Tangential Navier-Stokes equations on evolving surfaces: Analysis and simulations^††thanks: This work was partially supported by US National Science Foundation (NSF) through grants DMS-1953535 and DMS-2011444.

We now introduce a major simplification by assuming that the geometric evolution of $\Gamma$ is known. We make this more precise below and derive equations governing the unknown lateral motions of the surface fluid. To this end, consider a smooth velocity field $\mathbf{w}=\mathbf{w}(t,\mathbf{x})$ in $[0,T]\times\mathbb{R}^{3}$ that passively advects the embedded surface $\Gamma(t)$ given by

$$\Gamma(t)=\{\mathbf{y}\in\mathbb{R}^{3}~{}|~{}\mathbf{y}=\mathbf{x}(t,\mathbf{z}),~{}\mathbf{z}\in\Gamma_{0}\},$$

(2.4)

where the trajectories $\mathbf{x}(t,\mathbf{z})$ are the unique solutions of the Cauchy problem

$\displaystyle\begin{cases}\mathbf{x}(0,\mathbf{z})=\mathbf{z}\\ \frac{d}{dt}\mathbf{x}(t,\mathbf{z})=\mathbf{w}(t,\mathbf{x}(t,\mathbf{z})),\end{cases}$

(2.5)

for all $\mathbf{z}$ on an initial smooth connected surface $\Gamma_{0}=\Gamma(0)$ embedded in $\mathbb{R}^{3}$. We now assume that the normal material motion of $\Gamma$ is completely determined by the ambient flow $\mathbf{w}$ and the lateral material motion is free, i.e., for the given $\mathbf{w}$ the relation

$$\mathbf{u}\cdot\mathbf{n}=\mathbf{w}\cdot\mathbf{n}\quad\text{on}~{}\Gamma(t)$$

(2.6)

holds for the normal component¹¹1For velocity fields $\mathbf{v}\in\mathbb{R}^{3}$ defined on $\Gamma(t)$ we use a splitting into tangential and normal components $\mathbf{v}=\mathbf{v}_{T}+\mathbf{v}_{N}=\mathbf{v}_{T}+v_{N}\mathbf{n}$, with $v_{N}=\mathbf{v}\cdot\mathbf{n}$. $u_{N}=\mathbf{u}\cdot\mathbf{n}$, while the tangential component $\mathbf{u}_{T}$ of the surface fluid flow is unknown and depends on $\mathbf{w}$ only implicitly through the variation of $\Gamma(t)$ and conservation laws represented by equations (2.2). The resulting system can be seen as an idealized model for the motion of a fluid layer embedded in bulk fluid, where one neglects friction forces between the surface and bulk fluids as well as any effect of the layer on the bulk flow. In such a physical setting, (2.6) means non-penetration of the bulk fluid through the material layer.

Material trajectories of points on the surface are defined by the flow field $\mathbf{u}$, rather than $\mathbf{w}$. We are also interested in a derivative determined by the variation of a quantity along the so-called normal trajectories defined below.

It is clear from (2.8) that this normal time derivative is an intrinsic surface quantity. Similar to the material derivative in (2.1), it can be expressed in terms of bulk derivatives if one assumes a smooth extension of $\mathbf{v}$ from $\mathcal{S}$ to its neighborhood:

$$\partial^{\circ}\mathbf{v}(t,\mathbf{x})=\frac{d}{dt}\mathbf{v}(t,\mathbf{x}^{n}(t,\mathbf{z}))=\Big{(}\frac{\partial\mathbf{v}^{e}}{\partial t}+(\mathbf{w}_{N}\cdot\nabla)\mathbf{v}^{e}\Big{)}(t,\mathbf{x})$$

(2.9)

for $(t,\mathbf{x})\in\mathcal{S}$. Comparing the material and normal time derivatives of a flow field $\mathbf{v}$ on the surface we find the equality

$$\mathbf{P}\accentset{\mbox{\large.}}{\mathbf{v}}=\mathbf{P}\partial^{\circ}\mathbf{v}+(\nabla_{\Gamma}\mathbf{v})\mathbf{u}_{T}.$$

