Abstract
We consider the motion of an incompressible viscous fluid that completely covers a smooth, compact and embedded hypersurface \(\Sigma \) without boundary and flows along \(\Sigma \). Localintime wellposedness 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 \(\Sigma \) is a smooth, compact, connected, embedded (oriented) hypersurface in \({{\mathbb {R}}}^{d+1}\) without boundary. Then, we consider the motion of an incompressible viscous fluid that completely covers \(\Sigma \) and flows along \(\Sigma \).
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, 23, 24] and the references cited therein.
In this paper, we model the fluid by the ‘surface Navier–Stokes equations’ on \(\Sigma \), using as constitutive law the Boussinesq–Scriven surface stress tensor
where \(\mu _s\) is the surface shear viscosity, \(\lambda _s\) the surface dilatational viscosity, u the velocity field, \(\pi \) the pressure, and
the surface rateofstrain tensor. Here, \({{\mathcal {P}}}_\Sigma \) denotes the orthogonal projection onto the tangent bundle \(\mathsf{T}\Sigma \) of \(\Sigma \), \(\mathrm{div}_\Sigma \) the surface divergence, and \(\nabla _\Sigma \) the surface gradient. We refer to Prüss and Simonett [13] and Appendix for more background information 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.
For an incompressible fluid, i.e., \(\mathrm{div}_\Sigma u=0\), the Boussinesq–Scriven surface tensor simplifies to
We then consider the following surface Navier–Stokes equations for an incompressible viscous fluid
where \(\varrho \) is a positive constant. In the sequel, we will always assume that \(u_0\in \mathsf{T}\Sigma \), i.e., \(u_0\) is a tangential field.
Remark 1.1
Suppose \(u_0\in \mathsf{T}\Sigma \). If \((u(t),\pi (t))\) is a (sufficiently) smooth solution to (1.4) on some time interval [0, T), then we also have \(u(t)\in \mathsf{T}\Sigma \) for all \(t\in [0,T]\). This can readily be seen by taking the inner product of the first equation in (1.4) with \(\nu _\Sigma (p)\), yielding \((\partial _t u(t, p)  \nu _\Sigma (p))=0\) for \((t,p)\in [0,T)\times \Sigma \), where \(\nu _\Sigma \) is the unit normal field of \(\Sigma \). Hence, \((u(t,p)\nu _\Sigma (p))=(u_0(p)  \nu _\Sigma (p))=0\) for \((t,p)\in [0,T)\times \Sigma .\)
It will be shown in Appendix that (1.4) can we written in the form
where \(\Delta _\Sigma \) is the (negative) BochnerLaplacian, \(\kappa _\Sigma \) the dfold mean curvature of \(\Sigma \) (the sum of the principal curvatures), and \(L_\Sigma \) the Weingarten map. Here, we use the convention that a sphere has negative mean curvature.
We would like to emphasize that the tensor \((\kappa _\Sigma L_\Sigma  L^2_\Sigma )\) is an intrinsic quantity. In fact, we shall show in Proposition A.2 that
where \(\mathrm{Ric}_\Sigma \) is the Ricci tensor and \(K_\Sigma \) the Gaussian curvature of \(\Sigma \) (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 [23]. Indeed, this follows from (1.5)(1.6) and the relation
where \(\Delta _H\) denotes the Hodge Laplacian (acting on 1forms).
The plan of this paper is as follows. In Sect. 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 \(\Sigma \). We finish Sect. 2 with some observations concerning the motion of fluid particles in the case of a stationary velocity field.
In Sect. 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 \(\Sigma \) and we prove interpolation results for divergencefree vector fields on \(\Sigma \). We then establish local wellposedness of (1.5) in the (weighted) class of \(L_p\)\(L_q\)maximal regularity, see Theorem 3.5.
In Sect. 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 \({{\mathcal {E}}}\) coincides with the vector space of all Killing fields on \(\Sigma \). It then becomes an interesting question to know how many Killing fields a given manifold can support. In Sect. 4.1, we include some remarks about the dimension of \({{\mathcal {E}}}\) 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 [23], see also [11] and the comprehensive list of references in [4]. The authors in [11, 23] employ techniques of pseudodifferential 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 [23]. The author remarks that this assumption implies that the isometry group of \(\Sigma \) is discrete. In contrast, our stability result in Theorem 4.3 applies to any manifold.
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 \(\varrho =1\). 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 \((u,\pi )\) is a sufficiently smooth solution of (1.4) with initial value \(u_0\in \mathsf{T}\Sigma \), defined on some interval (0, T). Then,
Proof
By Remark 1.1, we know that \(u(t)\in \mathsf{T}\Sigma \) for each \(t \in (0,T)\). In order not to overburden the notation, we suppress the variables \((t,p)\in (0,T)\times \Sigma \) in the following computation. It follows from (A.17) and Lemma A.1 that
where \((\cdot  \cdot )\) denotes the Euclidean inner product. Hence, the surface divergence theorem (A.14) and the relation \(\mathrm{div}_\Sigma u=0\) yield
\(\square \)
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 \(\Sigma \).
Proposition 2.2

(a)
Let \({{\mathfrak {E}}}:=\{(u,\pi )\in C^2(\Sigma , \mathsf{T}\Sigma )\times C^1(\Sigma ): (u,\pi ) \text { is an equilibrium for}~(1.4)\}\). Then,
$$\begin{aligned} {{\mathfrak {E}}}=\Big \{(u,\pi ): \mathrm{div}_\Sigma u=0,\; {{\mathcal {D}}}_\Sigma (u)=0,\; \pi =\frac{1}{2}u^2+c\Big \}, \end{aligned}$$where c is an arbitrary constant.

(b)
Suppose \({{\mathcal {D}}}_\Sigma (u)=0\) for \(u\in C^1(\Sigma , \mathsf{T}\Sigma )\). Then, \(\mathrm{div}_\Sigma u=0\).

(c)
The \(C^1\)tangential fields satisfying the relation \({{\mathcal {D}}}_\Sigma (u)=0\) correspond exactly to the Killing fields of \(\Sigma \).
Proof

(a)
Suppose \((u,\pi )\) is an equilibrium of (1.4). By the second line in (1.4), \(\mathrm{div}_\Sigma u=0\). It follows from Proposition 2.1 that \({{\mathcal {D}}}_\Sigma (u)=0\) and the first line in (1.4) then implies
$$\begin{aligned} {{\mathcal {P}}}_\Sigma ( u\cdot \nabla _\Sigma u) +\nabla _\Sigma \pi =0.\ \end{aligned}$$This, together with \({{\mathcal {D}}}_\Sigma (u)=0\) and Lemma A.1(b),(c), yields
$$\begin{aligned} \nabla _\Sigma \pi = {{\mathcal {P}}}_\Sigma (u\cdot \nabla _\Sigma u) ={{\mathcal {P}}}_\Sigma ((\nabla _\Sigma u)^\mathsf{T}u) ={{\mathcal {P}}}_\Sigma ((\nabla _\Sigma u)u)=\frac{1}{2}\nabla _\Sigma u^2. \end{aligned}$$Analogous arguments show that the inverse implication also holds true.

(b)
This is a direct consequence of (A.17)\(_3\).

