Abstract
We consider a smooth, compact and embedded hypersurface \(\Sigma \) without boundary and show that the corresponding (shifted) surface Stokes operator admits a bounded \(H^\infty \)-calculus with angle smaller than \(\pi /2\). As an application, we consider critical spaces for the Navier–Stokes equations on the surface \(\Sigma \). In case \(\Sigma \) is two-dimensional, we show that any solution with a divergence-free initial value in \(L_2(\Sigma , \textsf{T}\Sigma )\) exists globally and converges exponentially fast to an equilibrium, that is, to a Killing field.
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1 Introduction
Suppose \(\Sigma \) is a smooth, compact, connected, embedded hypersurface in \({\mathbb R}^{d+1}\) without boundary. We then consider the motion of an incompressible viscous fluid that completely covers \(\Sigma \) and flows along \(\Sigma \). The motion can be modeled by the surface Navier–Stokes equations for an incompressible viscous fluid
Here, the density \(\varrho \) is a positive constant, \({{\mathcal {T}}}_\Sigma =2\mu _s {{\mathcal {D}}}_\Sigma (u) -\pi {{\mathcal {P}}}_\Sigma \), and
is the surface rate-of-strain tensor, with u the fluid velocity and \(\pi \) the pressure. Moreover, \({{\mathcal {P}}}_\Sigma \) denotes the orthogonal projection onto the tangent bundle \(\textsf{T}\Sigma \) of \(\Sigma \), \(\textrm{div}_\Sigma \) the surface divergence, and \(\nabla _\Sigma \) 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 \(\Sigma \), and that all of these are normally stable. Moreover, it was shown that (1.1) can be reformulated as
where \(\Delta _\Sigma \) is the (negative) Bochner-Laplacian and \(\textsf{Ric}_\Sigma \) the Ricci tensor. To be more precise, in slight abuse of the usual convention, we interpret \(\textsf{Ric}_\Sigma \) here as the (1, 1)-tensor given in local coordinates by \(\textsf{Ric}^\ell _k= g^{i\ell }R_{ik}\). We remind that in case \(d=2\), \(\textsf{Ric}_\Sigma u=K_\Sigma u\), with \(K_\Sigma \) being the Gaussian curvature of \(\Sigma \).
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 \(\Sigma \), but not on the ambient space. In an intrinsic formulation, the surface Navier Stokes equations (1.2) can be stated as
where \(\nabla \) is the covariant derivative (induced by the Levi-Civita connection of \(\Sigma \)), and \(\textsf{grad}\,\pi =\nabla _\Sigma \pi \). 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 \(\Sigma \) without boundary. We remind that the Bochner Laplacian \(\Delta _\Sigma \) is related to the Hodge Laplacian by the formula \(\Delta _\Sigma = \Delta _H +\textsf{Ric}_\Sigma ,\) 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).
In Theorem 3.1, we show that the surface Stokes operator
with \(P_{\Sigma , H}\) being the surface Helmholtz projection, has the property that \((\omega + A_{S,\Sigma }) \) admits a bounded \(H^\infty \)-calculus in \(L_{q,\sigma }(\Sigma , \textsf{T}\Sigma )\) with \(H^\infty \)-angle \(\phi _{A_{S,\Sigma }}^\infty <\pi /2\), provided \(\omega \) is larger than the spectral bound of \(-A_{S,\Sigma }\) and \(1<q<\infty \).
For the Stokes operator on domains in Euclidean space under various boundary conditions, the existence of an \(H^\infty \)-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.
Having established the existence of a bounded \(H^\infty \)-calculus allows us to employ the results in Prüss, Simonett, and Wilke [23, 25] to establish existence and uniqueness of solutions to the system (1.1), or (1.3), for initial values \(u_0\) in the critical spaces \(B^{d/q-1}_{qp,\sigma }(\Sigma , \textsf{T}\Sigma )\), see Theorem 4.1 and Theorem 4.3.
In particular, our results imply existence and uniqueness of solutions for initial values \(u_0\in L_{q,\sigma }(\Sigma , \textsf{T}\Sigma )\) 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 \(u_0\in L_{2,\sigma }(\Sigma , \textsf{T}\Sigma )\) 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 \(d\ge 3\), 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 [16, 32] 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 \({{\mathcal {T}}}_\Sigma \) 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 \(\nabla \) denotes the covariant derivative (or the Levi-Civita connection) on \(\Sigma \). In the following, we will occasionally take the liberty to use the shorter notation \(\nabla _u v\) and \(\nabla u\) mentioned in (1.4). We recall that for sufficiently smooth vectors fields u, say \(u \in C^1(\Sigma ,\textsf{T}\Sigma )\), one has \(\nabla u\in C(\Sigma , \textsf{T}^1_1\Sigma )\), the space of all (1, 1)-tensors on \(\Sigma \). As the Levi-Civita connection \(\nabla \) is a metric connection, we have
for tangential vector fields u, v, w on \(\Sigma \), where \((u|v):=g(u,v)\) is the Riemannian metric (induced by the Euclidean inner product of \({\mathbb R}^{d+1}\) in case \(\Sigma \) is embedded in \({\mathbb R}^{d+1}\)). Occasionally, we also write \(\textsf{grad}\,\varphi \) in lieu of \(\nabla _\Sigma \varphi \) for scalar functions \(\varphi \).
We use the notation
whenever the right hand side exists, say for \(u\in L_q(\Sigma , \textsf{T}\Sigma )\) and \(v\in L_{q'}(\Sigma , \textsf{T}\Sigma )\), where \(1/q+1/q'=1\). For \(k\in {\mathbb N}\) and \(q\in (1,\infty )\), the space \(H^k_q(\Sigma , \textsf{T}\Sigma )\) is defined as the completion of \( C^\infty (\Sigma , \textsf{T}\Sigma )\), the space of all smooth vector fields, in \(L_{1,\textrm{loc}}(\Sigma , \textsf{T}\Sigma )\) with respect to the norm
The Bessel potential spaces \(H^s_q(\Sigma , \textsf{T}\Sigma )\) and the Besov spaces \(B^s_{qp}(\Sigma , \textsf{T}\Sigma )\) 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).
2 The Surface Stokes Operator
In [24, Corollary 3.4] we showed that there exists a number \(\omega _0>0\) such that for \(\omega >\omega _0\), the system
admits a unique solution
if and only if
Moreover, the solution \((u,\pi )\) depends continuously on the given data \((f,u_0)\) in the corresponding spaces.
Let \(P_{H,\Sigma }\) denote the surface Helmholtz projection, defined by
where \(\nabla _\Sigma \psi _v\in L_q(\Sigma ,\textsf{T}\Sigma )\) is the unique solution of
see Lemma A.1. We note that \((P_{H,\Sigma }u|v)_\Sigma =(u|P_{H,\Sigma } v)_\Sigma \) for all \(u\in L_q(\Sigma ,\textsf{T}\Sigma )\), \(v\in L_{q'}(\Sigma ,\textsf{T}\Sigma )\), which follows directly from the definition of \(P_{H,\Sigma }\) (and for smooth functions from the surface divergence theorem). Indeed,
as \(\psi _u\in \dot{H}_q^1(\Sigma )\), \(\psi _v\in \dot{H}_{q'}^1(\Sigma )\). Let
The surface Stokes operator is defined by
Making use of the projection \(P_{H,\Sigma }\), (2.1) with \(\textrm{div}_\Sigma f=0\) is equivalent to the equation
Indeed, if \((u,\pi )\) is a solution to (2.1), then \(u=P_{H,\Sigma }u\) solves (2.3) as can be seen by applying \(P_{H,\Sigma }\) to the first equation in (2.1). Conversely, let u be a solution of (2.3). Then, by definition of \(P_{H,\Sigma }\),
where \(\psi _v\in \dot{H}_q^1(\Sigma )\) solves
with \(v:= \mathcal {P}_\Sigma \textrm{div}_\Sigma \mathcal {D}_\Sigma (u)\in L_q(\Sigma )\). Defining \(\pi :=2\mu _s\psi _v\), we see that \((u,\pi )\) is a solution of (2.1).
