Relative Entropy Inequality: Proof of Proposition 5
The general structure of the proof is in parts similar to the proof of [12, Proposition 10]. In what follows, we thus mainly focus on how to exploit the boundary conditions for the velocity fields (u, v) and a boundary adapted extension \(\xi \) of the strong interface unit normal in these computations.
Step 1: Since \(\rho (\chi _v)\) is an affine function of \(\chi _v\), it consequently satisfies
$$\begin{aligned} \int _{\Omega }\rho (\chi _v(\cdot , T')) \varphi (\cdot , T') \,\mathrm {d}x- \int _{\Omega }\rho (\chi ^0_v) \varphi (\cdot , 0) \,\mathrm {d}x= \int _{0}^T \int _{\Omega }\rho (\chi _v) (\partial _t\varphi + (v \cdot \nabla ) \varphi ) \,\mathrm {d}x\,\mathrm {d}t\end{aligned}$$
(46)
for almost every \(T' \in [0, T]\) and all \(\varphi \in C^\infty ({\overline{\Omega }} \times [0, T])\). By the regularity of v and an approximation argument, we may test this equation with \(v \cdot \eta \) for any \(\eta \in C^\infty ({\overline{\Omega }}\times [0,T];{\mathbb {R}}^d)\), yielding
$$\begin{aligned}&\int _{\Omega }\rho (\chi _v(\cdot , T')) v(\cdot , T') \cdot \eta (\cdot , T') \,\mathrm {d}x- \int _{\Omega }\rho (\chi ^0_v) v(\cdot , 0) \cdot \eta (\cdot , 0)\,\mathrm {d}x\nonumber \\&=\int _{0}^{T'}\int _{\Omega }\rho (\chi _v) (v \cdot \partial _t\eta + \eta \cdot \partial _tv) \,\mathrm {d}x\,\mathrm {d}t\nonumber \\&\quad + \int _{0}^{T'} \int _{\Omega }\rho (\chi _v)(\eta \cdot (v \cdot \nabla ) v + v \cdot (v \cdot \nabla ) \eta ) \,\mathrm {d}x\,\mathrm {d}t\end{aligned}$$
(47)
for almost every \(T' \in [0, T]\). Next, we subtract from (47) the equation for the momentum balance (37) of the strong solution. It follows that the velocity field v of the strong solution satisfies
$$\begin{aligned} 0 =&\int _{0}^{T'} \int _{\Omega }\rho (\chi _v) \eta \cdot \partial _tv \,\mathrm {d}x\,\mathrm {d}t+ \int _{0}^{T'} \int _{\Omega }\rho (\chi _v)\eta \cdot (v \cdot \nabla ) v \,\mathrm {d}x\,\mathrm {d}t\nonumber \\&+ \int _{0}^{T'} \int _{\Omega }\mu (\nabla v + \nabla v^T) : \nabla \eta \,\mathrm {d}x\,\mathrm {d}t- \sigma \int _{0}^{T'} \int _{I_v(t)} {\text {H}}_{I_v} \cdot \, \eta \,\mathrm {d}S\,\mathrm {d}t\end{aligned}$$
(48)
for almost every \(T' \in [0,T]\) and every test vector field \(\eta \in C^\infty ({\overline{\Omega }}\times [0,T];{\mathbb {R}}^d)\) such that \(\nabla \cdot \eta = 0\) and \((\eta \cdot n_{\partial \Omega }) |_{\partial \Omega } = 0\). For any such test vector field \(\eta \), note that by means of (16c), the incompressibility of \(\eta \) as well as \((\eta \cdot n_{\partial \Omega })|_{\partial \Omega } =0\), we may rewrite
$$\begin{aligned} - \sigma \int _{0}^{T'} \int _{I_v(t)} {\text {H}}_{I_v} \cdot \, \eta \,\mathrm {d}S\,\mathrm {d}t&= \sigma \int _{0}^{T'} \int _{I_v(t)} (\nabla \cdot \xi ) \eta \cdot n_{I_v} \,\mathrm {d}S\,\mathrm {d}t\nonumber \\&= - \sigma \int _{0}^{T'} \int _{\Omega }\chi _v (\eta \cdot \nabla ) (\nabla \cdot \xi ) \,\mathrm {d}x\,\mathrm {d}t. \end{aligned}$$
(49)
Hence, we deduce from inserting (49) back into (48) that
