof. Without loss of generality, we may assume (div Σ f, g, u 0 ) = 0. With the operator L ω defined above, we rewrite (3.9) as For the term of order zero on the right hand side, we have the estimate where C > 0 does not depend on ω > 0. By Theorem 3.1 the solution depends continuously on the data, hence there exists a constant M = M (ω 0 ) > 0 such that Therefore, a Neumann series argument yields the claim, if ω > 0 is chosen sufficiently large.
We will now define the Stokes operator on surfaces. Let P H,Σ denote the surface Helmholtz projection, defined by which follows directly from the definition of P H,Σ (and for smooth functions from the surface divergence theorem (A.14)). Define We would also like to refer to the survey article [6] for the Stokes operator in various other geometric settings. Making use of the projection P H,Σ , (3.9) with (div Σ f, g) = 0 is equivalent to the equation By Corollary 3.4, the operator A S,Σ has L p -maximal regularity, hence −A S,Σ generates an analytic C 0 -semigroup in X 0 , see for instance [13, Proposition 3.5.2].

Interpolation spaces.
In this subsection, we will determine the real and complex interpolation spaces (X 0 , X 1 ) α,p and (X 0 , X 1 ) α , respectively. To this end, let A Σ u := −2µ s P Σ div Σ D Σ (u) with domain D(A Σ ) := H 2 q (Σ, TΣ) and define a linear mapping Q on D(A Σ ) by for some fixed and sufficiently large ω > 0.
Then Q : D(A Σ ) → X 1 is a bounded projection, as Qu ∈ X 1 and for all u ∈ D(A Σ ). Furthermore, Q| X1 = I X1 and therefore Q : D(A Σ ) → X 1 is surjective. By a duality argument, there exists some constant C > 0 such that for all u ∈ D(A Σ ). In fact, for all u ∈ D(A Σ ) and φ ∈ L q ′ (Σ, TΣ), with Since D(A Σ ) is dense in L q (Σ, TΣ), there exists a unique bounded extensionQ : L q (Σ, TΣ) → X 0 of Q. Clearly,Q is a projection and as X 1 is dense in X 0 ,Q| X0 = I X0 .
In case that A S,Σ has a bounded H ∞ -calculus one may set µ = µ c in Theorem 3.5, thereby obtaining well-posedness in critical spaces. This will be studied in future work.

Stability of equilibria, examples
Consider the semilinear evolution equation in X 0 = L q,σ (Σ, TΣ). Define the set We show that the set E corresponds exactly to the set of equilibria for (4.1). To this end, let u * be an equilibrium of (4.1), i.e. u * ∈ X 1 satisfies A S,Σ u * = F Σ (u * ). Multiplying this equation by u * and integrating over Σ yields By Lemma A.1 and the surface divergence theorem (A.14), the last term vanishes, since div Σ u * = 0. Furthermore, (A.17) and again (A.14) show that Conversely, let u * ∈ E be given. Then A S,Σ u * = 0 and from Lemma A.1 we obtain that Summarizing, we have shown that Observe that the set E is a linear manifold, consisting exactly of the Killing fields on Σ, see Remark A.3.
Define an operator A 0 : X 1 → X 0 by . This operator is the full linearization of (4.1) at the equilibrium u * ∈ E. We collect some properties of A 0 in the following Proposition 4.1. Suppose u * ∈ E and let A 0 be given by (4.3). Then −A 0 generates a compact analytic C 0 -semigroup in X 0 which has L p -maximal regularity. The spectrum of A 0 consists only of eigenvalues of finite algebraic multiplicity and the kernel N (A 0 ) is given by Proof. By Subsection 3.4 the surface Stokes operator A S,Σ has the property of L pmaximal regularity in X 0 , hence −A S,Σ is R-sectorial in X 0 . Furthermore, the linear mapping [v → F ′ Σ (u * )v] is relatively bounded with respect to A S,Σ . An application of [13,Proposition 4.4.3] yields that −A 0 generates an analytic C 0 -semigroup in X 0 having L p -maximal regularity.
Since the domain X 1 of A 0 is compactly embedded into X 0 , the spectrum σ(A 0 ) is discrete and consists solely of eigenvalues of A 0 having finite algebraic multiplicity. Let λ ∈ σ(−A 0 ) and denote by v ∈ X 1 a corresponding eigenfunction. Multiplying the equation λv +A 0 v = 0 by the complex conjugatev and integrating over Σ yields Here, we used the identities since D Σ (u * ) = 0 and div Σ u * = 0, employing the surface divergence theorem (A.14). It follows that Reλ ≤ 0 and if Reλ = 0, then D Σ (v) = 0. Observe that the equations D Σ (v) = 0 = D Σ (u * ) then lead to the identity hence F ′ Σ (u * )v = P H,Σ (∇ Σ (u * |v)) = 0 and therefore A 0 v = 0.