In particular, the operator \(A_{S,\Sigma }\) has \(L_{p}\)-maximal regularity, hence \(-(\omega +A_{S,\Sigma })\) generates an analytic \(C_0\)-semigroup in \(X_0\) (see for instance [22, Proposition 3.5.2]) which is exponentially stable provided \(\omega >s(-A_{S,\Sigma })\), where \(s(\cdot )\) denotes the spectral bound of \(-A_{S,\Sigma }\). This readily implies that the operator \(\omega +A_{S,\Sigma }\) is sectorial with spectral angle \(\phi _{\omega +A_{S,\Sigma }}<\pi /2\).
3 \(H^\infty \)-calculus
In this section, we are going to prove the following result.
Theorem 3.1
Let \(q\in (1,\infty )\) and \(\Sigma \) be a smooth, compact, connected, embedded hypersurface in \({\mathbb R}^{d+1}\) without boundary. Let \(A_{S,\Sigma }\) be the surface Stokes operator in \(L_{q,\sigma }(\Sigma ,\textsf{T}\Sigma )\) defined in (2.2).
Then \(\omega +A_{S,\Sigma }\) admits a bounded \(H^\infty \)-calculus with \(H^\infty \)-angle \(\phi _{A_{S,\Sigma }}^\infty <\pi /2\) for each \(\omega >s(-A_{S,\Sigma })\), the spectral bound of \(-A_{S,\Sigma }\).
Remark 3.2
Let \({{\mathcal {E}}}\) denote the set of equilibria. We have shown in [24], Proposition 4.1, that \(s(-A_{S,\Sigma })=0\) provided \({{\mathcal {E}}}\ne \{0\}\) und \(s(-A_{S,\Sigma })<0\) if \({{\mathcal {E}}}=\{0\}\).
Therefore, it follows from Theorem 3.1 that for each \(\omega >0\) the operator \(\omega +A_{S,\Sigma }\) admits a bounded \(H^\infty \)-calculus with \(H^\infty \)-angle \(\phi _{A_{S,\Sigma }}^\infty <\pi /2\). In case \({{\mathcal {E}}}=\{0\}\) one may set \(\omega =0\).
3.1 Resolvent and Pressure Estimates
We consider the following resolvent problem
where \(\omega >s(-A_{S,\Sigma })\) and \(\lambda \in \Sigma _{\pi -\phi }\), \(\phi >\phi _{\omega +A_{S,\Sigma }}\). By sectoriality of the operator \(\omega +A_{S,\Sigma }\) (and since \(0\in \rho (\omega +A_{S,\Sigma })\)), it follows that for given \(f\in L_q(\Sigma ,\textsf{T}\Sigma )\) there exists a unique solution
of (3.1) and there is a constant \(C>0\) such that
for all \(\lambda \in \Sigma _{\pi -\phi }\). Note that without loss of generality, we may assume that \(P_0\pi =\pi \), where
for \(v\in L_1(\Sigma )\). Furthermore, if \({\text {div}}_\Sigma f=0\), the pressure \(\pi \) satisfies the estimate
for some constant \(C>0\). The proof of estimate (3.3) follows exactly the lines of the proof of [24, Proposition 3.3].
3.2 Localization
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 \(\{\psi _k^2\}_{k=1}^N\subset C^\infty (\Sigma )\) 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 \(\textsf{T}_p\Sigma \) 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 \(\textsf{T}^*_p\Sigma \) 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 the Appendix in [24]. 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
\(\ell \in \{1,\ldots ,d\}\), \(B_{(k)}\) collects all terms of order at most 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 \(\bar{g}_{(k)}^{ij}\in W_\infty ^2({\mathbb R}^d)\) and \(|\bar{g}_{(k)}^{ij}-\delta _j^i|_{L_\infty (\mathbb {R}^d)}\le \eta \), 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 (3.4), 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 (3.4) reads as
Let us remove the term \(H_{(k)}(\bar{u})\), since it is not of lower order. For that purpose, we solve the equation \(\textrm{div}( G_{(k)}\nabla \phi _k)=H_{(k)}(\bar{u})\) by Lemma A.2 to obtain a unique solution \(\nabla \phi _k\in H_q^2({\mathbb R}^d)^d\) with
where C is a positive constant. For this, observe that \(H_{(k)}(\bar{u})\) is compactly supported and we have \(\int _{{\mathbb R}^d}H_{(k)}(\bar{u})dx=0\). Therefore, \(H_{(k)}(\bar{u})\) induces a functional on \(\dot{H}_{q'}^1({\mathbb R}^d)\) with norm bounded by \(C|H_{(k)}(\bar{u})|_{L_q({\mathbb R}^d)}.\) To see this, choose \(R>0\) such that \(\textrm{supp}(H_{(k)}(\bar{u}))\subset B(0,R)\) and let \(\phi \in \dot{H}^1_{q'}({\mathbb R}^d).\) Then we have
where \(\hat{\phi }= |B(0,R)|^{-1} \int _{B(0,R)} \phi \,dx\). By the Poincaré-Wirtinger inequality,
Let
It follows from (3.5) that the functions \((\tilde{u}_{(k)}, \tilde{\pi }_{(k)})\) then solve the system
where \(\tilde{F}_{(k)}(\bar{u},\bar{\pi }):=F_{(k)}(\bar{u},\bar{\pi })+\mu _s (G_{(k)}\nabla |\nabla )(G_{(k)}\nabla \phi _k).\) In order to remove the pressure term in (3.7) we introduce the projections \(P_k^G\), defined by
Here, \(\nabla \Phi _k\in L_q({\mathbb R}^d)\) is the unique solution of \(\textrm{div}(G_{(k)}\nabla \Phi _k)=\textrm{div}\, v\) in \(\dot{H}_q^{-1}({\mathbb R}^d)\), established in Lemma A.2. It is readily seen that \(P_k^Gv=v\) if \(\textrm{div}\, v=0\) and \(P_k^G(G_{(k)}\nabla \tilde{\pi }_{(k)})=0\). Applying the projection \(P_k^G\) to equation (3.7) leads to
We claim that each of the operators \(\omega +A_{S,k}^G:=\omega -\mu _sP_k^G(G_{(k)}\nabla |\nabla )\) in (3.8) admits a bounded \(H^\infty \)-calculus in \(P_k^GL_q({\mathbb R}^d)=L_{q,\sigma }({\mathbb R}^d)\), provided \(\omega \) is sufficiently large. To see this, we write
where \(A_S=-\mu _sP_H\Delta u=-\mu _s\Delta u\) is the Stokes operator in \(L_{q,\sigma }({\mathbb R}^d)\) and \(P_H\) is the Helmholtz projection in \({\mathbb R}^d\).
Recall that each matrix \(G_{(k)}\) is a perturbation of the identity in \({\mathbb R}^{d\times d}\). Therefore,
where we may choose \(\eta >0\) as small as we wish. Furthermore,
by Lemma A.2, since
as \({\text {div}}\Delta u=\Delta {\text {div}} u=0\) in \({\mathbb R}^d\). As before, we may choose \(\eta >0\) as small as we wish.
Note that the shifted Stokes operator \(\omega +A_S\) admits a bounded \(H^\infty \)-calculus in \(L_{q,\sigma }({\mathbb R}^d)\) with angle \(\phi _{A_S}^\infty <\pi /2\), see e.g. [22, Theorem 7.1.2]. By the abstract perturbation result [7, Theorem 3.2] (see also [20, Section 3]), there exists \(\omega _0>0\) such that each of the operators \(\omega + A_{S,k}^G\) admits a bounded \(H^\infty \)-calculus in \(L_{q,\sigma }({\mathbb R}^d)\) if \(\omega \ge \omega _0\). Moreover, for any given \(\phi _0>\phi _{A_S}^\infty \) we may assume \(\phi _{A_{S,k}^G}^\infty \le \phi _0\), provided that \(|G_{(k)}-I|_\infty \) is sufficiently small.
