H ∞ -Calculus for the Surface Stokes Operator and Applications

. We consider a smooth, compact and embedded hypersurface Σ without boundary and show that the corresponding (shifted) surface Stokes operator admits a bounded H ∞ -calculus with angle smaller than π/ 2. As an application, we consider critical spaces for the Navier–Stokes equations on the surface Σ. In case Σ is two-dimensional, we show that any solution with a divergence-free initial value in L 2 (Σ , T Σ) exists globally and converges exponentially fast to an equilibrium, that is, to a Killing ﬁeld.


Introduction
Suppose Σ is a smooth, compact, connected, embedded hypersurface in R d+1 without boundary.We then consider the motion of an incompressible viscous fluid that completely covers Σ and flows along Σ.The motion can be modeled by the surface Navier-Stokes equations for an incompressible viscous fluid (1.1) Here, the density ̺ is a positive constant, T Σ = 2µ s D Σ (u) − πP Σ , and is the surface rate-of-strain tensor, with u the fluid velocity and π the pressure.Moreover, 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 Chapter 2 in [22] and the Appendix in [24] for more information concerning these objects.
The formulation (1.1) coincides with [12, formula (3.2)].In that paper, the equations were derived from fundamental continuum mechanical principles.The same equations were also derived in [14, formula (4.4)], based on global energy principles.We mention that the authors of [12,14] also consider material surfaces that may evolve in time.
Existence and uniqueness of solutions to the surface Navier-Stokes equations (1.1) was established in [24].It was shown that the set of equilibria consists of all Killing vector fields on Σ, and that all of these are normally stable.Moreover, it was shown that (1.1) can be reformulated as where ∆ Σ is the (negative) Bochner-Laplacian and Ric Σ the Ricci tensor.To be more precise, in slight abuse of the usual convention, we interpret Ric Σ here as the (1, 1)-tensor given in local coordinates by Ric ℓ k = g iℓ R ik .We remind that in case d = 2, Ric Σ u = K Σ u, with K Σ being the Gaussian curvature of Σ.
The formulation (1.2) shows that the surface Navier-Stokes equations can be formulated by intrinsic quantities that only depend on the geometry of the surface Σ, but not on the ambient space.In an intrinsic formulation, the surface Navier Stokes equations (1.2) can be stated as where ∇ is the covariant derivative (induced by the Levi-Civita connection of Σ), and grad π = ∇ Σ π.An inspection of the proofs then shows that all the results in [24], and the results of this paper, are also valid for (1.3) for any smooth Riemannian manifold Σ without boundary.We remind that the Bochner Laplacian ∆ Σ is related to the Hodge Laplacian by the formula ∆ Σ = ∆ H +Ric, with the usual identification of vector fields and one-forms by means of lowering or rising indices.
We would like to point out that several formulations for the surface Navier-Stokes equations have been considered in the literature, see [6] for a comprehensive discussion, and also [12,Section 3.2].It has been advocated in [8,Note added to Proof], and also in the more recent publications [6,30], that the surface Navier-Stokes equations on a Riemannian manifold ought to be modeled by the system (1.3).
For the Stokes operator on domains in Euclidean space under various boundary conditions, the existence of an H ∞ -calculus (or the related property of bounded imaginary powers) has been obtained by Giga [10], Abels [1], Noll and Saal [17], Saal [29], Prüss and Wilke [26], and Prüss [20].We also refer to the survey article by Hieber and Saal [11] for additional references and information concerning the Stokes operator on domains in Euclidean space.
In particular, our results imply existence and uniqueness of solutions for initial values u 0 ∈ L q,σ (Σ, TΣ) for q = d, see Corollary 4.4.Hence, the celebrated result of Kato [13] is also valid for the surface Navier-Stokes equations.
For d = 2, we show in Theorem 4.9 that any solution to (1.1) with initial value u 0 ∈ L 2,σ (Σ, TΣ) exists globally and converges exponentially fast to an equilibrium, that is, to a Killing field.The proof is based on an abstract result in [23], Korn's inequality (established in Theorem A.3), and an energy estimate.Moreover, in Remark 4.10 we show that in case d ≥ 3, any global solution converges to an equilibrium.
We refer to [23,26] for background information on critical spaces and for a discussion of the existing literature concerning critical spaces for the Navier-Stokes equations (and other equations) for domains in Euclidean space.