$$\begin{aligned} 0 =&\int _{0}^{T'} \int _{\Omega }\rho (\chi _v) \eta \cdot \partial _tv \,\mathrm {d}x\,\mathrm {d}t+ \int _{0}^{T'} \int _{\Omega }\rho (\chi _v)\eta \cdot (v \cdot \nabla ) v \,\mathrm {d}x\,\mathrm {d}t\nonumber \\&+ \int _{0}^{T'} \int _{\Omega }\mu (\nabla v + \nabla v^T) : \nabla \eta \,\mathrm {d}x\,\mathrm {d}t- \sigma \int _{0}^{T'} \int _{\Omega }\chi _v (\eta \cdot \nabla ) (\nabla \cdot \xi ) \,\mathrm {d}x\,\mathrm {d}t\end{aligned}$$
(50)
for almost every \(T' \in [0,T]\) and every test vector field \(\eta \in C^\infty ({\overline{\Omega }}\times [0,T];{\mathbb {R}}^d)\) such that \(\nabla \cdot \eta = 0\) and \((\eta \cdot n_{\partial \Omega }) |_{\partial \Omega } = 0\). The merit of rewriting (48) into the form (50) consists of the following observation. Consider a test vector field \(\eta \in C^\infty ([0,T];H^1(\Omega ;{\mathbb {R}}^d))\) such that \(\nabla \cdot \eta = 0\) and \((\eta \cdot n_{\partial \Omega }) |_{\partial \Omega } = 0\). Denoting by \(\psi \) a standard mollifier, for every \(k\in {\mathbb {N}}\) by \(\psi _k:=k^d\psi (k\cdot )\) its usual rescaling, and by \(P_{\Omega }\) the Helmholtz projection associated with the smooth domain \(\Omega \), it follows from standard theory (e.g., by a combination of [30] and standard \(W^{m,2}(\Omega )\)-elliptic regularity theory – see also Appendix A) that \(\eta _k := P_{\Omega }(\psi _k *\eta )\) is an admissible test vector field for (50). Moreover, taking the limit \(k\rightarrow \infty \) in (50) with \(\eta _k\) as test vector fields is admissible and results in
$$\begin{aligned} 0 =&\int _{0}^{T'} \int _{\Omega }\rho (\chi _v) \eta \cdot \partial _tv \,\mathrm {d}x\,\mathrm {d}t+ \int _{0}^{T'} \int _{\Omega }\rho (\chi _v)\eta \cdot (v \cdot \nabla ) v \,\mathrm {d}x\,\mathrm {d}t\nonumber \\&+ \int _{0}^{T'} \int _{\Omega }\mu (\nabla v + \nabla v^T) : \nabla \eta \,\mathrm {d}x\,\mathrm {d}t- \sigma \int _{0}^{T'} \int _{\Omega }\chi _v (\eta \cdot \nabla ) (\nabla \cdot \xi ) \,\mathrm {d}x\,\mathrm {d}t\end{aligned}$$
(51)
for almost every \(T' \in [0,T]\) and every test vector field \(\eta \in C^\infty ([0,T];H^1(\Omega ;{\mathbb {R}}^d))\) such that \(\nabla \cdot \eta = 0\) and \((\eta \cdot n_{\partial \Omega }) |_{\partial \Omega } = 0\). As an important consequence, because of the boundary condition for the velocity fields (u, v) and their solenoidality, we may choose (after performing a mollification argument in the time variable) \(\eta =u-v\) as a test function in (51) which entails for almost every \(T' \in [0,T]\)
$$\begin{aligned} 0 =&\int _{0}^{T'} \int _{\Omega }\rho (\chi _v) (u-v) \cdot \partial _tv \,\mathrm {d}x\,\mathrm {d}t+ \int _{0}^{T'} \int _{\Omega }\rho (\chi _v) (u-v) \cdot (v \cdot \nabla ) v \,\mathrm {d}x\,\mathrm {d}t\nonumber \\&+ \int _{0}^{T'} \int _{\Omega }\mu (\nabla v {+} \nabla v^{\mathsf {T}}) : \nabla (u{-}v) \,\mathrm {d}x\,\mathrm {d}t- \sigma \int _{0}^{T'} \int _{\Omega }\chi _v ((u{-}v)\cdot \nabla ) (\nabla \cdot \xi ) \,\mathrm {d}x\,\mathrm {d}t. \end{aligned}$$
(52)
We proceed by testing the analogue of (46) for the phase-dependent density \(\rho (\chi _u)\) with the test function \(\frac{1}{2}|v|^2\), obtaining for almost every \(T' \in [0,T]\)