With the splitting $\mathbf{v}=\mathbf{v}_{T}+\mathbf{v}_{N}$ we get

$$\mathbf{P}\accentset{\mbox{\large.}}{\mathbf{v}}=\mathbf{P}\partial^{\circ}\mathbf{v}_{T}+\mathbf{P}\partial^{\circ}\mathbf{v}_{N}+(\nabla_{\Gamma}\mathbf{v}_{T})\mathbf{u}_{T}+(\nabla_{\Gamma}\mathbf{v}_{N})\mathbf{u}_{T}.$$

(2.10)

Noting that $\mathbf{P}\mathbf{n}=0$ and $\mathbf{P}\partial^{\circ}\mathbf{n}=-\nabla_{\Gamma}w_{N}$ (cf. (2.16) in [16]), we rewrite $\mathbf{P}\partial^{\circ}\mathbf{v}_{N}$ as

$$\mathbf{P}\partial^{\circ}\mathbf{v}_{N}=\partial^{\circ}v_{N}\mathbf{P}\mathbf{n}+v_{N}\mathbf{P}\partial^{\circ}\mathbf{n}=v_{N}\mathbf{P}\partial^{\circ}\mathbf{n}=-v_{N}\nabla_{\Gamma}w_{N}.$$

We also have the relation $(\nabla_{\Gamma}\mathbf{v}_{N})\mathbf{u}_{T}=v_{N}\mathbf{H}\mathbf{u}_{T}$. Using these results and letting $\mathbf{v}=\mathbf{u}$ in (2.10) one obtains

$$\mathbf{P}\accentset{\mbox{\large.}}{\mathbf{u}}=\mathbf{P}\partial^{\circ}\mathbf{u}_{T}+(\nabla_{\Gamma}\mathbf{u}_{T})\mathbf{u}_{T}+w_{N}\mathbf{H}\mathbf{u}_{T}-\tfrac{1}{2}\nabla_{\Gamma}w_{N}^{2},$$

(2.11)

where we also used $u_{N}=w_{N}$. To derive an equation for the unknown tangential velocity $\mathbf{u}_{T}$, we apply the projection $\mathbf{P}$ to the first equation in (2.2). For $\mathbf{P}\accentset{\mbox{\large.}}{\mathbf{u}}$ we have the result (2.11). Note that the term $\tfrac{1}{2}\nabla_{\Gamma}w_{N}^{2}$ is known and can be treated as a source term. For a stationary surface ($w_{N}=0$) the normal time derivative is just the usual time derivative, $\mathbf{P}\partial^{\circ}\mathbf{u}_{T}=\frac{\partial\mathbf{u}_{T}}{\partial t}$. The term $(\nabla_{\Gamma}\mathbf{u}_{T})\mathbf{u}_{T}$ is the analog of the quadratic term in Navier-Stokes equations. Using ${\mathop{\,\rm div}}_{\Gamma}\mathbf{u}_{N}=u_{N}\kappa$ and $u_{N}=w_{N}$, the second equation in (2.2) yields ${\mathop{\,\rm div}}_{\Gamma}\mathbf{u}_{T}=-w_{N}\kappa$. We are not interested in variable density case and let $\rho=1$. Summarizing, from the surface Navier-Stokes equations (2.2) we get the following reduced system for $\mathbf{u}_{T}$ and $\pi$ which we call the tangential surface Navier-Stokes equations (TSNSE):

$$\left\{\begin{aligned} \mathbf{P}\partial^{\circ}\mathbf{u}_{T}+(\nabla_{\Gamma}\mathbf{u}_{T})\mathbf{u}_{T}+w_{N}\mathbf{H}\mathbf{u}_{T}-2\mu\mathbf{P}{\mathop{\,\rm div}}_{\Gamma}E_{s}(\mathbf{u}_{T})+\nabla_{\Gamma}\pi&=\mathbf{f}\\ {\mathop{\,\rm div}}_{\Gamma}\mathbf{u}_{T}&=g,\\ \end{aligned}\right.$$

(2.12)

with right-hand sides known in terms of geometric quantities, $w_{N}$ and the tangential component of the external area force $\mathbf{b}$:

$$g=-w_{N}\kappa,\qquad\mathbf{f}=\mathbf{b}_{T}+2\mu\mathbf{P}{\mathop{\,\rm div}}_{\Gamma}(w_{N}\mathbf{H})+\tfrac{1}{2}\nabla_{\Gamma}w_{N}^{2}.$$

(2.13)

In the remainder of this paper we study this TSNSE. Note that these equations have a structure similar to the standard incompressible Navier-Stokes equations in Euclidean domains. Important differences are that TSNSE is formulated on a space-time manifold that does not have an evident tensor product structure and, related to this, a normal time derivative $\mathbf{P}\partial^{\circ}$ instead of the standard time derivative is used and an additional term $w_{N}\mathbf{H}\mathbf{u}_{T}$ occurs. After some preliminary results in the next section, we introduce a well-posed weak formulation of the TSNSE in section 4.