(c)
This follows from Remark A.3(e).
\(\square \)
Suppose that \((u,\pi )\) is an equilibrium solution of (1.4). Then, we point out the following interesting observation.
Let \(\gamma (s)\) be the trajectory of a fluid particle on \(\Sigma \). Then, \(\gamma \) satisfies the differential equation
Using the assertion in Proposition 2.2(a), we obtain
as \({{\mathcal {D}}}_\Sigma (u)=0\). Hence, \(\pi \) is constant along stream lines of the flow.
Furthermore,
A short computation shows that
for tangential vector fields u, hence \((u\nabla _\Sigma P_\Sigma u)=(L_\Sigma uu)\nu _\Sigma \). Therefore, we obtain the relation
since \(\nabla _\Sigma \pi =P_\Sigma (u\cdot \nabla _\Sigma u)\) in an equilibrium. Let us compare this ODE with the following secondorder system with constraints:
Here, \(f:{{\mathbb {R}}}^{d}\rightarrow {{\mathbb {R}}}^{d}\) and \(g:{{\mathbb {R}}}^d\rightarrow {{\mathbb {R}}}^{dm}\) are smooth with \(\mathrm{rank}\ g'(x)=dm\) for each \(x\in g^{1}(0)\). In general, the ODE \(\ddot{x}=f(x,{\dot{x}})\) does not leave \(\Sigma :=g^{1}(0)\) 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 ODE
In other words, the effective force field is the sum of the tangential part of f on \(\Sigma \) and the constraint force \(({\dot{x}}\nabla _\Sigma P_\Sigma (x) {\dot{x}})\), which results from the geodesic flow, see also [17, Section 13.5].
Observe that the structure of (2.3) and (2.4) is the same. Of course, in our situation, it follows from the ODE \({\dot{\gamma }}=u(\gamma )\) that \(\gamma (s)\in \Sigma \), \({\dot{\gamma }}(s)\in {\mathsf {T}}_{\gamma (s)}\Sigma \), provided \(\gamma (0)=\gamma _0\in \Sigma \), since u is a tangential vector field on \(\Sigma \).
Finally, note that the energy \(E(s):=\frac{1}{2}{\dot{\gamma }}(s)^2+\pi (\gamma (s))\) is conserved, i.e., \({\dot{E}}(s)=0\), since \((u\nabla _\Sigma P_\Sigma u)\) is perpendicular to \({\mathsf {T}}\Sigma \) and \({\dot{\gamma }}\in {\mathsf {T}}\Sigma \). This known as Bernoulli’s principle.
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).
The principal linearization
We consider the following linear problem
where \(\omega >0\). Here and in the sequel, we assume without loss of generality that \(\varrho =1\). The main result of this section reads as follows.
Theorem 3.1
Suppose \(\Sigma \) is a smooth, compact, connected, embedded (oriented) hypersurface in \({{\mathbb {R}}}^{d+1}\) without boundary and let \(1<p,q<\infty \), \(\mu \in (1/p,1]\). Then, there exists \(\omega _0>0\) such that for each \(\omega >\omega _0\), problem (3.1) admits a unique solution
if and only if the data \((f,g,u_0)\) are subject to the following conditions

(1)
\(f\in L_{p,\mu }({{\mathbb {R}}}_+;L_q(\Sigma ,{\mathsf {T}}\Sigma ))=:{\mathbb {E}}_{0,\mu }(\Sigma )\)

(2)
\(g\in L_{p,\mu }({{\mathbb {R}}}_+;H_q^1(\Sigma ))\), \(g\in H_{p,\mu }^1({{\mathbb {R}}}_+;{\dot{H}}_q^{1}(\Sigma ))\)

(3)
\(u_0\in B_{qp}^{2\mu 2/p}(\Sigma ,{\mathsf {T}}\Sigma )\)

(4)
\(g(0)=\mathrm{div}_\Sigma u_0\).
Moreover, the solution \((u,\pi )\) 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 notations
$$\begin{aligned} {\dot{H}}_{q}^1(\Sigma ):=\{w\in L_{1,loc}(\Sigma ):\nabla _\Sigma w\in L_{q}(\Sigma ,{\mathsf {T}}\Sigma )\},\quad {\dot{H}}_q^{1}(\Sigma ):=({\dot{H}}_{q'}^1(\Sigma ))^* \end{aligned}$$and we identify g with the functional \([\phi \mapsto \int _\Sigma g\phi \,\mathrm{d}\Sigma ]\) on \({\dot{H}}_{q'}^1(\Sigma )\).

(b)
Note that the assumption \(g\in {\dot{H}}_q^{1}(\Sigma )\) includes the condition \(\int _\Sigma g\,\mathrm{d}\Sigma =0\).

(c)
The assertion \(\pi \in L_{p,\mu }({{\mathbb {R}}}_+;{\dot{H}}_q^1(\Sigma ))\) means that \(\pi \) is unique up to a constant.