We would now like to briefly compare the results of this paper with previous results by other authors.Existence and uniqueness of solutions for the Navier-Stokes equations (1.3) for initial data in Morrey and Besov-Morrey spaces was established by Taylor [32] and Mazzucato [16], respectively; see also [6] for a comprehensive list of references.The authors in [32,16] employ techniques of pseudo-differential operators and they make use of the property that the Hodge Laplacian commutes with the Helmholtz projection.In case d = 2, global existence is proved in [32,Proposition 6.5], but that result does not establish convergence of solutions.
The Boussinesq-Scriven surface stress tensor T Σ has also been employed in the situation where 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 for instance [5,22].Finally, we mention [12,18,27,28] and the references contained therein for interesting numerical investigations.
Notation: We now introduce some notation and some auxiliary results that will be used in the sequel.It follows from the considerations in [24, Lemma A.1 and Remarks A.3] that for tangential vector fields u, v, where ∇ denotes the covariant derivative (or the Levi-Civita connection) on Σ.In the following, we will occasionally take the liberty to use the shorter notation ∇ u v and ∇u mentioned in (1.4).We recall that for sufficiently smooth vectors fields u, say u ∈ C 1 (Σ, TΣ), one has ∇u ∈ C(Σ, T 1 1 Σ), the space of all (1, 1)-tensors on Σ.As the Levi-Civita connection ∇ is a metric connection, we have for tangential vector fields u, v, w on Σ, where (u|v) := g(u, v) is the Riemannian metric (induced by the Euclidean inner product of R d+1 in case Σ is embedded in R d+1 ).Occasionally, we also write grad ϕ in lieu of ∇ Σ ϕ for scalar functions ϕ.
We use the notation whenever the right hand side exists, say for u ∈ L q (Σ, TΣ) and v ∈ L q ′ (Σ, TΣ), where 1/q + 1/q ′ = 1.For k ∈ N and q ∈ (1, ∞), the space H k q (Σ, TΣ) is defined as the completion of C ∞ (Σ, TΣ), the space of all smooth vector fields, in L 1,loc (Σ, TΣ) with respect to the norm The Bessel potential spaces H s q (Σ, TΣ) and the Besov spaces B s qp (Σ, TΣ) can then be defined through interpolation, see for instance [3,Section 7].It is well-known that these spaces can be given equivalent norms by means of local coordinates, see for instance [3,Theorem 7.3] (for the more general context of singular and uniformly regular manifolds).

The surface Stokes operator
In [24,Corollary 3.4] we showed that there exists a number ω 0 > 0 such that for ω > ω 0 , the system admits a unique solution Moreover, the solution (u, π) depends continuously on the given data (f, u 0 ) in the corresponding spaces.

H ∞ -calculus
In this section, we are going to prove the following result.Theorem 3.1.Let q ∈ (1, ∞) and Σ be a smooth, compact, connected, embedded hypersurface in R d+1 without boundary.Let A S,Σ be the surface Stokes operator in L q,σ (Σ, TΣ) defined in (2.2).
3.2.Localization.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 the Appendix in [24].Then, with ū k and so on, the system (3.1) with respect to the local charts (U k , ϕ k ), k ∈ {1, . . ., N }, reads as follows. where ), ℓ ∈ {1, . . ., d}, B (k) collects all terms of order at most one and Here, upon translation and rotation, ḡij (k) (0) = δ i j and the coefficients have been extended in such a way that ḡij , 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 (3.4),we define vectors in R d as follows: With these notations, system (3.4) reads as Let us remove the term H (k) (ū), since it is not of lower order.For that purpose, we solve the equation div( where C is a positive constant.For this, observe that H (k) (ū) is compactly supported and we have where In order to remove the pressure term in (3.7) we introduce the projections P G k , defined by We claim that each of the operators provided ω is sufficiently large.To see this, we write Recall that each matrix G (k) is a perturbation of the identity in R d×d .Therefore, , where we may choose η > 0 as small as we wish.Furthermore, As before, we may choose η > 0 as small as we wish.