To define evolving Hilbert spaces based on standard Bochner spaces, we need a suitable pushforward map. In the context of this paper it is natural to use a surface Piola transform as pushforward map, since this transform conserves the solenoidal property of a tangential vector field.

To define a surface variant of the Piola transform based on the normal flow map $\Phi_{t}^{n}:\,\Gamma_{0}\to\Gamma(t)$, we need some further notation. Below we always take $\mathbf{z}\in\Gamma_{0}$ and $\mathbf{x}:=\Phi^{n}_{t}(\mathbf{z})\in\Gamma(t)$. Since for each $t\in[0,T]$ the map $\Phi_{t}^{n}:\,\Gamma_{0}\to\Gamma(t)$ is a $C^{2}$-diffeomorphism, the differential $D\Phi_{t}^{n}(\mathbf{z}):\,(T\Gamma_{0})_{\mathbf{z}}\to T\Gamma(t)_{\mathbf{x}}$, is invertible. Define $J=J(t,\mathbf{z}):=\det D\Phi_{t}^{n}(\mathbf{z})$, $J^{-1}=J^{-1}(t,\mathbf{x})=\det D\Phi_{-t}^{n}(\mathbf{x})=J(t,\mathbf{z})^{-1}$. Denote by $\mathbf{D}=\mathbf{D}(t,\mathbf{z})$ and $\mathbf{D}^{-1}=\mathbf{D}^{-1}(t,\mathbf{x})$ the matrices of linear mappings given by $D\Phi_{t}^{n}(\mathbf{z})\mathbf{P}(\mathbf{z}):\mathbb{R}^{3}\to\mathbb{R}^{3}$ and $D\Phi_{-t}^{n}(\mathbf{x})\mathbf{P}(\mathbf{x})=[D\Phi_{t}^{n}(\mathbf{z})]^{-1}\mathbf{P}(\mathbf{x}):\mathbb{R}^{3}\to\mathbb{R}^{3}$, respectively. Note that $\mathbf{D}^{-1}\mathbf{D}=\mathbf{P}(\mathbf{z})$ and $\mathbf{D}\mathbf{D}^{-1}=\mathbf{P}(\mathbf{x})$ hold. For these mappings the following useful identities hold:

$$\begin{split}D_{\Gamma}\,(\mathbf{v}\circ\Phi_{-t}^{n})=\left(D_{\Gamma_{0}}\mathbf{v}\right)\mathbf{D}^{-1}\quad\text{for}~{}\mathbf{v}\in C^{1}(\Gamma_{0})^{3},\\ \left(D_{\Gamma}\,\mathbf{v}\right)\mathbf{D}=D_{\Gamma_{0}}(\mathbf{v}\circ\Phi_{t}^{n})\quad\text{for}~{}\mathbf{v}\in C^{1}\big{(}\Gamma(t)\big{)}^{3}.\end{split}$$

(3.3)

We need the Piola transform for arbitrary, not necessarily tangential vectors. For this it is convenient to define an invertible operator ${\mathbf{A}}(t,\mathbf{z}):\mathbb{R}^{3}\to\mathbb{R}^{3}$ such that $\mathbf{A}|_{T\Gamma_{0}}=J^{-1}D\Phi_{t}^{n}:\,T\Gamma_{0}\to T\Gamma(t)$ and $\mathbf{A}:T\Gamma_{0}^{\perp}\to T\Gamma(t)^{\perp}$. We use the operator

$${\mathbf{A}}(t,\mathbf{z})\mathbf{v}:=J^{-1}(t,\mathbf{x})\mathbf{D}(t,\mathbf{z})\mathbf{v}+\mathbf{n}_{\Gamma(t)}(\mathbf{x})\mathbf{n}_{\Gamma_{0}}(\mathbf{z})\cdot\mathbf{v}\,,\quad\mathbf{v}\in\mathbb{R}^{3}.$$