(d)
Necessity of the conditions (1)–(4) in Theorem 3.1 is well known, we refer the reader, e.g., to the monograph [13, Chapter 7].
Pressure Regularity
It is a remarkable fact that the pressure \(\pi \) has additional timeregularity in some special cases.
Proposition 3.3
In the situation of Theorem 3.1, assume further
Then, \(P_0\pi \in {_0}H_{p,\mu }^\alpha ({{\mathbb {R}}}_+;L_q(\Sigma ))\), for \(\alpha \in (0,1/2]\), where
for \(v\in L_1(\Sigma )\). Furthermore, there exists a constant \(C>0\) such that the estimate
is valid.
Proof
Let \(\phi \in L_{q'}(\Sigma )\), \(1/q+1/q'=1\) and solve the equation
Here, \(\Delta _{\Sigma }^L\) denotes the (scalar) LaplaceBeltrami operator on \(\Sigma \). This yields a unique solution \(\psi \in H_{q'}^2(\Sigma )\) with
This follows, for instance, from [13, Theorem 6.4.3 (i)], and the fact 0 is in the resolvent set of \(\Delta _{\Sigma }^L\), acting on functions with zero average. We then obtain from the surface divergence theorem (A.14), (A.17), Proposition A.2 (a), and the fact that \((\mathrm{div}_\Sigma f,g)=0\)
where \((\cdot \cdot )_\Sigma \) denotes the inner product in \(L_2(\Sigma )\) or \(L_2(\Sigma ,{\mathsf {T}}\Sigma )\).
Noting that \({{\mathcal {D}}}_\Sigma (u)\in {_0}H^{1/2}_{p,\mu }({{\mathbb {R}}}_+; L_q(\Sigma ))\), we may apply the fractional timederivative \(\partial _t^\alpha \) to the result
since \(\partial _t^\alpha \) and \(L_\Sigma \) commute. This yields the claim. \(\square \)
Without loss of generality, we may always assume that \((\mathrm{div}_\Sigma f,g,u_0)=0\). To see this, let \((u,\pi )\) be a solution of (3.1) and solve the parabolic problem
by [13, Theorem 6.4.3 (ii)] to obtain a unique solution \(v\in {\mathbb {E}}_{1,\mu }(\Sigma )\). Next, we solve \(\Delta _{\Sigma }^L\Phi =\mathrm{div}_{\Sigma } vg\) in \(\dot{ H}_q^{1}(\Sigma )\) by [13, Theorem 6.4.3 (i)] to obtain a solution \(\Phi \) such that \(\nabla _\Sigma \Phi \) is unique with regularity
Note that \(\nabla _\Sigma \Phi (0)=0\) by the compatibility condition \(\mathrm{div}_\Sigma u_0=g(0)\). Define
where \(\nabla _\Sigma \psi \in L_q(\Sigma ,{\mathsf {T}}\Sigma )\) is the unique solution of
Then, \(({\tilde{u}},{\tilde{\pi }})\) solves (3.1) with \((\mathrm{div}_\Sigma f,g,u_0)=0\).
Of course, the converse is also true. If \(({\tilde{u}},{\tilde{\pi }})\) solves (3.1) with \((\mathrm{div}_\Sigma f,g,u_0)=0\), then one may construct a solution \((u,\pi )\) 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 \((u,\pi )\) be a solution of (3.1) with \((\mathrm{div}_\Sigma f,g,u_0)=0\).
By compactness of \(\Sigma \), there exists a family of charts \(\{(U_k,\varphi _k):k\in \{1,\ldots ,N\}\}\) such that \(\{U_k\}_{k=1}^N\) is an open covering of \(\Sigma \). Let \(\{\psi _k\}_{k=1}^N\subset C^\infty (\Sigma )\) be a partition of unity subordinate to the open covering \(\{U_k\}_{k=1}^N\). Note that without loss of generality, we may assume that \(\varphi _k(U_k)=B_{{\mathbb {R}}^d}(0,r)\). We call \(\{(U_k,\varphi _k,\psi _k):k\in \{1,\ldots ,N\}\}\) a localization system for \(\Sigma \).
Let \(\{\tau _{(k)j}(p)\}_{j=1}^d\) denote a local basis of the tangent space \({\mathsf {T}}_p\Sigma \) of \(\Sigma \) at \(p\in U_k\) and denote by \(\{\tau _{(k)}^j(p)\}_{j=1}^d\) the corresponding dual basis of the cotangent space \(\mathsf{T}^*_p\Sigma \) at \(p\in U_k\). Accordingly, we define \(g_{(k)}^{ij}=(\tau _{(k)}^i\tau _{(k)}^j)\) and \(g_{(k)ij}\) is defined in a very similar way, see also Appendix. Then, with \({\bar{u}}=u\circ \varphi _k^{1}\), \({\bar{\pi }}=\pi \circ \varphi _k^{1}\) and so on, the system (3.1) with respect to the local charts \((U_k,\varphi _k)\), \(k\in \{1,\ldots ,N\}\), reads as follows.
where
\(\ell \in \{1,\ldots ,d\}\), \(B_{(k)}\) is a linear differential operator of order one and
Here, upon translation and rotation, \({\bar{g}}_{(k)}^{ij}(0)=\delta _j^i\) and the coefficients have been extended in such a way that \({\bar{g}}_{(k)}^{ij}\delta _j^i_{L_\infty ({\mathbb {R}}^d)}\le \eta \), where \(\eta >0\) can be made as small as we wish, by decreasing the radius \(r>0\) of the ball \(B_{{\mathbb {R}}^d}(0,r)\).
In order to handle system (3.3), we define vectors in \({{\mathbb {R}}}^d\) as follows:
and
Moreover, we define the matrix \(G_{(k)}=({\bar{g}}_{(k)}^{ij})_{i,j=1}^d\in {{\mathbb {R}}}^{d\times d}\). With these notations, system (3.3) reads as
For each \(k\in \{1,\ldots ,N\}\), we define operators \(L_{k,\omega }\) 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 fullspace 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 \({{\mathbb {R}}}^{d\times d}\). With the operator \(L_{k,\omega }\) at hand, we may rewrite (3.5) in the more condensed form
Next, we remove the term \(H_{(k)}\), since it is not of lower order. For that purpose, solve the equation \(\mathrm{div}( G_{(k)}\nabla \phi _k)=H_{(k)}({\bar{u}})\). Since \(\int _{{{\mathbb {R}}}^d}H_{(k)}({\bar{u}})\mathrm{d}x=0\) (\(H_{(k)}({\bar{u}})\) is compactly supported), there exists a solution \(\phi _k\) such that \(\nabla \phi _k\) is unique, with regularity
Moreover, we have the estimates
see also [13, (7.41)]. Define
where \(({u}_{(k)}^0,{\pi }_{(k)}^0)=L_{k,\omega }^{1}({\bar{f}}_{(k)},0)\) and \(\Phi _k\) satisfies \(\mathrm{div}( G_{(k)}\nabla \Phi _k)=\mathrm{div}{\tilde{F}}_{(k)}({\bar{u}},{\bar{\pi }})\) in \({\dot{H}}_{q}^{1}({{\mathbb {R}}}^d)\), with
The couple \(({\tilde{u}}_{(k)},{\tilde{\pi }}_{(k)})\) then solves the equation
We note on the go that \(\mathrm{div}({\tilde{F}}_{(k)}({\bar{u}},{\bar{\pi }})G_{(k)}\nabla \Phi _k)=0\) and that the pressure \({\tilde{\pi }}_{(k)}\) 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 \(\omega \) into account. Set
and similarly for \(\Vert (u,\pi )\Vert _\omega \) on \(\Sigma \).
For each \(k\in \{1,\ldots ,N\}\), there exists \(\omega _0>0\) such that the operator \(L_{k,\omega }\) has the property of \(L_p\)\(L_q\)maximal regularity, provided \(\omega >\omega _0\). In particular, there exists a constant \(C>0\) such that
and
where for the last inequality, we made use of Proposition 3.3, implying the estimate
for some constant \(\gamma >0\), by interpolation between
In the same way, making also use of (3.6) and the definition of \(\Phi _k\), we obtain
Furthermore, we have
by (3.7), Proposition 3.3 for \({\bar{\pi }}\) and the Poincaré inequality for \(\pi _{(k)}^0\) and \(\Phi _k\), since we may assume without loss of generality, that \(\pi _{(k)}^0\) as well as \(\Phi _k\) have mean value zero on \(B_{{\mathbb {R}}^d}(0,r)\). By interpolation with (3.6), this yields
In conclusion, we obtain the estimate