4.1.
Existence of equilibria. According to (4.2), see also Proposition 2.2, the equilibria E of the evolution equation (4.1) correspond to the Killing fields of Σ.
It is then an interesting question to know how many Killing fields a given manifold can support, or to be more precise, what the dimension of the vector space of all Killing fields of Σ is. (In fact, it turns out that the Killing fields on a Riemannian manifold form a sub Lie-algebra of the Lie-algebra of all tangential fields).
It might also be worthwhile to recall that the Killing fields of a Riemannian manifold (M, g) are the infinitesimal generators of the isometries I(M, g) on (M, g), that is, the generators of flows that are isometries on (M, g), Moreover, in case (M, g) is complete, the Lie-algebra of Killings fields is isometric to the Lie-algebra of I(M, g), see for instance Corollary III.6.3 in [20].
It then follows from [20, Proposition III.6.5] that dim E ≤ d(d + 1)/2, where d is the dimension of Σ. For compact manifolds, equality holds if and only if Σ is isometric to S d , the standard d-dimensional Euclidean sphere in R d+1 .
On the other hand, if (M, g) is compact and the Ricci tensor is negative definite everywhere, then any Killing field on M is equal to zero and I(M, g) is a finite group, see [20,Proposition III.6.6]. In particular, if (M, g) is a two-dimensional Riemannian manifold with negative Gaussian curvature then any Killing field is 0. Then dim E = 3 and each equilibrium u * ∈ E corresponds to a rotation about an axis spanned by a vector ω = (ω 1 , ω 2 , ω 3 ) ∈ R 3 . Therefore, u * ∈ E is given by for some ω ∈ R 3 . According to Theorem 4.3, each equilibrium u * = ω × x is stable and each solution u of (4.1) that starts out close to u * converges at an exponential rate towards a (possibly different) equilibrium u ∞ = ω ∞ × x for some ω ∞ ∈ R 3 .
Another family of equilibria is obtained by rotation around the small circles with radius r in planes that contain the x 3 -axis. In local coordinates, u * is then given by u * = ω(− sin φ cos θ, − sin φ sin θ, cos φ), with ω ∈ R.
According to Theorem 4.3, all the linear combinations of these equilibria are stable.
Appendix A.
In this appendix we collect some results from differential geometry that are employed throughout the manuscript. We also refer to [13,Chapter 2] for complementary information. We will assume throughout that Σ is a smooth, compact, closed (that is, without boundary) hypersurface embedded in R d+1 . We mention on the go that these assumptions imply that Σ is orientable, see for instance [21].
Let ν Σ be the unit normal field of Σ (which is compatible with the chosen orientation). Then the orthogonal projection P Σ onto the tangent bundle of Σ is defined by P Σ = I − ν Σ ⊗ ν Σ .
We use the notation {τ 1 (p), · · · , τ d (p)} to denote a local basis of the tangent space T p Σ of Σ at p, and {τ 1 (p), · · · , τ d (p)} to denote the corresponding dual basis of the cotangent space T * p Σ at p. Hence we have (τ i (p)|τ j (p)) = δ i j , the Kronecker delta function. Note that In this manuscript we will occasionally not distinguish between vector fields and covector fields, that is, we identify where, as usual, the Einstein summation convention is employed throughout. The metric tensor is given by g ij = (τ i |τ j ), where (·|·) is the Euclidean inner product of R d+1 , and the dual metric g * on the cotangent bundle T * Σ is given by g ij = (τ i |τ j ). It holds that Hence, g is induced by the inner product (·|·), that is, we have It holds that where Λ k ij are the Christoffel symbols, l ij are the components of the second fundamental form, and l i j are the components of the Weingarten tensor L Σ ; that is, we have and If ϕ ∈ C 1 (Σ, R), the surface gradient of ϕ is defined by ∇ Σ ϕ = ∂ i ϕτ i . If u is a C 1 -vector field on Σ (not necessarily tangential) we define the surface gradient of u by It follows from (A.5) that for a tangential vector field u. The covariant derivative ∇ i u of a tangential vector field is defined by ∇ i u = P Σ ∂ i u. Hence we have The surface divergence div Σ u for a (not necessarily tangential) vector field u is defined by div Σ u = (τ i |∂ i u). (A.11) Then the d-fold mean curvature κ Σ of Σ is given by Hence, κ Σ is the trace of L Σ (which equals the sum of the principal curvatures). For a vector field u = v j τ j + wν Σ , it follows from (A.9), (A.12) and the fact that ν Σ and τ i are orthogonal For a tangent vector field u and a scalar function ϕ, the surface divergence theorem states that For a tensor K = k j i τ i ⊗ τ j , the surface divergence is defined by Hence, div Σ P Σ = ∂ i (τ j ⊗ τ j )τ i = κ Σ ν Σ . (A.16) Lemma A.1. Suppose ϕ is a C 1 -scalar function and u, v, w are C 1 -tangential vector fields on Σ. Then (a) div Σ (ϕP Σ ) = ∇ Σ ϕ + ϕκ Σ ν Σ .
The assertion is now a consequence of (A.16).
(a) Then we have the following representation: where ∆ Σ is the Bochner-Laplacian (also called the conformal Laplacian), defined in local coordinates by where Ric Σ is the Ricci tensor and K Σ the Gaussian curvature of Σ, respectively.

(b) The computations in (A.19)-(A.20) show that in local coordinates
(κ Σ L Σ − L 2 Σ )u = g jm (l jm l ik − l im l jk )u k τ i . By the Gauss equation, see for instance [20,Proposition II.3.8], this yields (κ Σ L Σ − L 2 Σ )u = g jm R jikm u k τ i = R ik u k τ i = Ric Σ u, where R jikm are the components of the curvature tensor and R ik the components of the Ricci (0, 2)-tensor. In case that Σ is a surface embedded in R 3 , one obtains (κ Σ L Σ − L 2 Σ )u = K Σ u, where K Σ is the Gauss curvature of Σ. This can, for instance, be seen as follows: (κ Σ L Σ − L 2 Σ )u = g jm (l jm l ik − l im l jk )u k τ i = (l j j l k i − l j i l k j )u k τ i = det(L Σ ) δ k i u k τ i = K Σ u. The proof of Proposition A.2 is now complete.