(3.4)

For ${\mathbf{A}}^{-1}(t,\mathbf{x}):=J(t,\mathbf{z})\mathbf{D}^{-1}(t,\mathbf{x})+\mathbf{n}_{\Gamma_{0}}(\mathbf{z})\mathbf{n}_{\Gamma(t)}(\mathbf{x})^{T}$ it holds ${\mathbf{A}}^{-1}(t,\mathbf{x}){\mathbf{A}}(t,\mathbf{z})=I_{\mathbb{R}^{3}}$. The matrices of $\mathbf{A}$ and $\mathbf{A}^{-1}$ in the standard basis are also denoted by $\mathbf{A}$ and $\mathbf{A}^{-1}$, respectively. Note that $\det\mathbf{A}=1$ holds. We define the surface Piola transform $P_{t}:\mathbb{R}^{3}\to\mathbb{R}^{3}$ by

$$(P_{t}\mathbf{v})(\mathbf{x}):=\mathbf{A}(t,\mathbf{z})\,\mathbf{v}(\mathbf{z}),\quad\mathbf{z}\in\Gamma_{0}.$$

(3.5)

This operator maps tangential vectors on $\Gamma_{0}$ to tangential vectors on $\Gamma(t)$ and for tangential vectors $\mathbf{v}$ it satisfies ${\mathop{\,\rm div}}_{\Gamma}P_{t}\mathbf{v}=0$ a.e. on $\Gamma(t)$ iff ${\mathop{\,\rm div}}_{\Gamma}\mathbf{v}=0$ a.e. on $\Gamma_{0}$, cf. [41].

We need some regularity properties of $\mathbf{D}$, $\mathbf{A}$ and $\mathbf{D}^{-1}$, $\mathbf{A}^{-1}$, which are collected in the following lemma. For a function $g\in C^{1}({\mathcal{S}}_{0})$ the maximum norm is $\|g\|_{C^{1}(\mathcal{S}_{0})}:=\max_{(t,\mathbf{z})\in\mathcal{S}_{0}}\big{(}|g(t,\mathbf{z})|+|\nabla_{S_{0}}g(t,\mathbf{z})|\big{)}$ and similarly for vector and matrix valued functions as well as for such functions on $\mathcal{S}$.

For constructing suitable evolving Hilbert spaces, we first define tangential velocity spaces on $\Gamma(t)$. The notation $(\cdot,\cdot)_{0,t}$ and $\|\cdot\|_{0,t}$ is used for the canonical inner product and norm in $L^{2}(\Gamma(t))$. We need the Sobolev spaces of order one

$$H^{1}(t):=\{\mathbf{v}\in H^{1}(\Gamma(t))^{3}~{}|~{}\mathbf{v}\cdot\mathbf{n}=0~{}~{}\text{on}~{}~{}\Gamma(t)\},$$

with the inner product $(\cdot,\cdot)_{1,t}:=(\cdot,\cdot)_{0,t}+(D_{\Gamma}\cdot,D_{\Gamma}\cdot)_{0,t}$, and its closed subspace of divergence free tangential fields,

$$V_{\hskip 0.0pt1}(t):=\{\,\mathbf{v}\in H^{1}(t)~{}|~{}{\mathop{\,\rm div}}_{\Gamma}\mathbf{v}=0~{}~{}\text{a.e. on}~{}~{}\Gamma(t)\,\}.$$

The space $V_{\hskip 0.0pt0}(t)$ is defined as closure of a space of smooth div-free tangential fields in the $L^{2}(\Gamma(t))$ norm:

$$V_{\hskip 0.0pt0}(t):=\overline{\mathcal{V}(t)}^{\|\cdot\|_{0,t}},\quad\mathcal{V}(t):=\{\,\mathbf{v}\in C^{1}(\Gamma(t))^{3}~{}|~{}\mathbf{v}\cdot\mathbf{n}=0,~{}{\mathop{\,\rm div}}_{\Gamma}\mathbf{v}=0~{}~{}\text{on}~{}~{}\Gamma(t)\,\}.\\$$