This yields the following representation of the resolvent \(u=(\lambda +\omega +A_{S,\Sigma })^{-1} f\).
where \(\{e_\ell \}_{\ell =1}^d\) is the standard basis in \({\mathbb R}^d\) and
In a next step, we will estimate the term \(R(\lambda )f\) in \(L_{q}(\Sigma ,\textsf{T}\Sigma )\). To this end, observe that the operators \(P_{H,\Sigma }\) and \(P_k^G\) are bounded in \(L_{q}(\Sigma ,\textsf{T}\Sigma )\) and \(L_{q}({\mathbb R}^d)\), respectively. This, together with (3.2), (3.3), (3.6) and the fact that each of the operators \(\omega +A_{S,k}^G\) is sectorial in \(L_{q,\sigma }({\mathbb R}^d)\) for \(\omega \) 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 \(H_q^1(\Sigma )=[L_q(\Sigma ),H_q^2(\Sigma ))]_{1/2}\).
Next, observe that \(T:L_{q,\sigma }(\Sigma ,\textsf{T}\Sigma )\rightarrow H_q^1(\Sigma ,\textsf{T}\Sigma )\cap L_{q,\sigma }(\Sigma ,\textsf{T}\Sigma )\), since \(G_{(k)}\in W_\infty ^2({\mathbb R}^d)^{d\times d}\) and \(\nabla \phi _{k}\in H_q^1({\mathbb R}^d)^d\) for each \(k\in \{1,\ldots ,N\}\), hence T is compact in \(L_{q,\sigma }(\Sigma ,\textsf{T}\Sigma )\). Consequently, \(I-T\) is a Fredholm operator with index 0. In particular, \(\ker (I-T)\) is finite dimensional and the range \(\textrm{ran}(I-T)\) is closed in \(L_q(\Sigma )\). Let \(\{v_1,\ldots ,v_m\}\) be an orthonormal basis of \(\ker (I-T)\) and define
by
Then it can be readily checked that Q is a projection onto \(\ker (I-T)\) and it is continuous in \(L_{q,\sigma }(\Sigma ,\textsf{T}\Sigma )\) for any \(q\in (1,\infty )\), since \(v_k\in H_q^1(\Sigma ,\textsf{T}\Sigma )\) for any \(q\in (1,\infty )\) (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 \(\textrm{ran}(I-T)\), hence
Let \(\phi _0\in (\phi _{A_S}^\infty ,\pi /2)\) such that \(\phi _{A_{S,k}^G}^\infty \le \phi _0\) for each \(k\in \{1,\ldots ,N\}\). For \(h\in H_0(\Sigma _{\phi })\), \(\phi \in (\phi _0,\pi /2)\), we then obtain
with \(\Gamma =(\infty ,0]e^{-i(\pi -\theta )}\cup [0,\infty )e^{i(\pi -\theta )}\), \(\theta \in (\phi _0,\phi )\). Estimate (3.9) then yields
since each of the operators \(\omega +A_{S,k}^G\) has a bounded \(H^\infty \)-calculus. The remaining part \(Qh(\omega +A_{S,\Sigma })\) 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 \(\omega +A_{S,\Sigma }\) admits a bounded \(H^\infty \)-calculus in \(L_{q,\sigma }(\Sigma ,\textsf{T}\Sigma )\) with angle \(\phi _{A_{S,\Sigma }}^\infty <\pi /2\) provided \(\omega >0\) is large enough. An application of [22, Corollary 3.3.15] finally yields that it is enough to require \(\omega >s(-A_{S,\Sigma })\). This completes the proof of Theorem 3.1.
4 Critical Spaces
4.1 Strong Setting
We consider the abstract system
where \(F_\Sigma (u):=-P_{H,\Sigma }\mathcal {P}_\Sigma (u\cdot \nabla _\Sigma u)= -P_{H,\Sigma }\nabla _u u\). Let \(q\in (1,\infty )\),
Furthermore, let \(X_\beta =[X_0,X_1]_\beta \) for some \(\beta \in (0,1)\), where \([\cdot ,\cdot ]_\beta \) denotes the complex interpolation functor. Then, by Theorem 3.1, it holds that \(X_\beta =D((\omega +A_{S,\Sigma })^\beta )\) for \(\omega >s(-A_{S,\Sigma })\). In [24, Section 3.5], we determined the real and complex interpolation spaces \((X_0,X_1)_{\alpha ,p}\) and \([X_0,X_1]_\alpha \) as
for \(\alpha \in (0,1)\) and \(p\in (1,\infty ),\) where \(H_{q,\sigma }^{s}(\Sigma ,\textsf{T}\Sigma ):= H_{q}^{s}(\Sigma ,\textsf{T}\Sigma )\cap L_{q,\sigma }(\Sigma ,\textsf{T}\Sigma )\) and \(B_{qp,\sigma }^{s}(\Sigma ,\textsf{T}\Sigma ):= B_{qp}^{s}(\Sigma ,\textsf{T}\Sigma )\cap L_{q,\sigma }(\Sigma ,\textsf{T}\Sigma )\) for \(s\in (0,2)\).
By Hölder’s inequality, the estimate
holds. We choose \(r,r'\in (1,\infty )\), \(1/r+1/r'=1\), 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 \(q\in (d/3,d)\). For \(q\in (d/3,d)\) we define the critical weight by
with \(2/p+d/q\le 3\), so that \(\mu _c\in (1/p,1]\). We consider now the problem
where \(\omega >s(-A_{S,\Sigma })\). It is clear that u is a solution of (4.1) if and only if u is a solution to (4.2). Note that for each \(\omega >s(-A_{S,\Sigma })\), the operator \(\omega +A_{S,\Sigma }\) admits a bounded \(H^\infty \)-calculus in \(X_0\) with \(H^\infty \)-angle \(\phi _{A_{S,\Sigma }}^\infty <\pi /2\). We may therefore apply [23, Theorem 1.2] to (4.2) which yields the following result.
Theorem 4.1
Let \(p\in (1,\infty )\), \(q\in (d/3,d)\) such that \(\frac{2}{p}+\frac{d}{q}\le 3\). Then for any initial value \(u_0\in B_{qp,\sigma }^{d/q-1}(\Sigma ,\textsf{T}\Sigma )\) there exists a unique strong solution
of (4.1) for some \(a=a(u_0)>0\), with \(\mu _c=1/p+d/(2q)-1/2\). The solution exists on a maximal time interval \([0,t_+(u_0))\) and depends continuously on \(u_0\). In addition, we have
which means that the solution regularizes instantaneously if \(2/p+d/q<3\).
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.
4.2 Weak Setting
In order to cover the case \(q\ge d\), we proceed as follows. Let \(A_0=\omega +A_{S,\Sigma }\), \(\omega >s(-A_{S,\Sigma })\) and recall that \(X_0=L_{q,\sigma }(\Sigma ,\textsf{T}\Sigma )\). By [2, Theorems V.1.5.1 & V.1.5.4], the pair \((X_0,A_0)\) generates an interpolation-extrapolation scale \((X_\alpha ,A_\alpha )\), \(\alpha \in \mathbb {R}\), with respect to the complex interpolation functor. Note that for \(\alpha \in (0,1)\), \(A_\alpha \) is the \(X_\alpha \)-realization of \(A_0\) (the restriction of \(A_0\) to \(X_\alpha \)) and
since \(A_0\) admits a bounded \(H^\infty \)-calculus.
Let \(X_0^\sharp :=(X_0)'\) and \(A_0^\sharp :=(A_0)'\) with \(D(A_0^\sharp )=:X_1^\sharp \). Then \((X_0^\sharp ,A_0^\sharp )\) generates an interpolation-extrapolation scale \((X_\alpha ^\sharp ,A_\alpha ^\sharp )\), the dual scale, and by [2, Theorem V.1.5.12], it holds that
for \(\alpha \in \mathbb {R}\). Choosing \(\alpha =1/2\) in the scale \((X_\alpha ,A_\alpha )\), we obtain an operator
where \(X_{-1/2}=(X_{1/2}^\sharp )'\) (by reflexivity) and, since also \(A_0^\sharp \) has a bounded \(H^\infty \)-calculus,
with \(p'=p/(p-1)\) being the conjugate exponent to \(p\in (1,\infty )\). Moreover, we have \(A_{-1/2}=(A_{1/2}^\sharp )'\) and \(A_{1/2}^\sharp \) is the restriction of \(A_0^\sharp \) to \(X_{1/2}^\sharp \). Thus, the operator \(A_{-1/2}:X_{1/2}\rightarrow X_{-1/2}\) inherits the property of a bounded \(H^\infty \)-calculus with \(H^\infty \)-angle \(\phi _{A_{-1/2}}^\infty <\pi /2\) from the operator \(A_0\).