valid for each \(k\in \{1,\ldots ,N\}\) and \(C>0\) does not depend on \(\omega >0\). Here, \(\{\chi _k\}_{k=1}^N\subset C^\infty (\Sigma )\) such that \(\chi _k=1\) on \(\mathrm{supp}(\psi _k)\) and \(\mathrm{supp}(\chi _k)\subset U_k\). As usual, we have set \({\bar{\chi }}_k=\chi _k\circ \varphi ^{1}_k\).
For the components \({\bar{u}}_{(k)}^\ell \circ \varphi _k\) of the vector \({\bar{u}}_{(k)}\circ \varphi _k\in {{\mathbb {R}}}^d\), we derive from (3.4)
hence
Since \(\chi _k=1\) on \(\mathrm{supp}(\psi _k)\), this finally yields the estimate
valid for all \(\omega >\omega _0\). Choosing \(\omega _0>0\) 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 \((\mathrm{div}_\Sigma f,g,u_0)=0\). Solve the parabolic problem
by [13, Theorem 6.4.3 (ii)] to obtain a unique solution \(v\in {_0}{\mathbb {E}}_{1,\mu }(\Sigma )\). Next, we solve \(\Delta _{\Sigma }^L\phi =\mathrm{div}_{\Sigma } v\) by [13, Theorem 6.4.3 (i)] to obtain a solution \(\phi \) such that \(\nabla _\Sigma \phi \) is unique with regularity
Define \({\tilde{u}}=v\nabla _\Sigma \phi \) and \({\tilde{\pi }}=(\partial _t+\omega )\phi \). It follows that
where
is defined by
Making use of local coordinates, one can show that
where \({\mathcal {A}}_\Sigma \) is a secondorder operator. Setting \({\hat{u}}={\tilde{u}}\), \({\hat{\pi }}={\tilde{\pi }}\mu _s\Delta _{\Sigma }^L\phi \) and \(Sf:=({\hat{u}},{\hat{\pi }})\), we obtain
with \(Rf:=\mu _s{\mathcal {A}}_\Sigma \phi \). Since \({\mathcal {A}}_\Sigma \) is of second order, this yields
where the constant \(C>0\) does not depend on \(\omega \). We note that \((\Delta _\Sigma ^L)^{1}\), acting on functions with average zero, is well defined. A Neumann series argument implies that \((I+R)\) is invertible provided \(\omega >0\) is sufficiently large. Hence, the operator \(S(I+R)^{1}\) is a right inverse for \(L_\omega \), which means that \(L_\omega \) is surjective. This completes the proof of Theorem 3.1.
The surface Stokes operator
By Proposition A.2, \({\mathcal {P}}_\Sigma \mathrm{div}_\Sigma {\mathcal {D}}_\Sigma (u)\) is a lower perturbation of \(\Delta _\Sigma u\) if \(\mathrm{div}_\Sigma u\) is prescribed. This implies the following result for the system
Corollary 3.4
Under the assumptions in Theorem 3.1, there exists \(\omega _0>0\) such that for each \(\omega >\omega _0\), problem (3.9) admits a unique solution
if and only if the data \((f,g,u_0)\) are subject to the conditions (1)–(4) in Theorem 3.1. Moreover, the solution \((u,\pi )\) depends continuously on the given data \((f,g,u_0)\) in the corresponding spaces.
Proof
Without loss of generality, we may assume \((\mathrm{div}_\Sigma f,g,u_0)=0\). With the operator \(L_\omega \) 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 \(\omega >0\). By Theorem 3.1, the solution depends continuously on the data; hence, there exists a constant \(M=M(\omega _0)>0\) such that
Therefore, a Neumann series argument yields the claim, if \(\omega >0\) is chosen sufficiently large. \(\square \)
We will now define the Stokes operator on surfaces. Let \(P_{H,\Sigma }\) denote the surface Helmholtz projection, defined by
where \(\nabla _\Sigma \psi \in L_q(\Sigma ,{\mathsf {T}}\Sigma )\) is the unique solution of
We note that \((P_{H,\Sigma }uv)_\Sigma =(uP_{H,\Sigma } v)_\Sigma \) for all \(u\in L_q(\Sigma ,{\mathsf {T}}\Sigma )\), \(v\in L_{q'}(\Sigma ,{\mathsf {T}}\Sigma )\), which follows directly from the definition of \(P_{H,\Sigma }\) (and for smooth functions from the surface divergence theorem (A.14)). Define
and \(X_1:=H_q^2(\Sigma ,{\mathsf {T}}\Sigma )\cap L_{q,\sigma }(\Sigma ,{\mathsf {T}}\Sigma )\). The surface Stokes operator is defined by
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 \(P_{H,\Sigma }\), (3.9) with \((\mathrm{div}_\Sigma f,g)=0\) is equivalent to the equation
By Corollary 3.4, the operator \(A_{S,\Sigma }\) has \(L_{p}\)maximal regularity, hence \(A_{S,\Sigma }\) 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 \((X_0,X_1)_{\alpha ,p}\) and \((X_0,X_1)_\alpha \), respectively. To this end, let \(A_\Sigma u:=2\mu _s{{\mathcal {P}}}_\Sigma \mathrm{div}_\Sigma {{\mathcal {D}}}_\Sigma (u)\) with domain \(D(A_\Sigma ):=H_q^2(\Sigma ,{\mathsf {T}}\Sigma )\) and define a linear mapping Q on \(D(A_\Sigma )\) by
for some fixed and sufficiently large \(\omega >0\).
Then, \(Q:D(A_\Sigma )\rightarrow X_1\) is a bounded projection, as \(Qu\in X_1\) and
for all \(u\in D(A_\Sigma )\). Furthermore, \(Q_{X_1}=I_{X_1}\) and therefore \(Q:D(A_\Sigma )\rightarrow X_1\) is surjective. By a duality argument, there exists some constant \(C>0\) such that
for all \(u\in D(A_\Sigma )\). In fact,
implies
for all \(u\in D(A_\Sigma )\) and \(\phi \in L_{q'}(\Sigma ,{\mathsf {T}}\Sigma )\), with
Since \(D(A_\Sigma )\) is dense in \(L_q(\Sigma ,{\mathsf {T}}\Sigma )\), there exists a unique bounded extension \({\tilde{Q}}:L_{q}(\Sigma ,{\mathsf {T}}\Sigma )\rightarrow X_0\) of Q. Clearly, \({\tilde{Q}}\) is a projection and as \(X_1\) is dense in \(X_0\), \({\tilde{Q}}_{X_0}=I_{X_0}\).
It follows that
since \({\tilde{Q}}D(A_\Sigma )=D(A_\Sigma )\cap R({\tilde{Q}})=D(A_{S,\Sigma })=X_1\). Moreover, with the help of the projection \({\tilde{Q}}\) and the relation \(R({\tilde{Q}})=L_{q,\sigma }(\Sigma ,{\mathsf {T}}\Sigma )\), we may now compute
as well as
for \(\alpha \in (0,1)\) and \(p\in (1,\infty )\), see [25, Theorem 1.17.1.1].
Nonlinear wellposedness
We will show that there exists a unique localintime solution to (1.5). Observe that the semilinear problem (1.5) is equivalent to the abstract semilinear evolution equation
where \(F_\Sigma (u):=P_{H,\Sigma }{\mathcal {P}}_\Sigma (u\cdot \nabla _\Sigma u)\). In order to solve this equation in the maximal regularity class \({\mathbb {E}}_{1,\mu }(\Sigma )\), we will apply Theorem 2.1 in [9]. To this end, let \(q\in (1,d)\) and
with \(2/p+d/q<3\), so that \(\mu _c\in (1/p,1)\). We will show that for each \(\mu \in (\mu _c,1]\) there exists \(\beta \in (\mu 1/p,1)\) with \(2\beta 1<\mu 1/p\) such that \(F_\Sigma \) satisfies the estimate
for all \(u,v\in X_\beta :=(X_0,X_1)_{\beta ,p}\).
By Hölders inequality, the estimate
holds. We choose \(r,r'\in (1,\infty )\) in such a way that
which is feasible if \(q\in (1,d)\). Next, by Sobolev embedding, we have
provided
The condition \(\beta <1\) requires \(q>d/3\), hence \(q\in (d/3,d)\). Note that
since \(\mu >\mu _c\). This implies that \(1>\beta >(d/q+1)/4\) can be chosen in such a way that the inequalities \(2\beta 1<\mu 1/p\) and \(\mu 1/p<\beta \) are satisfied.
In case \(q\ge d\), we may choose any \(\mu \in (1/p,1]\), since
provided \(2\beta >1\).
Since \(F_\Sigma \) is bilinear, it follows that the estimate (3.15) holds and, moreover, that \(F_\Sigma \in C^\infty (X_\beta ,X_0)\). Therefore, Theorem 2.1 in [9] yields the following result.
Theorem 3.5
Let \(p,q\in (1,\infty )\). Suppose that one of the following conditions holds:

(a)
\(q\in (d/3,d)\), \(2/p+d/q<3\) and \(\mu \in (\mu _c,1],\) where \(\mu _c\) is defined in (3.14).

(b)
\(q\ge d\) and \(\mu \in (1/p,1]\).
Then, for any initial value \(u_0\in B_{qp}^{2\mu 2/p}(\Sigma ,{\mathsf {T}}\Sigma )\cap L_{q,\sigma }(\Sigma ,{\mathsf {T}}\Sigma )\), there exists a number \(a=a(u_0)>0\) such that (1.5) admits a unique solution
Moreover,
Remark 3.6

(a)
The number \(\mu _c\in (1/p,1]\) defined in (3.14) is called the critical weight and was introduced in [14,15,16]. It has been shown in [14] that the ‘critical spaces’ \((X_0,X_1)_{\mu _c1/p,p}\) correspond to scaling invariant spaces in case the underlying equations enjoy scaling invariance.

(b)
In future work, we plan to show that \(A_{S,\Sigma }\) has a bounded \(H^\infty \)calculus. Then, one may set \(\mu =\mu _c\) in Theorem 3.5, thereby obtaining wellposedness in critical spaces.

(c)
In case \(d=2\), global existence has been obtained by Taylor [23, Proposition 6.5]. An alternative proof can be based on the approach via critical spaces mentioned above.
Stability of equilibria, examples
Consider the semilinear evolution equation
in \(X_0=L_{q,\sigma }(\Sigma ,{\mathsf {T}}\Sigma )\). Define the set
We show that the set \({{\mathcal {E}}}\) corresponds exactly to the set of equilibria for (4.1). To this end, let \(u_*\) be an equilibrium of (4.1), i.e., \(u_*\in X_1\) satisfies \(A_{S,\Sigma }u_*=F_\Sigma (u_*)\). Multiplying this equation by \(u_*\) and integrating over \(\Sigma \) yields
By Lemma A.1 and the surface divergence theorem (A.14), the last term vanishes, since \(\mathrm{div}_\Sigma u_*=0\). Furthermore, (A.17) and again (A.14) show that
which implies \({\mathcal {D}}_\Sigma (u_*)=0\), hence \(u_*\in {{\mathcal {E}}}\).
Conversely, let \(u_*\in {{\mathcal {E}}}\) be given. Then, \(A_{S,\Sigma }u_*=0\) and from Lemma A.1, we obtain that
Summarizing, we have shown that
Observe that the set \({{\mathcal {E}}}\) is a linear manifold, consisting exactly of the Killing fields on \(\Sigma \), see Remark A.3.
Define an operator \(A_0:X_1\rightarrow X_0\) by
where \(F_\Sigma '(u_*)v:=P_{H,\Sigma }{\mathcal {P}}_\Sigma \left( v\cdot \nabla _\Sigma u_*+u_*\cdot \nabla _\Sigma v\right) \). This operator is the full linearization of (4.1) at the equilibrium \(u_*\in {{\mathcal {E}}}\). We collect some properties of \(A_0\) in the following
Proposition 4.1
Suppose \(u_*\in {{\mathcal {E}}}\) 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
If \({{\mathcal {E}}}\ne \{0\}\), then \(u_*\in {{\mathcal {E}}}\) is normally stable, i.e.,

(i)
\(\mathrm{Re}\ \sigma (A_0)\le 0\) and \(\sigma (A_0)\cap i{{\mathbb {R}}}=\{0\}\).

(ii)
\(\lambda =0\) is a semisimple eigenvalue of \(A_0\).

(iii)
The kernel \(N(A_0)\) is isomorphic to \({\mathsf {T}}_{u_*}{{\mathcal {E}}}\).
In case \({{\mathcal {E}}}=\{0\}\), it holds that \(\mathrm{Re}\ \sigma (A_0)<0\).
Proof
By Sect. 3.4, the surface Stokes operator \(A_{S,\Sigma }\) has the property of \(L_p\)maximal regularity in \(X_0\), hence \(A_{S,\Sigma }\) is \({\mathcal {R}}\)sectorial in \(X_0\). Furthermore, the linear mapping \([v\mapsto F_\Sigma '(u_*)v]\) is relatively bounded with respect to \(A_{S,\Sigma }\). 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 \(\sigma (A_0)\) is discrete and consists solely of eigenvalues of \(A_0\) having finite algebraic multiplicity.
We first consider the case \({\mathcal {E}}\ne \{0\}\). Let \(\lambda \in \sigma (A_0)\) and denote by \(v\in X_1\) a corresponding eigenfunction. Multiplying the equation \(\lambda v+A_0v=0\) by the complex conjugate \({\bar{v}}\) and integrating over \(\Sigma \) yields
Here, we used the identities
and
since \({\mathcal {D}}_\Sigma (u_*)=0\) and \(\mathrm{div}_\Sigma u_*=0\), employing the surface divergence theorem (A.14). It follows that \(\mathrm{Re}\lambda \le 0\) and if \(\mathrm{Re}\lambda =0\), then \({\mathcal {D}}_\Sigma (v)=0\). Observe that the equations \({\mathcal {D}}_\Sigma (v)=0={\mathcal {D}}_\Sigma (u_*)\) then lead to the identity
hence \(F_\Sigma '(u_*)v=P_{H,\Sigma }(\nabla _\Sigma (u_*v))=0\) and therefore \(A_0v=0\).
The above calculations show that \(\sigma (A_0)\cap i{{\mathbb {R}}}=\{0\}\) and
wherefore \(N(A_0)\cong {{\mathcal {E}}}\cong {\mathsf {T}}_{u_*}{{\mathcal {E}}}\).
We will now prove that \(\lambda =0\) is semisimple. To this end, it suffices to prove that \(N(A_0^2)\subset N(A_0)\). Let \(w\in N(A_0^2)\) and \(v:=A_0w\). Then, \(v\in N(A_0)\) and we obtain
by Lemma A.1, (A.17) and the property \({\mathcal {D}}_\Sigma (v)=0\), which implies \(w\in N(A_0)\).
Finally, we consider the case \({\mathcal {E}}=\{0\}\). If \(\lambda \in \sigma (A_0)\) with eigenfunction \(v\ne 0\), it follows from (4.4) that \({\text {Re}}\lambda \le 0\) and if \({\text {Re}}\lambda =0\), then \({\mathcal {D}}_\Sigma (v)=0\) by (4.4), hence \(v\in {\mathcal {E}}=\{0\}\), a contradiction. Therefore, in this case, \(\mathrm{Re}\ \sigma (A_0)<0\). \(\square \)
Remark 4.2
The above computations show that the operator \(A_0\) from (4.3) is not necessarily symmetric in case \(u_*\ne 0\). In fact, we have
Since \(F_\Sigma \) is bilinear, we obtain from [13, Theorem 5.3.1] and the proof of Proposition 5.1 as well as Theorem 5.2 in [10] the following result.
Theorem 4.3
Suppose p, q and \(\mu \) satisfy the assumptions of Theorem 3.5.
Then, each equilibrium \(u_*\in {{\mathcal {E}}}\) is stable in \(X_{\gamma ,\mu }:=B_{qp}^{2\mu 2/p}(\Sigma ,{\mathsf {T}}\Sigma )\cap L_{q,\sigma }(\Sigma ,{\mathsf {T}}\Sigma )\) and there exists \(\delta >0\) such that the unique solution u(t) of (4.1) with initial value \(u_0\in X_{\gamma ,\mu }\) satisfying \(u_0u_*_{X_{\gamma ,\mu }}<\delta \) exists on \({{\mathbb {R}}}_+\) and converges at an exponential rate in \(X_{\gamma ,1}\) to a (possibly different) equilibrium \(u_\infty \in {{\mathcal {E}}}\) as \(t\rightarrow \infty \).
Existence of equilibria
According to (4.2), see also Proposition 2.2, the equilibria \({{\mathcal {E}}}\) of the evolution equation (4.1) correspond to the Killing fields of \(\Sigma \).
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 \(\Sigma \) is. (In fact, it turns out that the Killing fields on a Riemannian manifold form a sub Liealgebra of the Liealgebra 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 Liealgebra of Killings fields is isometric to the Liealgebra of I(M, g), see for instance Corollary III.6.3 in [20].
It then follows from [20, Proposition III.6.5] that \(\mathrm{dim}\,{{\mathcal {E}}}\le d(d+1)/2,\) where d is the dimension of \(\Sigma \). For compact manifolds, equality holds if and only if \(\Sigma \) is isometric to \({\mathbb {S}}^d\), the standard ddimensional Euclidean sphere in \({{\mathbb {R}}}^{d+1}\).
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 twodimensional Riemannian manifold with negative Gaussian curvature then any Killing field is 0.
Example 4.4

(a)
Let \(\Sigma ={\mathbb {S}}^2\). Then, \(\mathrm{dim}\,{{\mathcal {E}}}=3\) and each equilibrium \(u_* \in {{\mathcal {E}}}\) corresponds to a rotation about an axis spanned by a vector \(\upomega =(\upomega _1,\upomega _2,\upomega _3)\in {{\mathbb {R}}}^3\). Therefore, \(u_*\in {{\mathcal {E}}}\) is given by
$$\begin{aligned} u_*(x)=\upomega \times x,\quad x\in {{\mathbb {S}}}^2, \end{aligned}$$for some \(\upomega \in {{\mathbb {R}}}^3\). According to Theorem 4.3, each equilibrium \(u_*=\upomega \times x \) is stable and each solution u of (4.1) that starts out close to \(u_*\) converges at an exponential rate toward a (possibly different) equilibrium \(u_\infty =\upomega _\infty \times x\) for some \(\upomega _\infty \in {{\mathbb {R}}}^3\).