The space of smooth functions $\mathcal{V}(t)$ is dense not only in $V_{\hskip 0.0pt0}(t)$ but also in $V_{\hskip 0.0pt1}(t)$. Indeed, for any tangential velocity field $\mathbf{u}\in L^{2}(\Gamma(t))^{3}$ on the $C^{2}$ smooth surface $\Gamma(t)$ we have a Helmholz decomposition $\mathbf{u}=\nabla_{\Gamma}\psi+\mathbf{n}\times(\nabla_{\Gamma}\phi)+\mathbf{h}$ with some $\psi,\phi\in H^{1}(\Gamma(t))$ and a harmonic field $\mathbf{h}\in C^{1}(\Gamma(t))^{3}$ [35]. For $\mathbf{u}\in V_{\hskip 0.0pt1}(t)$ we have $\psi=0$ and the result follows from the density of $C^{2}$-smooth functions in $H^{1}(\Gamma(t))$. Therefore, endowed with canonical scalar products, the spaces form a Gelfand triple $V_{\hskip 0.0pt1}(t)\hookrightarrow V_{\hskip 0.0pt0}(t)\hookrightarrow V_{\hskip 0.0pt1}(t)^{\prime}$. We also have that the dense embedding $V_{\hskip 0.0pt1}(t)\hookrightarrow V_{\hskip 0.0pt0}(t)$ is compact. Here and later $H^{\prime}$ always denotes a dual of a Hilbert space $H$, and we adopt the common notation $H^{-1}(t)$ for $H^{1}(t)^{\prime}$.

For the space $L^{2}(t):=\{\mathbf{v}\in L^{2}(\Gamma(t))^{3}~{}|~{}\mathbf{v}\cdot\mathbf{n}=0~{}\text{a.e. on}~{}\Gamma(t)\}$ we define a pushforward map $\phi_{t}:L^{2}(0)\to L^{2}(t)$, based on the Piola transform, by

$$(\phi_{t}\mathbf{v})(\mathbf{x})=(P_{t}\mathbf{v})(\mathbf{x})=\mathbf{A}(t,\mathbf{z})\mathbf{v}(\mathbf{z}),\quad\mathbf{v}\in L^{2}(0),~{}\mathbf{x}=\Phi_{t}^{n}(\mathbf{z}),\,\mathbf{z}\in\Gamma_{0}.$$

(3.8)

The inverse map (pullback) is given by $(\phi_{-t}\mathbf{v})(\mathbf{z})=\mathbf{A}^{-1}(t,\mathbf{x})\mathbf{v}(\mathbf{x})$, $\mathbf{v}\in L^{2}(t)$. Since $\mathbf{A}\in C^{1}({\mathcal{S}}_{0})^{3\times 3}$, the restriction of $\phi_{t}$ to $H^{1}(0)$ is a pushforward map from $H^{1}(0)$ to $H^{1}(t)$. Because $\phi_{t}$ is based on the Piola transform and thus conserves the solenoidal property, we also have that $\phi_{t}$ is a pushforward map from $V_{\hskip 0.0pt0}(0)$ to $V_{\hskip 0.0pt0}(t)$, and from $V_{\hskip 0.0pt1}(0)$ to $V_{\hskip 0.0pt1}(t)$. For this pushforward map we have for $\mathbf{v}\in H^{1}(0)$:

$$\begin{split}\|\phi_{t}\mathbf{v}\|_{1,t}&=\left(\|\mathbf{A}\,\mathbf{v}\circ\Phi_{-t}^{n}\|^{2}_{0,t}+\|D_{\Gamma}(\mathbf{A}\,\mathbf{v}\circ\Phi_{-t}^{n})\|^{2}_{0,t}\right)^{\frac{1}{2}}\\ &\leq\left(\|\mathbf{A}\|_{C(\Gamma(t))}+\|D_{\Gamma}\mathbf{A}\|_{C(\Gamma(t))}\right)\|\mathbf{v}\circ\Phi_{-t}^{n}\|_{0,t}+\|\mathbf{A}\|_{C(\Gamma(t))}\|D_{\Gamma}(\mathbf{v}\circ\Phi_{-t}^{n})\|_{0,t}\\ &\leq\left(\|\mathbf{A}\|_{C(\Gamma(t))}+\|D_{\Gamma}\mathbf{A}\|_{C(\Gamma(t))}\right)\|J\|_{C(\Gamma_{0})}^{\frac{1}{2}}\|\mathbf{v}\|_{0,0}\\ &\qquad+\|\mathbf{A}\|_{C(\Gamma(t))}\|\mathbf{D}^{-1}\|_{C(\Gamma(t))}\|J\|_{C(\Gamma_{0})}^{\frac{1}{2}}\|D_{\Gamma}\mathbf{v}\|_{0,0}.\end{split}$$