Since \(A_{-1/2}\) is the closure of \(A_0\) in \(X_{-1/2}\) it follows that \(A_{-1/2}u=A_0u\) for \(u\in X_1=D(A_0)=H_{q,\sigma }^2(\Sigma ,\textsf{T}\Sigma )\) and thus, for all \(v\in X_{1/2}^\sharp \), it holds that
where we made use of the surface divergence theorem. Using that \(X_1\) is dense in \(X_{1/2}\), we obtain the identity
valid for all \((u,v)\in X_{1/2}\times X_{1/2}^\sharp \). We call the operator \(A_{S,\Sigma }^\textsf{w}:=A_{-1/2}-\omega \) the weak surface Stokes operator, given by its representation
Multiplying (4.1) by a function \(\phi \in X_{1/2}^\sharp = H_{q',\sigma }^1(\Sigma ,\textsf{T}\Sigma )\) and using the surface divergence theorem, we obtain the weak formulation
in \(X_{-1/2}\), where
Note that u is a solution of (4.3) if and only if u solves
in \(X_{-1/2}\). We will apply Theorem 1.2 in [23] to (4.4) with the choice
For that purpose, we will first characterize some relevant interpolation spaces. Let
By similar arguments as in [26, Section 2.3] we obtain
Next we show that the nonlinearity \(F_\Sigma ^\textsf{w}:X_\beta ^\textsf{w} \rightarrow X_{0}^\textsf{w} \) is well defined, where \(X_\beta ^\textsf{w}\) denotes the complex interpolation space, that is, \(X^\textsf{w}_\beta :=[X_0^\textsf{w},X_1^\textsf{w}]_{\beta } \) for \(\beta \in (1/2,1).\) By (4.5), (4.6) and Sobolev embedding, we have
provided that \(2\beta -1\ge \frac{d}{2q}\). From now on, we assume \(2\beta -1=\frac{d}{2q}\), which means \(q>d/2\) as \(\beta <1\). Then, by Hölder’s inequality and (4.7), we obtain
showing that
For \(2\beta -1=d/2q\), the critical weight \(\mu _c\in (1/p,1]\) is given by \(\mu _c:=\mu ^\textsf{w}_c=1/p+d/2q\) and the corresponding critical trace space in the weak setting reads
Note that in case \(q\in (d/2,d)\), the critical spaces in the weak and strong setting coincide.
The existence and uniqueness result for (4.3) in critical spaces reads as follows.
Theorem 4.3
Let \(p\in (1,\infty )\) and \(q\in (d/2,\infty )\) such that \(\frac{2}{p}+\frac{d}{q}\le 2\). Then for any initial value \(u_0\in B_{qp,\sigma }^{d/q-1}(\Sigma ,\textsf{T}\Sigma )\) there exists a unique solution
of (4.3) for some \(a=a(u_0)>0\), with \(\mu _c =1/p+d/2q\). The solution exists on a maximal time interval \([0,t_+(u_0))\) and depends continuously on \(u_0\). Moreover,
Suppose \(p>2\) and \(q\ge d\). Then each solution with \(u_0\in B_{qp,\sigma }^{d/q-1}(\Sigma ,\textsf{T}\Sigma )\) satisfies in addition
Hence, in this case, each solution regularizes instantaneously and becomes a strong solution.
Proof
Since \(A_{-1/2}=\omega +A_{S,\Sigma }^\textsf{w}\) admits a bounded \(H^\infty \)-calculus in \(X_{-1/2}\) with \(H^\infty \)-angle \(\phi _{A_{-1/2}}^\infty <\pi /2\), the first assertion follows readily from [23, Theorem 1.2].
Suppose that \(p>2\) and \(q\ge d\). Then \(\mu _c=1/p + d/2q<1\) and [23, Theorem 1.2] yields \(u\in C((0,t_+); B^{1-2/p}_{qp,\sigma } (\Sigma ,\textsf{T}\Sigma )).\) Therefore,
for \(t\in (0,t_+(u_0))\). Noting that for \(\mu \in (1/p,1/2]\) we have the embedding
at our disposal, we may now solve (4.1) by [24, Theorem 3.5] with initial value \(u(t_0)\), \(t_0\in (0,t_+(u_0))\), to obtain a strong solution. The assertion of the theorem follows now from uniqueness, see [15, Theorem 3.4(c)]. \(\square \)
Corollary 4.4
Suppose \(d\ge 2\). Then equation (4.3) has for each initial value \(u_0\in L_{d,\sigma }(\Sigma , \textsf{T}\Sigma )\) a unique solution which enjoys the regularity properties of Theorem 4.3 with \(q=d\) for each fixed p with \(p>2\) and \(p\ge d\).
Proof
Suppose \(p>2\) and \(p\ge d\). Then the embedding \( L_{d,\sigma }(\Sigma , \textsf{T}\Sigma )\hookrightarrow B^0_{dp,\sigma }(\Sigma , \textsf{T}\Sigma ) \) holds, see for instance [4, Theorem 6.2.4 and Theorem 6.4.4]. The assertion follows now from Theorem 4.3 with \(q=d\). \(\square \)
We now consider the limiting case \(p=q=d=2\). In this case, we have \(\mu _c =1\) and the corresponding critical trace space is given by
see for example [33, Remark 1.18.10.3].
We may now also extend each weak solution to a strong solution. To this end, observe that
for any \(r\in (2,\infty )\). Then we solve (4.3) by Theorem 4.3 with \(u_0\in (X_{-1/2},X_{1/2})_{1/2,r}\) and \(\mu _c=1/r+1/2\) (choosing \(p=r\) and \(q=d=2\)). This yields
with \(X_1^\textsf{w}=H_{2,\sigma }^1(\Sigma ,\textsf{T}\Sigma )\) and \(X_0^\textsf{w}=H^{-1}_{2,\sigma }(\Sigma ,\textsf{T}\Sigma )\). Since now \(u(t)\in B_{2r,\sigma }^{1-2/r}(\Sigma ,\textsf{T}\Sigma )\) for \(t\in (0,a)\), we may argue as above to conclude that the weak solution regularizes to a strong solution
Moreover,
for \(t\in (0,a)\), provided \(q\ge 2\), \(p\ge r\) and \(2-2/r-d/2 \ge 2\mu -2/p\ - d/q\). For \(d=2\) this means
We now solve (4.1) with initial value \(u(t_0)\in B^{2\mu -2/p}_{qp,\sigma }(\Sigma ,\textsf{T}\Sigma )\) by [24, Theorem 3.5] to obtain a solution
As in the proof of [15, Theorem 3.4(c)] we conclude that uniqueness holds, that is, \(v(t)=u(t_0 +t)\). As \(t_0\) can be chosen arbitrarily small, this implies that u shares the regularity properties of v for \( t>0\). We have shown the following result.
Theorem 4.5
Let \(d=2\). Then for any \(u_0\in L_{2,\sigma }(\Sigma ,\textsf{T}\Sigma )\), problem (4.3) admits a unique solution
for some \(a=a(u_0)>0\). The solution exists on a maximal time interval \([0,t^+(u_0))\). In addition, we have
Furthermore, each solution satisfies
for any fixed \(q\in (1,\infty )\) and any fixed \(p\in (2,\infty )\). Therefore, any solution with initial value \(u_0\in L_{2,\sigma }(\Sigma ,\textsf{T}\Sigma )\) regularizes instantaneously and becomes a strong \(L_p\)-\(L_q\) solution.