Note that the shifted Stokes operator ω + A S admits a bounded This yields the following representation of the resolvent u = (λ + ω + A S,Σ ) −1 f .
where {e ℓ } d ℓ=1 is the standard basis in R d and In a next step, we will estimate the term R(λ)f in L q (Σ, TΣ).To this end, observe that the operators P H,Σ and P G k are bounded in L q (Σ, TΣ) and L q (R d ), respectively.This, together with (3.2), (3.3), (3.6) and the fact that each of the operators ω+A G S,k is sectorial in L q,σ (R d ) for ω sufficiently large, yields the estimate for some constant C > 0. Indeed, by (3.2), (3.3), (3.6), we obtain Here we have also used complex interpolation (Σ, TΣ).Consequently, I −T is a Fredholm operator with index 0.In particular, ker(I − T ) is finite dimensional and the range ran(I − T ) is closed in L q (Σ).Let {v 1 , . . ., v m } be an orthonormal basis of ker(I − T ) and define Then it can be readily checked that Q is a projection onto ker(I − T ) and it is continuous in L q,σ (Σ, TΣ) for any q ∈ (1, ∞), since v k ∈ H 1 q (Σ, TΣ) for any q ∈ (1, ∞) (using a bootstrap argument).Consequently, the operator is invertible with bounded inverse.
We use the resolvent representation to conclude that the right hand side belongs to ran(I − T ), hence φ ∈ (φ 0 , π/2), we then obtain , θ ∈ (φ 0 , φ).Estimate (3.9) then yields since each of the operators ω + A G S,k has a bounded H ∞ -calculus.The remaining part Qh(ω + A S,Σ ) may be treated as follows.
For the last integral, we employ the definition of the projection Q from above to obtain By (3.2), we therefore obtain for some constant C > 0. Consequently, the operator ω + A S,Σ admits a bounded H ∞ -calculus in L q,σ (Σ, TΣ) with angle φ ∞ AS,Σ < π/2 provided ω > 0 is large enough.An application of [22,Corollary 3.3.15]finally yields that it is enough to require ω > s(−A S,Σ ).This completes the proof of Theorem 3.1.
Remark 4.2.Note that in case d = 3 and p = q = 2, the initial value belongs to Hence, the celebrated result of Fujita & Kato [9] holds true for the surface Navier-Stokes equations.
The existence and uniqueness result for (4.3) in critical spaces reads as follows.

Energy estimates and global existence.
In [24] we showed that the set of equilibria E for (1.1), respectively (4.1), consists exactly of the Killing vector fields on Σ, that is, We recall that the condition D Σ (u) = 0 implies that u is divergence free (which follows from the relation div Σ u = tr D Σ (u)).Moreover, one can show that any vector field u ∈ H 1 q (Σ, TΣ) satisfying D Σ (u) = 0 is already smooth, see for instance [19,Lemma 3].Lastly, we recall that E is a finite dimensional vector space.If fact, dim E ≤ d(d + 1)/2, with equal sign for the case where Σ is isometric to a Euclidean sphere, see for instance the remarks in Section 4.1 of [24].
Let us define the space Note that V j 2 (Σ) is a closed subspace of H j 2,σ (Σ, TΣ), and hence is a Banach space.Moreover, H j 2,σ (Σ, TΣ) = E ⊕ V j 2 (Σ), see Remark 4.10(a).From now on we assume that d = 2, and we show that any solution of (4.3) with initial value v 0 ∈ L 2,σ (Σ) being orthogonal to E will remain orthogonal for all later times.Moreover, we establish an energy estimate for such solutions.
Proof.(a) According to Theorem 4.5, we know that where the time derivative exists for almost all t ∈ (0, t + (v 0 )).For the last equal sign we employed the property that A S,Σ is symmetric on L 2,σ (Σ) and N (A S,Σ ) = E, see [24,Proposition 4.1].In a next step we show that (∇ v v|z) Σ = 0 for all v ∈ H 1 2,σ (Σ).Indeed, this follows from where we used (1.5), the surface divergence theorem and the property that z is a Killing vector field, (which implies (b) Similarly as in part (a), one shows (suppressing the variable t) that d dt The assertion in part (a) and Korn's inequality (A.2) readily imply with an appropriate constant α > 0. Integration yields the assertion in (b), as Proposition 4.7.Suppose that d = 2 and v 0 ∈ V 0 2 (Σ).Then problem (4.3) admits a unique global solution v enjoying the regularity properties stated in Theorem (4.5), with t + (v 0 ) = ∞.