The result (3.6) implies that the norms $\|\mathbf{A}\|_{C(\Gamma(t))}$, $\|\mathbf{D}^{-1}\|_{C(\Gamma(t))}$, $\|D_{\Gamma}\mathbf{A}\|_{C(\Gamma(t))}$, $\|J\|_{C(\Gamma_{0})}$ are uniformly bounded in $t$ and thus

$$\sup_{t\in[0,T]}\|\phi_{t}\mathbf{v}\|_{1,t}\leq C\|\mathbf{v}\|_{1,0}$$

holds with some $C$ independent of $\mathbf{v}\in H^{1}(0)$. With similar arguments one easily shows that $\|\phi_{-t}\mathbf{v}\|_{1,0}\leq C\|\mathbf{v}\|_{1,t}$ holds for all $\mathbf{v}\in H^{1}(t)$, with $C$ independent of $\mathbf{v}$ and $t$. These bounds remain obviously true if $H^{1}(0)$, $H^{1}(t)$ and the corresponding norms are replaced by $V_{\hskip 0.0pt0}(0)$, $V_{\hskip 0.0pt0}(t)$ and the corresponding norms. Using (3.6) one shows that the maps $t\to\|\phi_{t}\mathbf{v}\|_{1,t}$ and $t\to\|\phi_{t}\mathbf{v}\|_{0,t}$ are continuous. These properties imply that $\{V_{\hskip 0.0pt0}(t),\phi_{t}\}_{t\in[0,T]}$, $\{H^{1}(t),\phi_{t}\}_{t\in[0,T]}$, and $\{V_{\hskip 0.0pt1}(t),\phi_{t}\}_{t\in[0,T]}$ are “compatible pairs” in the sense of Definition 2.4 in [1]. This compatibility structure induces natural properties of the evolving spaces defined as follows:

$$\begin{split}L_{V_{\hskip 0.0pt1}}^{2}&:=\{\,\mathbf{v}:\,[0,T]\to\bigcup_{t\in[0,T]}\{t\}\times V_{\hskip 0.0pt1}(t),\,t\to(t,\bar{\mathbf{v}}(t))~{}|~{}\phi_{-(\cdot)}\bar{\mathbf{v}}(\cdot)\in L^{2}(0,T;V_{\hskip 0.0pt1}(0))\,\},\\ L_{V_{\hskip 0.0pt1}^{\prime}}^{2}&:=\{\,\mathbf{g}:\,[0,T]\to\bigcup_{t\in[0,T]}\{t\}\times V_{\hskip 0.0pt1}(t)^{\prime},\,t\to(t,\bar{\mathbf{g}}(t))~{}|~{}\phi_{(\cdot)}^{\ast}\bar{\mathbf{g}}(\cdot)\in L^{2}(0,T;V_{\hskip 0.0pt1}(0)^{\prime})\,\},\\ L^{2}_{V_{\hskip 0.0pt0}}&:=\{\,\mathbf{v}:\,[0,T]\to\bigcup_{t\in[0,T]}\{t\}\times V_{\hskip 0.0pt0}(t),\,t\to(t,\bar{\mathbf{v}}(t))~{}|~{}\phi_{-(\cdot)}\bar{\mathbf{v}}(\cdot)\in L^{2}(0,T;V_{\hskip 0.0pt0}(0))\,\},\end{split}$$

where $\phi_{t}^{\ast}$ is the dual of $\phi_{t}$. We shall also need the spaces $L_{V_{\hskip 0.0pt0}}^{\infty}$, $L^{2}_{H^{1}}$, $L^{2}_{H^{-1}}$, which are defined analogously, and the spaces of smooth space-time functions

$$\begin{split}\mathcal{D}&=\{\,\mathbf{v}\in L_{V_{\hskip 0.0pt1}}^{2}~{}|~{}\phi_{-(\cdot)}\bar{\mathbf{v}}(\cdot)\in C^{\infty}\big{(}[0,T];\mathcal{V}(0))\,\},\\ \mathcal{D}_{0}&=\{\,\mathbf{v}\in L_{V_{\hskip 0.0pt1}}^{2}~{}|~{}\phi_{-(\cdot)}\mathbf{v}(\cdot)\in C_{0}^{\infty}\big{(}(0,T);\mathcal{V}(0))\,\}.\end{split}$$