Proof
According to the considerations preceding the Theorem, the assertions hold true for a fixed \(q\ge 2\) and \(p\in (2, \infty ).\) The case \(q\in (1,2)\) follows readily from the result in the \(L_2\)-case and the embedding \(L_2(\Sigma ,\textsf{T}\Sigma )\hookrightarrow L_q(\Sigma ,\textsf{T}\Sigma )\). \(\square \)
4.3 Energy Estimates and Global Existence
In [24] we showed that the set of equilibria \({\mathcal {E}}\) for (1.1), respectively (4.1), consists exactly of the Killing vector fields on \(\Sigma \), that is,
We recall that the condition \({\mathcal {D}}_\Sigma (u)=0\) implies that u is divergence free (which follows from the relation \(\textrm{div}_\Sigma \,u = \textsf{tr}\,{\mathcal {D}}_\Sigma (u)\)). Moreover, one can show that any vector field \(u\in H^1_q(\Sigma ,\textsf{T}\Sigma )\) satisfying \({\mathcal {D}}_\Sigma (u)=0\) is already smooth, see for instance [19, Lemma 3]. Lastly, we recall that \({\mathcal {E}}\) is a finite dimensional vector space. If fact, \(\textrm{dim}\,{\mathcal {E}}\le d(d+1)/2\), with equal sign for the case where \(\Sigma \) is isometric to a Euclidean sphere, see for instance the remarks in Section 4.1 of [24].
Let us define the space
Note that \(V^j_2(\Sigma )\) is a closed subspace of \( H^j_{2,\sigma }(\Sigma ,\textsf{T}\Sigma )\), and hence is a Banach space. Moreover, \(H^j_{2,\sigma }(\Sigma , \textsf{T}\Sigma )={\mathcal {E}} \oplus V^j_2(\Sigma )\), 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 \(v_0\in L_{2,\sigma }(\Sigma )\) being orthogonal to \({\mathcal {E}}\) will remain orthogonal for all later times. Moreover, we establish an energy estimate for such solutions.
Proposition 4.6
Let \(d=2\). Suppose \(v_0\in V^0_2(\Sigma )\) and let v be the solution of (4.3) established in Theorem 4.5. Then
-
(a)
\(v(t)\in V^0_2(\Sigma )\) for \(t\in [0,t^+(v_0))\) and \(v(t)\in V^1_2(\Sigma )\) for \(t\in (0, t^+(v_0)).\)
-
(b)
There exists a universal constant \(M>0\) such that
$$\begin{aligned} |v(t)|^2_{L_2(\Sigma )}+ \int _0^t |v(s)|^2_{H^1_2(\Sigma )}\, ds \le M |v_0|^2_{L_2(\Sigma )}, \quad t\in (0,t^+(v_0)). \end{aligned}$$
Proof
(a) According to Theorem 4.5, we know that
for \(p>2\). Pick any \(z\in {\mathcal {E}}\). Then
where the time derivative exists for almost all \(t\in (0, t^+(v_0))\). For the last equal sign we employed the property that \(A_{S,\Sigma }\) is symmetric on \(L_{2,\sigma }(\Sigma )\) and \(N(A_{S,\Sigma })={\mathcal {E}}\), see [24, Proposition 4.1]. In a next step we show that
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 \((\nabla _v z| v) + (v | \nabla _v z)=0)\). Hence, we have shown that \(\frac{d}{dt} ( v(t)|z )_\Sigma =0\) for almost all \(t\in (0, t^+(v_0))\). (4.8) now implies
(b) Similarly as in part (a), one shows (suppressing the variable t) that
The assertion in part (a) and Korn’s inequality (A. 2) readily imply
with an appropriate constant \(\alpha >0\). Integration yields the assertion in (b), as \(|v|^2_{L_2(\Sigma )}\) is absolutely continuous on \([0,t^+(v_0))\). \(\square \)
Proposition 4.7
Suppose that \(d=2\) and \(v_0\in V^0_2(\Sigma )\).
Then problem (4.3) admits a unique global solution v enjoying the regularity properties stated in Theorem (4.5), with \(t^+(v_0)=\infty \).
Moreover, there exists a constant \(\alpha >0\) 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:
Observe that in case \(p=q=d=2\), it holds \(\mu _c=1/p+d/(2q)=1\), hence if
then the weak solution exists globally in time. Proposition 4.6 guarantees that any solution v with initial value \(v_0\in V^0_2(\Sigma )\) satisfies
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.
Finally, we conclude from (4.10) that \( \frac{d}{dt} |v(t)|^2_{L_2(\Sigma )} + \alpha \, |v(t)|^2_{L_2(\Sigma )}\le 0\) for any \(t>0\) and this implies the estimate in (4.11). \(\square \)
Remark 4.8
Suppose \(u_*\in {\mathcal {E}}\) and \(v_0\in V^0_2(\Sigma )\). Then the assertions of Theorem 4.1, Theorem 4.5 as well as Propositions 4.6 and 4.7 hold true for solutions of
respectively its weak formulation
where \(\langle F^\textsf{w}_\Sigma (v),\phi \rangle = ( v | \nabla _v \phi )_\Sigma + (v| \nabla _{u_*}\phi )_\Sigma + (u_* | \nabla _v \phi )_\Sigma \) for \(\phi \in H^1_{2,\sigma }(\Sigma ,\textsf{T}\Sigma )\).
In particular, each solution of(4.13) with initial value \(v_0\in V^0_2(\Sigma )\) exists globally and there exists a positive constant \(\alpha \) 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 \(v(t)\in V^0_2(\Sigma )\) for \(t\in [0,t^+(v_0))\). Let \(z\in {\mathcal {E}}\). Following the proof of Proposition 4.6(a), we obtain
According to the proof of Proposition 4.6(a), \((\nabla _v v|z)=0\) and it remains to show that \((\nabla _{u_*}v+ \nabla _{v}u_* | z)_\Sigma =0\) for any \(v\in H^1_{2,\sigma }(\Sigma , \textsf{T}\Sigma ).\) 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. \(\square \)
Theorem 4.9
(Global existence). Suppose \(d=2\).
Then any solution of (4.3) with initial value \(u_0\in L_{2,\sigma }(\Sigma ,\textsf{T}\Sigma )\) exists globally, has the regularity properties listed in Theorem 4.5, and converges at an exponential rate to the equilibrium \(u_* = P_{{{\mathcal {E}}}} u_0\) in the topology of \(H_q^{2}(\Sigma ,\textsf{T}\Sigma )\) for any fixed \(q\in (1,\infty )\), where \(P_{{{\mathcal {E}}}}\) is the orthogonal projection of \(u_0\) onto \({{\mathcal {E}}}\) with respect to the \(L_2(\Sigma , \textsf{T}\Sigma )\) inner product.
Proof
Let \(u_0\in L_{2,\sigma }(\Sigma ,\textsf{T}\Sigma )\) be given. Then there exist unique elements \(u_*\in {\mathcal {E}}\) and \(v_0\in V^0_2(\Sigma )\) such that \(u_0=u_*+v_0 \). Let v be the unique (global) solution of problem (4.13), respectively (4.12), whose existence has been asserted in Remark 4.8 Then
yields a (unique) global solution of (4.1), respectively (4.3), with initial value \(u_0\). For this, we just need to observe that
Indeed, this follows from the relations \(N(A_{S,\Sigma })={\mathcal {E}}\) and \(\nabla _{u_*}u_*=\frac{1}{2} \textrm{grad}(u_* | u_*)\), with the latter assertion implying \(P_{H,\Sigma } \nabla _{u_*} u_*=0\). It follows readily that \(u(t)\rightarrow u_*\) as \(t\rightarrow \infty \) at an exponential rate in \(L_2(\Sigma ,\textsf{T}\Sigma )\). To prove convergence in the stronger topology \(H_q^{2}(\Sigma ,\textsf{T}\Sigma )\), we proceed as follows.