Moreover, there exists a constant α > 0 such that Proof.By the abstract result [23,Theorem 2.4] on global-in-time existence, the maximal time of existence t + (u 0 ) satisfies the following property: 2 (Σ, TΣ)), then the weak solution exists globally in time.Proposition 4.6 guarantees that any solution v with initial value v 0 ∈ V 0 2 (Σ) satisfies v ∈ L 2 ((0, t + (v 0 )), H 1 2 (Σ, TΣ)) and, hence, global existence of the weak solution follows.Since we know that the weak solution in this case regularizes to a strong solution, we obtain global in time existence of strong solutions for q = d = 2 as well.
respectively its weak formulation where In particular, each solution of (4.13) with initial value v 0 ∈ V 0 2 (Σ) exists globally and there exists a positive constant α such that Proof.One readily verifies that the assertions of Theorems 4.1 and 4.5 remain valid for problem (4.12) and (4.13), respectively.In fact, one only needs to verify that the terms on the right hand side can be estimated in the same way as in the proof of Theorems 4.1 and 4.5.
Next we show that v(t) ∈ V 0 2 (Σ) for t ∈ [0, t + (v 0 )).Let z ∈ E. Following the proof of Proposition 4.6(a), we obtain According to the proof of Proposition 4.6(a), (∇ v v|z) = 0 and it remains to show that (∇ u * v + ∇ v u * |z) Σ = 0 for any v ∈ H 1 2,σ (Σ, TΣ).This follows from where we used (1.5), the surface divergence theorem and the property that z is a Killing vector field.The same arguments as in the proof of Propositions 4.6 and 4.7 yield the remaining assertions.
Then any solution of (4.3) with initial value u 0 ∈ L 2,σ (Σ, TΣ) exists globally, has the regularity properties listed in Theorem 4.5, and converges at an exponential rate to the equilibrium u * = P E u 0 in the topology of H 2 q (Σ, TΣ) for any fixed q ∈ (1, ∞), where P E is the orthogonal projection of u 0 onto E with respect to the L 2 (Σ, TΣ) inner product.
(iv) We will now consider (4.1) in the spaces , where (X α , A α ) is the interpolation-extrapolation scale with respect to the complex interpolation functor, based on X 0 = L s,σ (Σ, TΣ), s ∈ (1, ∞), introduced at the beginning of Section 4.2.We note that A 1/2 , the realization of A 0 = ω + A S,Σ in X 1/2 , has exactly the same properties as A 0 .For β = 1/2 we obtain
Since we can identify E ⊂ C ∞ (Σ, TΣ) as a subspace of F s (Σ), the expression v, z Σ is defined for every (v, z) ∈ F s (Σ) × E and (4.19) is, therefore, meaningful.
Proof.We will provide a proof of (4.19).As E is finite dimensional, we can find a basis {z 1 , . . ., z m } for E which has the property that (z i |z j ) Σ = δ ij .With this at hand, we define the projection onto E. This yields the direct topological decomposition F s (Σ) = E ⊕ V s (Σ), where V s (Σ) = (I − P s E )F s (Σ).In order to justify (4.19), it suffices to show that V s (Σ) = V s (Σ) := {v ∈ F s (Σ) | v, z Σ = 0, z ∈ E}.
(b) It is interesting to note that the assertions of Propositions 4.6 and 4.7 remain valid in case d > 2, with the following modifications: Let p > 2 and q ≥ d > 2. Suppose v 0 ∈ B d/q−1 qp,σ (Σ, TΣ) satisfies v 0 , z Σ = 0 for every z ∈ E, where v 0 , z Σ has the same meaning as in (a).Let v be the unique solution of (4.3) with initial value v 0 .
Next we study existence and uniqueness as well as regularity properties of solutions to some second order differential equations on R d .
For that purpose, we set for functions f ∈ L q (R d ).For λ > 0, k ∈ {−1, 0, 1} we then define the function spaces equipped with the parameter-dependent norms We are ready to prove the following result.q ′ (R d ).It is known that the operator ∆ : E −1 λ → F −1 λ is an isomorphism, see for instance [34], Theorem 5.2.3.1(i) and the remarks in Section 5. is an isomorphism as well, provided η ∈ (0, 1).