$$\begin{aligned}&\int _{\Omega }\frac{1}{2}\rho (\chi _u(\cdot , T')) |v(\cdot , T')|^2 \,\mathrm {d}x- \int _{\Omega }\frac{1}{2} \rho (\chi ^0_u) |v_0(\cdot )|^2 \,\mathrm {d}x\nonumber \\&=\int _{0}^{T'} \int _{\Omega }\rho (\chi _u) v \cdot \partial _tv \,\mathrm {d}x\,\mathrm {d}t+ \int _{0}^{T'} \int _{\Omega }\rho (\chi _u) v \cdot (u \cdot \nabla ) v \,\mathrm {d}x\,\mathrm {d}t. \end{aligned}$$
(53)
We next want to test (39) with the fluid velocity v. Modulo a mollification argument in the time variable, we have to argue that \(\nabla v\) does not jump across the interface so that v is an admissible test function. Indeed, since the tangential derivative \((\tau _{I_v}\cdot \nabla )v\) is continuous across the interface it follows from \(\nabla \cdot v=0\) that also \(n_{I_v}\cdot (n_{I_v}\cdot \nabla )v\) does not jump across \(I_v\). The only component which may jump is thus \(\tau _{I_v}\cdot (n_{I_v}\cdot \nabla )v\). However, this is ruled out by the equilibrium condition for the stresses along \(I_v\) together with having \(\mu _+=\mu _-\). In summary, using v in (39) implies
$$\begin{aligned} -&\int _{\Omega }\rho (\chi _u(\cdot , T')) u(\cdot , T') \cdot v (\cdot , T') \,\mathrm {d}x+ \int _{\Omega }\rho (\chi _u^0)) u_0 \cdot v_0 (\cdot ) \,\mathrm {d}x\nonumber \\ -&\int _{0}^{T'} \int _{\Omega }\mu (\nabla u + \nabla u^{\mathsf {T}}) : \nabla v \,\mathrm {d}x\,\mathrm {d}t\nonumber \\ =&-\int _{0}^{T'} \int _{\Omega }\rho (\chi _u) u \cdot \partial _t v \,\mathrm {d}x\,\mathrm {d}t- \int _{0}^{T'} \int _{\Omega }\rho (\chi _u) u \cdot ( u \cdot \nabla )v \,\mathrm {d}x\,\mathrm {d}t\nonumber \\ {}&+ \sigma \int _{0}^{T'} \int _{{\overline{\Omega }} \times {\mathbb {S}}^{d-1}} ({\text{ Id }}- s \otimes s) : \nabla v \,\mathrm {d}V_t(x,s)\,\mathrm {d}t \end{aligned}$$
(54)
for almost every \(T' \in [0,T]\). We finally use \(\sigma (\nabla \cdot \xi ) \) as a test function in the transport equation (40) for the indicator function \(\chi _u\) of the varifold solution. Hence, we obtain
$$\begin{aligned}&\sigma \int _{\Omega }\chi _u(\cdot , T') (\nabla \cdot \xi )(\cdot , T') \,\mathrm {d}x- \int _{\Omega }\chi ^0_u (\nabla \cdot \xi )(\cdot , 0) \,\mathrm {d}x\\&= \sigma \int _{0}^{T'} \int _{\Omega }\chi _u (\nabla \cdot \partial _t\xi + (u \cdot \nabla ) (\nabla \cdot \xi ) ) \,\mathrm {d}x\,\mathrm {d}t. \end{aligned}$$
for almost every \(T' \in [0,T]\). Based on the boundary condition (16b), which in turn in particular implies \((\partial _t\xi \cdot n_{\partial \Omega }) |_{ \partial \Omega } =\partial _t(\xi \cdot n_{\partial \Omega })|_{ \partial \Omega }=0\), we may integrate by parts to upgrade the previous display to
$$\begin{aligned}&- \sigma \int _{\Omega }n_u(\cdot , T') \cdot \xi (\cdot , T') \,\mathrm {d}|\nabla \chi _u(\cdot , T)|+ \int _{\Omega }n^0_u \cdot \xi (\cdot , 0) \,\mathrm {d}|\nabla \chi _u(\cdot , 0)|\nonumber \\&=- \sigma \int _{0}^{T'} \int _{\Omega }n_u \cdot \partial _t\xi \,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t+ \sigma \int _{0}^{T'} \int _{\Omega }\chi _u (u \cdot \nabla ) (\nabla \cdot \xi ) \,\mathrm {d}x\,\mathrm {d}t\end{aligned}$$
(55)
for almost every \(T' \in [0,T]\).