(i) First, we note that \(L_2(\Sigma ,\textsf{T}\Sigma )\hookrightarrow B_{2r}^0(\Sigma ,\textsf{T}\Sigma )\) for any fixed \(r\ge 2\). For fixed, but arbitrary \(t_1>0\), we solve (4.3) with initial value \(u_1:=u(t_1)\in B_{2r,\sigma }^0(\Sigma ,\textsf{T}\Sigma )\). Choosing \(\mu _c=1/r+1/2\), it follows from [23, Theorem 1.2] that there exist positive numbers \(\tau =\tau (u_*)\), \(\varepsilon =\varepsilon (u_*)\) and \(C_1=C_1(u_*)\) such that
for all \({\hat{u}}_1\in B_{2r,\sigma }^0(\Sigma ,\textsf{T}\Sigma )\) with \(|{\hat{u}}_1-u_*|_{B_{2r,\sigma }^0(\Sigma )}<\varepsilon ,\) where
It should be observed here that \(u(t, u_*)=u_*\), as \(u_*\) is an equilibrium.
By following the arguments in [22, page 228] and employing
see (4.6), we conclude with (4.14) that there is a constant \(C_2=C_2(u_*)\) such that
Letting \(C_3\) be the embedding constant of \(L_2(\Sigma ,\textsf{T}\Sigma )\hookrightarrow B^0_{2r}(\Sigma ,\textsf{T}\Sigma )\) we choose \(t_1\) large enough such that \(|u(t_1)-u_*|_{L_2(\Sigma )}<\varepsilon /C_3\). Setting \(t_2=\tau \), we may deduce from the estimates above that
with \(C=C(u_*)=C_2C_3 \). Here we have also used the uniqueness of solutions, cf. [15, Theorem 3.4(c)].
(ii) For fixed, but arbitrary \(r>2\), we now choose a weight \(\mu \in (\frac{1}{r},\frac{1}{2}]\) so that
Solving (4.1) with initial value \(u_2:=u(t_2,u_1)\in B_{2r,\sigma }^{2\mu -2/r}(\Sigma ,\textsf{T}\Sigma )\) and repeating the above procedure in the ‘strong’ spaces \((X_0,X_1)\), we obtain the estimate
for some \(t_3=t_3(u_*)>0\).
(iii) Next, we use the Sobolev embedding
valid for \(s\ge 2\) and \(\mu \in (\frac{1}{r},\frac{1}{2}+\frac{1}{s}]\). Choosing \(t_1>0\) from above sufficiently large, we infer from (4.16) that \(u(t_3,u_2)\) is close to \(u_*\) in the topology of \(B_{sr}^{2\mu -2/r}(\Sigma ,\textsf{T}\Sigma )\). Solving (4.1) with initial value \(u_3:=u(t_3,u_2)\in B_{sr,\sigma }^{2\mu -2/r}(\Sigma ,\textsf{T}\Sigma )\) and repeating the above procedure in the ‘strong’ spaces \((X_0,X_1)\), we obtain the estimate
for some \(t_4=t_4(u_*)>0\).
(iv) We will now consider (4.1) in the spaces
where \((X_\alpha , A_\alpha )\) is the interpolation-extrapolation scale with respect to the complex interpolation functor, based on \(X_0=L_{s,\sigma }(\Sigma , \textsf{T}\Sigma )\), \(s\in (1,\infty )\), introduced at the beginning of Section 4.2. We note that \(A_{1/2}\), the realization of \(A_0=\omega + A_{S,\Sigma }\) in \(X_{1/2}\), has exactly the same properties as \(A_0\). For \(\beta =1/2\) we obtain
It is easy to see that the nonlinearity
is bilinear and bounded. We can employ [23, Theorem 1.2 or 2.1] for problem (4.1) to obtain a solution \(u(\cdot , u_4)\) with initial value \(u_4=u(t_4,u_3)\in B_{sr}^{2-2/r}(\Sigma ,\textsf{T}\Sigma ).\) To do so, we choose \(\mu =1/2\) and verify that
Noting that the numbers \(\mu =1/2\) and \(\beta =1/2\) satisfy the assumptions of [23] Theorem 1.2 or 2.1, we can once more repeat the procedure outlined in step (i) to obtain
for some \(t_5=t_5(u_*)>0\), provided \(t_1\) is chosen sufficiently large. Combining the estimates (4.15)–(4.18) and using the semiflow property, we obtain the estimate
Since \(|u(t_1,u_0)-u_*|_{L_2(\Sigma )}\rightarrow 0\) at an exponential rate as \(t_1\rightarrow \infty \), we conclude that \(u(t,u_0)\) converges to \(u_*\) at an exponential rate in the topology of \(B^{3-2/r}_{sr,\sigma }(\Sigma , \textsf{T} \Sigma )\) as well, as \(t\rightarrow \infty \).
(iv) Finally, given \(q\in (1,\infty )\), the embedding \(B_{sr,\sigma }^{3-2/r}(\Sigma ,\textsf{T}\Sigma )\hookrightarrow H_{q,\sigma }^{2}(\Sigma ,\textsf{T}\Sigma ),\) being valid for sufficiently large parameters s and r, yields the last assertion of the theorem. \(\square \)
Remarks 4.10
(a) Let \(d\ge 2\) and let \({{\mathfrak {F}}}^s(\Sigma )\) denote any of the spaces
Then we have
Here, \(\langle v, z \rangle _\Sigma : = (v|z)_\Sigma \) if \(s>0\). In case \(s\le 0\),
denotes the duality pairing, induced by \((\cdot | \cdot )_\Sigma \), where
Since we can identify \({{\mathcal {E}}}\subset C^\infty (\Sigma , \textsf{T}\Sigma )\) as a subspace of \({{\mathfrak {F}}}^s(\Sigma )\), the expression \(\langle v, z \rangle _\Sigma \) is defined for every \((v,z)\in {{\mathfrak {F}}}^s(\Sigma )\times {{\mathcal {E}}}\) and (4.19) is, therefore, meaningful.
Proof
We will provide a proof of (4.19). As \({{\mathcal {E}}}\) is finite dimensional, we can find a basis \(\{z_1,\ldots , z_m\}\) for \({{\mathcal {E}}}\) which has the property that \((z_i | z_j)_\Sigma =\delta _{ij}\). With this at hand, we define the projection
onto \({{\mathcal {E}}}\). This yields the direct topological decomposition \({{\mathfrak {F}}}^s(\Sigma )= {{\mathcal {E}}}\oplus V^s(\Sigma )\), where \( V^s(\Sigma )= (I-P^s_{{\mathcal {E}}}) {{\mathfrak {F}}}^s(\Sigma ).\) In order to justify (4.19), it suffices to show that
Suppose \(v\in V^s(\Sigma )\). Then \(v=(I-P^s_{{\mathcal {E}}})v\) and we obtain
showing that \(v\in {\hat{V}}^s(\Sigma ).\) Suppose now that \(v \in {\hat{V}}^s(\Sigma )\). Since \({{\mathfrak {F}}}^s(\Sigma )= {{\mathcal {E}}}\oplus V^s(\Sigma )\), there are unique elements \((z,w)\in {{\mathcal {E}}}\times V^s(\Sigma )\) such that \(v=z+w\). Then, by the first step,
hence \(z=0\) and therefore \(v=w\in V^s(\Sigma )\). \(\square \)
(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 \(q\ge d > 2\). Suppose \(v_0\in B^{d/q-1}_{qp,\sigma }(\Sigma , \textsf{T}\Sigma ) \) satisfies \(\langle v_0, z\rangle _\Sigma =0\) for every \(z\in {\mathcal {E}}\), where \(\langle v_0, z\rangle _\Sigma \) has the same meaning as in (a). Let v be the unique solution of (4.3) with initial value \(v_0\).
Then there exists a constant \(\alpha >0\) such that
for fixed \(\tau \in [0,t^+(v_0))\), where \(\tau \in (0 ,t^+(v_0))\) in case \(v_0\notin L_{2,\sigma }(\Sigma , \textsf{T}\Sigma )\).