(b)
Suppose \(\Sigma ={{\mathbb {T}}}^2\), say with parameterization
$$\begin{aligned} \begin{aligned} x_1&=(R+r\cos \phi )\cos \theta \\ x_2&=(R+r\cos \phi )\sin \theta \\ x_3&=r\sin \phi , \end{aligned} \end{aligned}$$(4.5)where \(\phi ,\theta \in [0,2\pi )\) and \(0<r<R\). Then, one readily verifies that the velocity field \(u_*=\omega e_3\times x\), with \(\omega \in {{\mathbb {R}}}\), is an equilibrium. Hence, the fluid on the torus rotates about the \(x_3\)axis with angular velocity \(\omega \). According to Theorem 4.3, all of these equilibria are stable.
With the above parameterization, one shows that
$$\begin{aligned} K=\frac{\cos \phi }{r(R+r\cos \phi )}, \end{aligned}$$where K is the Gauss curvature. By Gauss’ Theorema Egregium, K is invariant under (local) isometries, and this implies that rotations around the \(x_3\)axis are the only isometries on \({\mathbb {T}}^2\).
References
 1.
M. Arnaudon, A.B. Cruzeiro, Lagrangian Navier–Stokes diffusions on manifolds: variational principle and stability. Bulletin des Sciences Mathématiques 136 (2012), 857–881.
 2.
D. Bothe, J. Prüss, On the twophase NavierStokes equations with BoussinesqScriven surface fluid. J. Math. Fluid Mech. 12 (2010),133150.
 3.
J. Boussinesq, Sur léxistence d‘une viscosité superficielle, dans la mince couche de transition séparant un liquide d’une autre fluide contigu. Ann. Chim. Phys. 29 (1913), 349–357.
 4.
C.H. Chan, M. Czubak, M. Disconzi, The formulation of the NavierStokes equations on Riemannian manifolds. J. Geom. Phys. 121 (2017), 335346.
 5.
D.G. Ebin, J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid. Annals of Mathematics 92 (1970), 102163.
 6.
M. Hieber, J. Saal, The Stokes equation in the \(L^p\)setting: wellposedness and regularity properties. In Handbook of mathematical analysis in mechanics of viscous fluids, 117206, Springer, Cham, 2018.
 7.
T. Jankuhn, M.A. Olshanskii, A. Reusken, Incompressible fluid problems on embedded surfaces: modeling and variational formulations. Interfaces Free Bound. 20 (2018), 353377.
 8.
H. Koba, C. Liu, Y. Giga, Energetic variational approaches for incompressible fluid systems on an evolving surface. Quart. Appl. Math. 75 (2017), 359389.
 9.
J. LeCrone, J. Prüss, and M. Wilke. On quasilinear parabolic evolution equations in weighted \(L_p\)spaces II. J. Evol. Equ., 14(3), 509–533, 2014.
 10.
G. Mazzone, J. Prüss, G. Simonett, A maximal regularity approach to the study of motion of a rigid body with a fluidfilled cavity. J. Math. Fluid Mech. 21 (2019), Paper No. 44.
 11.
A. Mazzucato, BesovMorrey spaces: function space theory and applications to nonlinear PDE. Trans. Amer. Math. Soc. 355 (2003), 12971364.
 12.
M.A. Olshanskii, A. Quaini, A. Reusken, V. Yushutin, A finite element method for the surface Stokes problem. SIAM J. Sci. Comput. 40 (2018), A2492–A2518.
 13.
J. Prüss, G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations. Monographs in Mathematics 105, Birkhäuser 2016.
 14.
J. Prüss, G. Simonett M. Wilke, Critical spaces for quasilinear parabolic evolution equations and applications. J. Differential Equations 264 (2018), 20282074 .
 15.
J. Prüss, M. Wilke, Addendum to the paper “On quasilinear parabolic evolution equations in weighted \(L_p\)spaces II”, J. Evol. Equ. 17 (2017), 13811388.
 16.
J. Prüss, M. Wilke, On critical spaces for the NavierStokes equations. J. Math. Fluid Mech. 20 (2018), 733755.
 17.
J. Prüss, M. Wilke, Gewöhnliche Differentialgleichungen und dynamische Systeme. Grundstudium Mathematik. Birkhäuser Verlag, Basel, 2019, 2nd edition.
 18.
A. Reusken,Y. Zhang, Numerical simulation of incompressible twophase flows with a BoussinesqScriven interface stress tensor. Internat. J. Numer. Methods Fluids 7 (2013), 1042–1058.
 19.
S. Reuther, A. Voigt, Solving the incompressible surface NavierStokes equation by surface elements, Phys. Fluids 30 (2018), 012107.
 20.
T. Sakai, Riemannian geometry. Translations of Mathematical Monographs, 149. American Mathematical Society, Providence, RI, 1996.
 21.
H. Samelson, Orientability of hypersurfaces in \({{\mathbb{R}}}^n\). Proc. Amer. Math. Soc. 22, 301–302 (1969).
 22.
L. E. Scriven, Dynamics of a fluid interface, Chem. Eng. Sci. 12 (1960), 98–108.
 23.
M. E. Taylor, Analysis on Morrey spaces and applications to NavierStokes and other evolution equations. Comm. Partial Differential Equations 17 (1992), 14071456.
 24.
R. Temam, Infinitedimensional dynamical systems in mechanics and physics. Springer, New York, 1988.
 25.
H. Triebel. Interpolation theory, function spaces, differential operators. NorthHolland Mathematical Library, 18. NorthHolland Publishing Co., AmsterdamNew York, 1978.
Acknowledgements
This is the last joint work with our close friend Jan Prüss, who passed away before this manuscript was completed. We are grateful for his input and ideas which are now incorporated in this manuscript. He is deeply missed. Lastly, we would also like to express our thanks to Marcelo Disconzi for helpful discussions.
Funding
Open Access funding enabled and organized by Projekt DEAL.
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Dedicated to Matthias Hieber on the occasion of his 60th birthday
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Gieri Simonett: Supported by a grant from the Simons Foundation (#426729).
Appendix A
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 \(\Sigma \) is a smooth, compact, closed (that is, without boundary) hypersurface embedded in \({{\mathbb {R}}}^{d+1}\). We mention on the go that these assumptions imply that \(\Sigma \) is orientable, see for instance [21].
Let \(\nu _\Sigma \) be the unit normal field of \(\Sigma \) (which is compatible with the chosen orientation). Then, the orthogonal projection \({{\mathcal {P}}}_\Sigma \) onto the tangent bundle of \(\Sigma \) is defined by \({{\mathcal {P}}}_\Sigma = I\nu _\Sigma \otimes \nu _\Sigma \).
We use the notation \(\{\tau _1(p), \cdots ,\tau _d(p)\}\) to denote a local basis of the tangent space \(\mathsf{T}_p\Sigma \) of \(\Sigma \) at p, and \(\{\tau ^1(p), \cdots ,\tau ^d(p)\}\) to denote the corresponding dual basis of the cotangent space \(\mathsf{T}^*_p\Sigma \) at p. Hence, we have \((\tau ^i(p)  \tau _j(p))=\delta ^i_j\), 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 \(g_{ij}=(\tau _i \tau _j),\) where \((\cdot  \cdot )\) is the Euclidean inner product of \({{\mathbb {R}}}^{d+1}\), and the dual metric \(g^*\) on the cotangent bundle \(\mathsf{T}^*\Sigma \) is given by \(g^{ij}=(\tau ^i\tau ^j)\). It holds that
Hence, g is induced by the inner product \((\cdot  \cdot )\), that is, we have
whereas
It holds that
where \(\Lambda ^k_{ij}\) 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 \(L_\Sigma \); that is, we have
and
If \(\varphi \in C^1(\Sigma , {{\mathbb {R}}})\), the surface gradient of \(\varphi \) is defined by \(\nabla _\Sigma \varphi =\partial _i\varphi \tau ^i\). If u is a \(C^1\)vector field on \(\Sigma \) (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 \(\nabla _i u\) of a tangential vector field is defined by \(\nabla _i u =P_\Sigma \partial _i u.\) Hence, we have
The surface divergence \(\mathrm{div}_\Sigma u\) for a (not necessarily tangential) vector field u is defined by
Then, the dfold mean curvature \(\kappa _\Sigma \) of \(\Sigma \) is given by
Hence, \(\kappa _\Sigma \) is the trace of \(L_\Sigma \) (which equals the sum of the principal curvatures). For a vector field \(u=v^j\tau _j+w\nu _\Sigma \), it follows from (A.9), (A.12) and the fact that \(\nu _\Sigma \) and \(\tau ^i\) are orthogonal
For a tangent vector field u and a scalar function \(\varphi \), the surface divergence theorem states that
For a tensor \(K=k^j_i \tau ^i\otimes \tau _j\), the surface divergence is defined by
Hence,
Lemma A.1
Suppose \(\varphi \) is a \(C^1\)scalar function and u, v, w are \(C^1\)tangential vector fields on \(\Sigma \). Then,