(3.9)

Note that functions in $\mathcal{D}_{0}$ have zero traces on $\partial\mathcal{S}$. With a slight abuse of notation we identify $\bar{\mathbf{v}}(t)$ with $\mathbf{v}(t)=(t,\bar{\mathbf{v}}(t))$.

In [1] it is shown that if $V_{\hskip 0.0pt1}(0)\hookrightarrow V_{\hskip 0.0pt0}(0)\hookrightarrow V_{\hskip 0.0pt1}(0)^{\prime}$ is a Gelfand triple, with a compact embedding $V_{\hskip 0.0pt1}(0)\hookrightarrow V_{\hskip 0.0pt0}(0)$, and both $\{V_{\hskip 0.0pt0}(t),\phi_{t}\}_{t\in[0,T]}$ and $\{V_{\hskip 0.0pt1}(t),\phi_{t}\}_{t\in[0,T]}$ are compatible pairs, then the $L$-spaces inherit certain properties of the standard Bochner spaces. In particular, cf. Section 2 in [1], the spaces $L_{V_{\hskip 0.0pt1}}^{2}$ and $L^{2}_{V_{\hskip 0.0pt0}}$ with

$$(\mathbf{u},\mathbf{v})_{\bf 1}=\int_{0}^{T}(\mathbf{u}(t),\mathbf{v}(t))_{1,t}\,dt,\quad(\mathbf{u},\mathbf{v})_{\bf 0}=\int_{0}^{T}(\mathbf{u}(t),\mathbf{v}(t))_{0,t}\,dt,$$

are separable Hilbert spaces, homeomorphic to $L^{2}(0,T;V_{\hskip 0.0pt1}(0))$ and $L^{2}(0,T;V_{\hskip 0.0pt0}(0))$ respectively. Furthermore, the embedding $L_{V_{\hskip 0.0pt1}}^{2}\hookrightarrow L^{2}_{V_{\hskip 0.0pt0}}$ is dense and compact, the dual space $(L_{V_{\hskip 0.0pt1}}^{2})^{\prime}$ is isometrically isomorphic to $L_{V_{\hskip 0.0pt1}^{\prime}}^{2}$, and

$$L_{V_{\hskip 0.0pt1}}^{2}\hookrightarrow L^{2}_{V_{\hskip 0.0pt0}}\hookrightarrow L_{V_{\hskip 0.0pt1}^{\prime}}^{2}$$

is a Gelfand triple. The space $\mathcal{D}_{0}$ is dense in $L_{V_{\hskip 0.0pt1}}^{2}$ and so $\mathcal{D}_{0}$ is dense also in $L^{2}_{V_{\hskip 0.0pt0}}$. By the same arguments $L_{H^{1}}^{2}$ is also a Hilbert space with inner product $(\cdot,\cdot)_{\bf 1}$. The subspace of smooth functions

$$\widehat{\mathcal{D}}_{0}=\{\,\mathbf{v}\in L_{H^{1}}^{2}~{}|~{}\phi_{-(\cdot)}\mathbf{v}(\cdot)\in C_{0}^{\infty}\big{(}(0,T);C^{1}(\Gamma_{0},T\Gamma_{0})\big{)}\,\}$$

(3.10)

is dense in $L_{H^{1}}^{2}$ and $\left(L_{H^{1}}^{2}\right)^{\prime}\simeq L_{H^{-1}}^{2}$ holds. Note that $L_{V_{\hskip 0.0pt1}}^{2}$ is a closed subspace of $L_{H^{1}}^{2}$ and that functions in ${\mathcal{D}}_{0}$ are solenoidal and functions in $\widehat{\mathcal{D}}_{0}$ are not necessarily solenoidal.