Proof
According to Theorem 4.3, there exists a unique solution v to problem (4.3) with regularity
for each fixed \(a\in (0, t^+)\). For \(z\in {{\mathcal {E}}}\) we obtain, as in the proof of Theorem 4.6,
for all \(t\in (0 ,t^+(v_0))\). Therefore, \(\langle v(\cdot ), z\rangle _\Sigma \in H^1_{p,\mu _c}(0, a)\) and \(\frac{d}{dt} \langle v(t) | z\rangle _\Sigma =0\) for \(t\in (0,t^+(v_0))\). By [21, Lemma 2.1(b)], or [22, Lemma 3.2.5(b)], we infer that \(\langle v(\cdot ), z\rangle _\Sigma \in H^1_{1,\textrm{loc}}([0, a])\) for any fixed \(a\in (0, t^+(v_0))\). We can now conclude from the fundamental theorem that
Let \(\tau \in (0,t^+(v_0))\) be fixed. As \(p,q>2\), we conclude from (4.22) and (4.23) that \(v(t)\in V^1_2(\Sigma )=\{u\in H^1_2(\Sigma , \textsf{T}\Sigma )\mid (u |z)_\Sigma =0,\ z\in {{\mathcal {E}}}\}\) for any \(t\in [\tau ,t^+(v_0))\). Hence, Korn’s inequality (A. 2) holds true for v(t) with \(t\in [\tau , t^+(v_0))\), and we can now follow the proof of Propositions 4.6(b) and 4.7 to obtain the assertion in (4.21). \(\square \)
(c) Suppose \(d>2\) and \(q\ge d\). Then (4.21) holds true with \(\tau =0\) for initial values \(v_0\in B^{2\mu -2/p}_{qp,\sigma }(\Sigma , \textsf{T}\Sigma )\) satisfying the assumptions of [24, Theorem 3.5(b)] and \((v_0 | z)_\Sigma =0\) for all \(z\in {{\mathcal {E}}}\).
(d) Suppose \(p>2\) and \(q\ge d>2\). Then every global solution of (4.3), respectively (4.1), converges exponentially fast to an equilibrium, namely to \(P_{{\mathcal {E}}}u_0\) (where \(P_{{\mathcal {E}}}\) is the projection defined in (4.20)), provided the initial value \(u_0\) satisfies the assumptions of Theorem 4.3 or [24, Theorem 3.5(b)].
Proof
This follows by similar arguments as in the proof of Remark 4.8 and Theorem 4.9. \(\square \)
(e) According to Theorem 4.3 in [24], any solution with initial value \(u_0\) sufficiently close to an equilibrium exists globally and, hence, converges to \(P_{{\mathcal {E}}}u_0\) at an exponential rate.
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Appendix A
Appendix A
1.1 A.1. Auxilliary Results
We first consider a weak elliptic problem on compact manifolds without boundary.
Lemma A.1
Let \(1<q<\infty \). For each \(v\in L_q(\Sigma ,\textsf{T}\Sigma )\) there exists a unique solution \(\nabla _\Sigma \psi \in L_q(\Sigma ,\textsf{T}\Sigma )\) of
Proof
For \(\lambda >0\), let \(A_0:=\lambda -\Delta _\Sigma \) in \(X_0=L_{q}(\Sigma )\) with domain \(X_1=H_q^2(\Sigma )\). By [2, Theorem V.1.5.1], the pair \((X_0,A_0)\) generates an interpolation-extrapolation scale \((X_\alpha ,A_\alpha )\), \(\alpha \in {\mathbb R}\), with respect to the complex interpolation functor \([\cdot ,\cdot ]_\theta \), \(\theta \in (0,1)\). Let \(X_0^\sharp =(X_0)'=L_{q'}(\Sigma )\) and denote by \(A_0^\sharp \) the dual operator of \(A_0\) in \(X_0^\sharp \) with domain \(X_1^\sharp :=H_{q'}^2(\Sigma )\). We write \((X_\alpha ^\sharp ,A_\alpha ^\sharp )\), \(\alpha \in {\mathbb R}\), for the dual interpolation-extrapolation scale generated by \((X_0^\sharp ,A_0^\sharp )\). Then, \(A_{-1/2}:X_{1/2}\rightarrow X_{-1/2}\) is a linear isomorphism, where
and
We claim
for all \((u,\phi )\in X_{1/2}\times X_{1/2}^\sharp \). Indeed, for \(u\in X_1\), it holds that \(A_{-1/2}u=A_0 u\), hence
The surface divergence theorem as well as the density of \(X_1\) in \(X_{1/2}\) yield the claim.
Define an operator \(B:X_{1/2}\rightarrow X_{-1/2}\) by \(Bu=A_{-1/2}u-\lambda u\). Since the embedding \(X_{1/2}\hookrightarrow X_{-1/2}\) is compact, the spectrum \(\sigma (B)\) consists solely of eigenvalues with finite multiplicity. Furthermore, \(\sigma (B)\) is invariant with respect to q and for each eigenfunction u of B it holds that \(u\in H_r^1(\Sigma )\) for any \(r\in (1,\infty )\).
We show that \(\mu =0\) is a semi-simple eigenvalue of B. The equation \(Bu=0\) in \(X_{-1/2}\) is equivalent to
for all \(\phi \in X_{1/2}^\sharp =H_{q'}^1(\Sigma )\). Choosing \(\phi =u\), we obtain \(\nabla _\Sigma u=0\), hence u is constant. This shows
We show \(N(B^2)=N(B)\). For that purpose, let \(u\in N(B^2)\) and define \(v:=Bu\). Then \(v\in N(B)\), hence v is constant. For \(\lambda >0\) we have
Solve \(A_0w=\lambda u+v\) to obtain a unique solution \(w\in H_q^2(\Sigma )\). Since \(A_0 w=A_{-1/2} w\), this yields \(A_{-1/2} u=A_{-1/2}w\) and therefore \(u=w\) by injectivity of \(A_{-1/2}\). This proves \(u\in H_q^2(\Sigma )\) which in turn yields \(A_0 u=\lambda u+v\) or equivalently \(-\Delta _\Sigma u=v\). Integrating the last equation over \(\Sigma \), yields \(v=0\) as v is constant. This shows \(u\in N(B)\), hence \(N(B^2)\subset N(B)\). Since the converse inclusion is obvious, we obtain the assertion.
We have shown so far that \(\mu =0\) is a semi-simple eigenvalue of B. In particular, this implies
Consequently, the restricted operator \(B:X_{1/2}\cap R(B)\rightarrow R(B)\) is invertible. Note that
For \(v\in L_q(\textsf{T}\Sigma )\), define \(f\in X_{-1/2}\) by \(\langle f,\phi \rangle =(v|\nabla _\Sigma \phi )_\Sigma \). Then obviously \(f\in R(B)\), hence there exists a unique solution \(u\in X_{1/2}\cap R(B)\) of the equation \(Bu=f\). The proof is completed. \(\square \)
Next we study existence and uniqueness as well as regularity properties of solutions to some second order differential equations on \({\mathbb R}^d\).
For that purpose, we set
for functions \(f\in L_q({\mathbb R}^d)\). For \(\lambda >0\), \(k\in \{-1,0,1\}\) we then define the function spaces
equipped with the parameter-dependent norms
We are ready to prove the following result.
Lemma A.2
Let \(1<q<\infty \), \(k\in \{-1,0,1\}\) and \(G\in W_\infty ^{1+k}({\mathbb R}^d)^{d\times d}\). Then there exist \(\eta ,\lambda _0>0\) such that
is an isomorphism, provided \(|G-I|_{L_\infty ({\mathbb R}^d)}\le \eta \) and \(\lambda \ge \lambda _0\).
Moreover, in this case there exists a constant \(c(\lambda _0)>0\) such that the unique solution of \(\textrm{div}\,(G\nabla u)=f\) satisfies the estimate
for any \(f\in \mathbb F^k_\lambda \).