(a)
\(\mathrm{div}_\Sigma (\varphi {{\mathcal {P}}}_\Sigma ) = \nabla _\Sigma \varphi + \varphi \kappa _\Sigma \nu _\Sigma \).

(b)
\( (u\cdot \nabla _\Sigma v) = (\nabla _\Sigma v)^\mathsf{T} u\).

(c)
\(\nabla _\Sigma (uv)= (\nabla _\Sigma u)v + (\nabla _\Sigma v)u\).

(d)
\(\big (u \big \nabla _\Sigma (vw)\big ) =\big (u\cdot \nabla _\Sigma v \big  w \big ) + \big (u\cdot \nabla _\Sigma w \big  v \big ) .\)
Proof

(a)
It follows from (A.15) that
$$\begin{aligned} \mathrm{div}_\Sigma (\varphi {{\mathcal {P}}}_\Sigma ) =\partial _i (\varphi {{\mathcal {P}}}_\Sigma )\tau ^i =(\partial _i\varphi ) {{\mathcal {P}}}_\Sigma \tau ^i +\varphi \partial _i ({{\mathcal {P}}}_\Sigma )\tau ^i = \partial _i\varphi \tau ^i +\varphi \mathrm{div}_\Sigma {{\mathcal {P}}}_\Sigma . \end{aligned}$$The assertion is now a consequence of (A.16).

(b)
Using local coordinates, we obtain
$$\begin{aligned} u\cdot \nabla _\Sigma v = u^i\partial _i v= (\partial _i v\otimes \tau ^i)u=(\nabla _\Sigma v)^\mathsf{T}u. \end{aligned}$$ 
(c)
In local coordinates, \(\partial _i (uv) =(\partial _i u  v) +(u  \partial _i v)\). It is now easy to conclude that
$$\begin{aligned} \nabla _\Sigma (uv) = \partial _i (uv) \tau ^i = (\tau ^i \otimes \partial _i u) v + (\tau ^i \otimes \partial _i v)u = (\nabla _\Sigma u)v + (\nabla _\Sigma v)u. \end{aligned}$$ 
(d)
This follows from the assertions in (b) and (c).
\(\square \)
Let
Suppose \(u\in C^2(\Sigma , \mathsf{T}\Sigma )\), \(v\in C^1(\Sigma , \mathsf{T}\Sigma )\). Then, one shows that
where \({{\mathcal {D}}}_\Sigma (u):\nabla _\Sigma v=({{\mathcal {D}}}_\Sigma (u) \tau ^j  (\nabla _\Sigma v)^\mathsf{T} \tau _j)\), \({{\mathcal {D}}}_\Sigma (u)^2={{\mathcal {D}}}_\Sigma (u):{{\mathcal {D}}}_\Sigma (u)\), and \(\mathrm{tr}\,{{\mathcal {D}}}_\Sigma (u)= ({{\mathcal {D}}}_\Sigma (u) \tau ^j  \tau _j)\).
Proposition A.2
Suppose \(u\in C^2(\Sigma ,\mathsf{T}\Sigma )\).

(a)
Then, we have the following representation:
$$\begin{aligned} 2{{\mathcal {P}}}_\Sigma \,\mathrm{div}_\Sigma \, {{\mathcal {D}}}_\Sigma (u) =\Delta _\Sigma u+ \nabla _\Sigma \, \mathrm{div}_\Sigma u + (\kappa _\Sigma L_\Sigma  L^2_\Sigma )u, \end{aligned}$$where \(\Delta _\Sigma \) is the BochnerLaplacian (also called the conformal Laplacian), defined in local coordinates by
$$\begin{aligned} \Delta _\Sigma =g^{ij}(\nabla _i\nabla _j  \Lambda ^k_{ij}\nabla _k). \end{aligned}$$ 
(b)
It holds that
$$\begin{aligned} \begin{aligned}&(\kappa _\Sigma L_\Sigma  L^2_\Sigma ) u = \mathrm{Ric}_\Sigma u \\&(\kappa _\Sigma L_\Sigma  L^2_\Sigma ) u = K_\Sigma u\quad \text {in case d=2,} \end{aligned} \end{aligned}$$where \(\mathrm{Ric}_\Sigma \) is the Ricci tensor and \(K_\Sigma \) the Gaussian curvature of \(\Sigma \), respectively.
Proof

(a)
We note that in local coordinates, \({{\mathcal {D}}}_\Sigma (u)\) is given by
$$\begin{aligned} \begin{aligned} 2{{\mathcal {D}}}_\Sigma (u)&= \tau ^i\otimes P_\Sigma \partial _i u + P_\Sigma \partial _i u\otimes \tau ^i = \tau ^i\otimes \nabla _i u + \nabla _i u\otimes \tau ^i. \end{aligned} \end{aligned}$$(A.18)From (A.15) and the relation, \({{\mathcal {P}}}_\Sigma = I\nu _\Sigma \otimes \nu _\Sigma \) follows
$$\begin{aligned} \begin{aligned} {{\mathcal {P}}}_\Sigma \mathrm{div}_\Sigma ( \nabla _i u\otimes \tau ^i )&= {{\mathcal {P}}}_\Sigma \partial _j ( \tau ^i \otimes {{\mathcal {P}}}_\Sigma \partial _iu ) \tau ^j \\&= ({{\mathcal {P}}}_\Sigma \partial _j\tau ^i \otimes {{\mathcal {P}}}_\Sigma \partial _i u)\tau ^j + (\tau ^i\otimes \partial _j {{\mathcal {P}}}_\Sigma \partial _i u)\tau ^j\\&= (\partial _i u  \tau ^j){{\mathcal {P}}}_\Sigma \partial _j\tau ^i + \tau ^i\big (\partial _j(\partial _i u [\nu _\Sigma \otimes \nu _\Sigma ]\partial _i u)  \tau ^j\big ) \\&= (\partial _i u  \tau ^j){{\mathcal {P}}}_\Sigma \partial _j\tau ^i + \tau ^i (\partial _j\partial _i u\tau ^j)  \tau ^i(\partial _j\nu _\Sigma  \tau ^j)(\nu _\Sigma \partial _i u), \end{aligned} \end{aligned}$$where, in the last line, we employed the relation \((\nu _\Sigma  \tau ^j)=0\). Next, we observe that
$$\begin{aligned} \begin{aligned} (\partial _j\partial _i u\tau ^j)\tau ^i&= \partial _i (\partial _j u \tau ^j) \tau ^i  (\partial _j u  \partial _i \tau ^j)\tau ^i \\&= \nabla _\Sigma \mathrm{div}_\Sigma u  (\partial _j u \Lambda ^j_{ik}\tau ^k +l^j_i\nu _\Sigma )\tau ^i \\&= \nabla _\Sigma \mathrm{div}_\Sigma u (\partial _j u \tau ^k) {{\mathcal {P}}}_\Sigma \partial _k\tau ^j  l^j_i (\partial _j u  \nu _\Sigma )\tau ^i \\&= \nabla _\Sigma \mathrm{div}_\Sigma u (\partial _j u \tau ^k) {{\mathcal {P}}}_\Sigma \partial _k\tau ^j  L^2_\Sigma u, \end{aligned} \end{aligned}$$where we used (A.9) and the relations \(L_\Sigma \tau ^m = l^m_r\tau ^r\) as well as \(L_\Sigma \tau _m=l_{m r}\tau ^r\) to deduce
$$\begin{aligned} l^j_i(\partial _j u  \nu _\Sigma )\tau ^i =l^j_i l_{jk} u^k \tau ^i = L_\Sigma (u^k l_{jk}\tau ^j ) = L^2_\Sigma u_k\tau ^k =L^2_\Sigma u. \end{aligned}$$(A.19)$$\begin{aligned}  (\partial _j\nu _\Sigma  \tau ^j)(\nu _\Sigma \partial _i u)\tau ^i = l^j_j l_{ik} u^k \tau ^i = \kappa _\Sigma L_\Sigma u^k\tau _k = \kappa _\Sigma L_\Sigma u. \end{aligned}$$(A.20)Summarizing we have shown that
$$\begin{aligned} {{\mathcal {P}}}_\Sigma \mathrm{div}_\Sigma ( \nabla _i u\otimes \tau ^i ) = \nabla _\Sigma \mathrm{div}_\Sigma u + \kappa _\Sigma L_\Sigma u L^2_\Sigma u . \end{aligned}$$Moreover, we have
$$\begin{aligned} \begin{aligned} {{\mathcal {P}}}_\Sigma \mathrm{div}_\Sigma (\tau ^i \otimes \nabla _iu)&= {{\mathcal {P}}}_\Sigma \partial _j (\nabla _iu \otimes \tau ^i)\tau ^j \\&= g^{ij}{{\mathcal {P}}}_\Sigma \partial _j \nabla _iu + (\partial _j \tau ^i\tau ^j) \nabla _i u \\&= g^{ij}(\nabla _i \nabla _j u \Lambda ^k_{ij}\nabla _k u) =\Delta _\Sigma u. \end{aligned} \end{aligned}$$ 
(b)
The computations in (A.19), (A.20) show that in local coordinates
$$\begin{aligned} (\kappa _\Sigma L_\Sigma L^2_\Sigma )u=g^{jm}(l_{jm}l_{ik} l_{im} l_{jk})u^k \tau ^i. \end{aligned}$$By the Gauss equation, see for instance [20, Proposition II.3.8], this yields
$$\begin{aligned} \begin{aligned} (\kappa _\Sigma L_\Sigma L^2_\Sigma )u&=g^{jm}R_{jikm} u^k\tau ^i =R_{ik} u^k\tau ^i = \mathrm{Ric}_\Sigma u, \end{aligned} \end{aligned}$$where \(R_{jikm}\) are the components of the curvature tensor and \(R_{ik}\) the components of the Ricci (0, 2)tensor. In case that \(\Sigma \) is a surface embedded in \({{\mathbb {R}}}^3\), one obtains
$$\begin{aligned} (\kappa _\Sigma L_\Sigma L^2_\Sigma )u= K_\Sigma u, \end{aligned}$$where \(K_\Sigma \) is the Gauss curvature of \(\Sigma \). This can, for instance, be seen as follows:
$$\begin{aligned} \begin{aligned} (\kappa _\Sigma L_\Sigma L^2_\Sigma )u&=g^{jm}(l_{jm}l_{ik} l_{im} l_{jk})u^k \tau ^i \\&= (l^j_j l^k_il^j_i l^k_j)u_k\tau ^i = \det (L_\Sigma )\, \delta ^k_i u_k\tau ^i = K_\Sigma u. \end{aligned} \end{aligned}$$The proof of Proposition A.2 is now complete.
\(\square \)
Remark A.3

(a)
We note that in local coordinates,
$$\begin{aligned} \begin{aligned} 2{{\mathcal {D}}}_\Sigma (u) =(\partial _i u_j \Lambda ^k_{ij}u_k)\tau ^i\otimes \tau ^j + (\partial _i u_j \Lambda ^k_{ij}u_k)\tau ^j\otimes \tau ^i \end{aligned} \end{aligned}$$for \(u=u_j\tau ^j\in C^1(\Sigma , \mathsf{T}^*\Sigma )\) and
$$\begin{aligned} \begin{aligned} 2{{\mathcal {D}}}_\Sigma (u) =(\partial _i u^j +\Lambda ^j_{ik}u^k)\tau ^i\otimes \tau _j + (\partial _i u^j +\Lambda ^j_{ik}u^k)\tau _j\otimes \tau ^i \end{aligned} \end{aligned}$$for \(u=u^j\tau _j\in C^1(\Sigma , \mathsf{T}\Sigma )\).

(b)
Suppose u, v are tangential fields on \(\Sigma \). Then,
$$\begin{aligned} \nabla _v u := (\nabla _i u\otimes \tau ^i)v \end{aligned}$$(A.21)coincides with the Levi–Civita connection \(\nabla \) of \(\Sigma \).
We note that \(\nabla _i =\nabla _{\tau _i}=\nabla _{\frac{\partial \;}{\partial x^i}}\) and \(\nabla u= \nabla _i u\otimes \tau ^i\) in local coordinates.

(c)
A straightforward computation shows that in local coordinates
$$\begin{aligned} g^{ij}(\nabla _i\nabla _j u  \Lambda ^k_{ij}\nabla _ku) =g^{ij}(\nabla ^2 u)(\tau _i,\tau _j),\quad u\in C^2(\Sigma , \mathsf{T}\Sigma ). \end{aligned}$$Hence, \(\Delta _\Sigma =\mathrm{tr}_g\,(\nabla ^2 u)\).

(d)
Let \(\nabla \) be the Levi–Civita connection of \(\Sigma \). Then, it follows from (b) and (A.18)
$$\begin{aligned} {{\mathcal {D}}}_\Sigma (u)= \frac{1}{2}\left( \nabla u + [\nabla u]^\mathsf{T}\right) . \end{aligned}$$ 
(e)
Suppose u, v, w are \(C^1\)tangential fields. Employing (A.18) and (A.21), one readily verifies that
$$\begin{aligned} ({{\mathcal {D}}}_\Sigma (u)v  w) + ({{\mathcal {D}}}_\Sigma (u)wv)= (\nabla _v u  w) + (\nabla _w u  v). \end{aligned}$$We remind that a tangential field u on \(\Sigma \) is called a Killing field if
$$\begin{aligned} (\nabla _v u w) + (\nabla _w uv)=0\quad \text {for all tangential fields }v,w\text { on }\Sigma , \end{aligned}$$see for instance [20, Lemma III.6.1]. This implies for a \(C^1\)tangent field u
$$\begin{aligned} {{\mathcal {D}}}_\Sigma (u)=0\;\; \Longleftrightarrow \;\; u\text { is a Killing field}. \end{aligned}$$
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Prüss, J., Simonett, G. & Wilke, M. On the Navier–Stokes equations on surfaces. J. Evol. Equ. (2020). https://doi.org/10.1007/s00028020006480
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Mathematics Subject Classification
 35Q35
 35Q30
 35B40
Keywords
 Surface Navier–Stokes equations
 Boussinesq–Scriven surface stress tensor
 Killing vector fields
 Stability of equilibria