Proof
-
(i)
We start with the case \(k=-1\). For \(u\in \mathbb {E}_\lambda ^{-1}\) we write
$$\begin{aligned} \langle {\text {div}}\,(G\nabla u),\phi \rangle =\langle \Delta u,\phi \rangle +\langle {\text {div}}((G-I)\nabla u),\phi \rangle , \end{aligned}$$where
$$\begin{aligned} \langle {\text {div}} v,\phi \rangle :=-\int _{{\mathbb R}^d}v\cdot \nabla \phi \, dx \end{aligned}$$for \((v,\phi )\in L_q({\mathbb R}^d)^d\times \dot{H}_{q'}^1({\mathbb R}^d)\). It is known that the operator \(\Delta :\mathbb {E}_\lambda ^{-1}\rightarrow \mathbb {F}_\lambda ^{-1}\) is an isomorphism, see for instance [34], Theorem 5.2.3.1(i) and the remarks in Section 5.2.5 concerning duality, or [31, Lemma 3.3]. Furthermore, the estimate
$$\begin{aligned} |\langle {\text {div}}((G-I)\nabla u),\phi \rangle |\le \eta |u|_{\dot{H}_q^1({\mathbb R}^d)}|\phi |_{\dot{H}_{q'}^1({\mathbb R}^d)} \end{aligned}$$holds. A Neumann series argument yields that the operator \({{\text {div}}(G\nabla \cdot ):\mathbb {E}_\lambda ^{-1}\rightarrow \mathbb {F}_\lambda ^{-1}}\) is an isomorphism as well, provided \(\eta \in (0,1)\).
-
(ii)
Let \(k=0\). Then we have
$$\begin{aligned} {\text {div}}(G\nabla u)=\Delta u+{\text {div}}((G-I)\nabla u). \end{aligned}$$Also in this case, \(\Delta :\mathbb {E}_\lambda ^0\rightarrow \mathbb {F}_\lambda ^0\) is an isomorphism. Furthermore, there exists a constant \(C>0\) such that \(|\Delta ^{-1}f|_{\mathbb {E}_\lambda ^0}\le C|f|_{\mathbb {F}_\lambda ^0}\) for all \(f\in {\mathbb {F}_\lambda ^0}\) and all \(\lambda \in (0,\infty )\). To see this, note that
$$\begin{aligned} |\nabla (\Delta ^{-1}F_f)|_{L_q({\mathbb R}^d)}\le C|F_f|_{\dot{H}_q^{-1}({\mathbb R}^d)} \end{aligned}$$and
$$\begin{aligned} |\nabla ^2(\Delta ^{-1}f)|_{L_q({\mathbb R}^d)}\le C|f|_{L_q({\mathbb R}^d)}. \end{aligned}$$The definition of the norms in \(\mathbb {E}_\lambda ^0\) and \(\mathbb {F}_\lambda ^0\) yields the claim. We then estimate as follows
$$\begin{aligned} |{\text {div}}((G-I)\nabla u)|_{L_q({\mathbb R}^d)}&\le |G|_{W_\infty ^1({\mathbb R}^d)}|\nabla u|_{L_q({\mathbb R}^d)}+|G-I|_{L_\infty ({\mathbb R}^d)}|\nabla ^2 u|_{L_q({\mathbb R}^d)}\\&\le C(\lambda ^{-1/2}+\eta )|u|_{\mathbb {E}_{\lambda }^0}. \end{aligned}$$Furthermore, as in the case \(k=-1\), we have
$$\begin{aligned} \lambda ^{1/2}|{\text {div}}((G-I)\nabla u)|_{\dot{H}_q^{-1}({\mathbb R}^d)}\le \eta |u|_{\mathbb {E}_\lambda ^0}. \end{aligned}$$Once again, a Neumann series argument shows that \({\text {div}}(G\nabla \cdot ):\mathbb {E}_\lambda ^0\rightarrow \mathbb {F}_\lambda ^0\) is an isomorphism, provided \(\eta \) and \(\lambda ^{-1/2}\) are chosen sufficiently small.
-
(iii)
The proof for the remaining case \(k=1\) follows literally the same strategy and is therefore omitted.
-
(iv)
The assertion in (A. 1) follows from the steps above and the definition of the norm of \(\mathbb E^k_\lambda \). \(\square \)
1.2 A.2. Korn’s Inequality
We will establish an appropriate version of Korn’s inequality for compact surfaces \(\Sigma \) without boundary. For this, we use the notation from Section 4.3.
Theorem A.3
(Korn’s inequality). There exists a constant \(C>0\) such that
Proof
Let \(Lu:=2{{\mathcal {P}}}_\Sigma \textrm{div}_\Sigma {\mathcal {D}}_\Sigma (u) \) for \(u\in H^2_{2,\sigma }(\Sigma ,\textsf{T}\Sigma )\). We know from Proposition A.2 in [24] that
where \(\Delta _\Sigma \) is the Bochner (the connection) Laplacian and \(\textsf{Ric}_\Sigma \,u\) is the Ricci (1, 1)-tensor, given in local coordinates by \((\textsf{Ric}_\Sigma )^\ell _k= g^{i\ell }R_{ik}\), so that \( \textsf{Ric}_\Sigma \,u = R^\ell _k u^k\tau _\ell .\) It is well-known that \((\Delta _\Sigma u|u)_\Sigma = - |\nabla u|^2_{L_2(\Sigma )}\) for \(u\in H^2_2(\Sigma ,\textsf{T}\Sigma )\), see for instance [30, Lemma 3.5]. Let \(u\in H^2_{2,\sigma }(\Sigma ,\textsf{T}\Sigma )\). Then
Here we used that on the compact surface \(\Sigma \), the Ricci tensor \( \textsf{Ric}_\Sigma \) can be bounded uniformly, yielding
with an appropriate positive constant \(C_1\). By density of the space \(H^2_{2,\sigma }(\Sigma ,\textsf{T}\Sigma )\) in \(H^1_{2,\sigma }(\Sigma ,\textsf{T}\Sigma )\) we obtain
for an appropriate constant \(C_2\). The assertion in (A. 2) now follows from the Petree-Tartar Lemma. For the readers’ convenience, we include the proof here.
Suppose (A. 2) does not hold. Then there exists a sequence \((v_n)_{n\in {\mathbb N}}\) in \(V^1_2(\Sigma )\), see (4.9), such that \(| v_n |_{H^1_2(\Sigma )}=1\) for \(n\in {\mathbb N}\) and \(|{\mathcal {D}}_\Sigma (v_n)|_{L_2(\Sigma )}\rightarrow 0\ \text {as}\ n\rightarrow \infty .\) Since \(H^1_2(\Sigma ,\textsf{T}\Sigma )\) is compactly embedded in \(L_2(\Sigma ,\textsf{T}\Sigma )\), there exists a subsequence, still denoted by \((v_n)_{n\in {\mathbb N}}\), which converges to an element \(v\in L_2(\Sigma ,\textsf{T}\Sigma )\). From (A. 3) follows that \( (v_n)_{n\in {\mathbb N}}\) is a Cauchy sequence in \(V^1_2(\Sigma )\). Completeness of \(V^1_2(\Sigma )\) shows that \(v\in V^1_2(\Sigma )\) and \(v_n\rightarrow v\) in \(V^1_2(\Sigma )\). Consequently, \( {\mathcal {D}}_\Sigma (v_n)\rightarrow {\mathcal {D}}_\Sigma (v)\) as well as \( {\mathcal {D}}_\Sigma (v_n) \rightarrow 0\) in \(L_2(\Sigma , \textsf{T}^1_1 \Sigma )\). This implies \({\mathcal {D}}_\Sigma (v)=0\), and then \(v\in {\mathcal {E}}\cap V^1_2(\Sigma )=\{0\}\), in contradiction to the assumption \(|v|_{H^1_2(\Sigma )}=1\). \(\square \)
Remark A.4
Korn’s inequality for embedded surfaces has also been established in [12, Lemma 4.1]. The proof given here is considerably shorter.
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Simonett, G., Wilke, M. \(H^\infty \)-Calculus for the Surface Stokes Operator and Applications. J. Math. Fluid Mech. 24, 109 (2022). https://doi.org/10.1007/s00021-022-00742-y
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DOI: https://doi.org/10.1007/s00021-022-00742-y
Keywords
- Surface Navier–Stokes equations
- surface Stokes operator
- \(H^\infty \)-calculus
- critical spaces
- Killing vector fields
- Korn’s inequality
- global existence