Step 2: Summing (52), (53), (41) as well as (54), we obtain
$$\begin{aligned}&LHS_{kin}(T') + LHS_{visc} + LHS_{surEn}(T') \nonumber \\&\le RHS_{kin}(0) + RHS_{surEn}(0) + RHS_{dt} + RHS_{adv} + RHS_{surTen}, \end{aligned}$$
(56)
where the individual terms are given by (cf. the proof of [12, Proposition 10])
$$\begin{aligned} LHS_{kin}(T')&:= \int _{\Omega }\frac{1}{2}\rho (\chi _u(\cdot , T')) |u{-}v|^2(\cdot , T') \,\mathrm {d}x,\, \end{aligned}$$
(57)
$$\begin{aligned} RHS_{kin}(0)&:= \int _{\Omega }\frac{1}{2}\rho (\chi ^0_u) |u_0-v_0|^2\,\mathrm {d}x, \end{aligned}$$
(58)
$$\begin{aligned} LHS_{surEn}(T')&:= \sigma |\nabla \chi _u(\cdot , T')|(\Omega ) + \sigma \int _{\Omega }(1- \theta _{T'}) \, \mathrm {d} |V_{T'}|_{{\mathbb {S}}^{d-1}}(x), \end{aligned}$$
(59)
$$\begin{aligned} RHS_{surEn}(0)&:= \sigma |\nabla \chi ^0_u(\cdot )|(\Omega ), \end{aligned}$$
(60)
$$\begin{aligned} LHS_{visc}&:= \int _{0}^{T'} \int _{\Omega }\frac{\mu }{2}|\nabla (u-v) + \nabla (u-v)^{\mathsf {T}} |^2 \,\mathrm {d}x\,\mathrm {d}t, \end{aligned}$$
(61)
$$\begin{aligned} RHS_{dt}&:=- \int _{0}^{T'} \int _{\Omega }(\rho (\chi _v) - \rho (\chi _u)) (u-v) \cdot \partial _tv \,\mathrm {d}x\,\mathrm {d}t, \end{aligned}$$
(62)
$$\begin{aligned} RHS_{adv}:=&- \int _{0}^{T'} \int _{\Omega }(\rho (\chi _u) - \rho (\chi _v)) (u-v) \cdot (v \cdot \nabla ) v \,\mathrm {d}x\,\mathrm {d}t\nonumber \\ {}&- \int _{0}^{T'} \int _{\Omega }\rho (\chi _u) (u-v) \cdot ((u-v) \cdot \nabla ) v \,\mathrm {d}x\,\mathrm {d}t, \end{aligned}$$
(63)
$$\begin{aligned} RHS_{surTen}&:= - \sigma \int _{0}^{T'} \int _{\Omega }\chi _v ((u{-}v)\cdot \nabla ) (\nabla \cdot \xi ) \,\mathrm {d}x\,\mathrm {d}t\nonumber \\&\quad + \sigma \int _{0}^{T'} \int _{{\overline{\Omega }} \times {\mathbb {S}}^{d-1}} ({\text {Id}}- s \otimes s) : \nabla v \,\mathrm {d}V_t(x,s)\,\mathrm {d}t. \end{aligned}$$
(64)
Adding zeros, \(\nabla \cdot v =0\), the boundary condition \(n_{\partial \Omega } \cdot (\nabla v {+} (\nabla v)^{\mathsf {T}})\xi = n_{\partial \Omega } \cdot (\nabla v {+} (\nabla v)^{\mathsf {T}}) (\mathrm {Id} - n_{\partial \Omega }\otimes n_{\partial \Omega })\xi = 0\) due to (36) and (16b), and the compatibility condition (42) allow to rewrite the second term of (64) as follows
$$\begin{aligned}&\sigma \int _{0}^{T'} \int _{{\overline{\Omega }} \times {\mathbb {S}}^{d-1}} ({\text{ Id }}- s \otimes s) : \nabla v \,\mathrm {d}V_t(x,s)\,\mathrm {d}t\nonumber \\ {}&= - \sigma \int _{0}^{T'} \int _{{\overline{\Omega }} \times {\mathbb {S}}^{d-1}} (s - \xi ) \cdot ((s - \xi ) \cdot \nabla ) v \,\mathrm {d}V_t(x,s)\,\mathrm {d}t\nonumber \\ {}&\quad - \sigma \int _{0}^{T'} \int _{{\overline{\Omega }} \times {\mathbb {S}}^{d-1}} s \cdot (\nabla v + (\nabla v)^{\mathsf {T}})\xi \,\mathrm {d}V_t(x,s)\,\mathrm {d}t\nonumber \\ {}&\quad + \sigma \int _{0}^{T'} \int _{{\overline{\Omega }} \times {\mathbb {S}}^{d-1}} \xi \cdot ( \xi \cdot \nabla ) v \,\mathrm {d}V_t(x,s)\,\mathrm {d}t\nonumber \\ {}&= - \sigma \int _{0}^{T'} \int _{{\overline{\Omega }} \times {\mathbb {S}}^{d-1}} (s - \xi ) \cdot ((s - \xi ) \cdot \nabla ) v \,\mathrm {d}V_t(x,s)\,\mathrm {d}t\nonumber \\ {}&\quad - \sigma \int _{0}^{T'} \int _{\Omega }\xi \cdot (n_u \cdot \nabla ) v \,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t- \sigma \int _{0}^{T'} \int _{\Omega }n_u \cdot (\xi \cdot \nabla ) v \,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t\nonumber \\ {}&\quad + \sigma \int _{0}^{T'} \int _{{\overline{\Omega }}} \xi \cdot ( \xi \cdot \nabla ) v \,\mathrm {d}|V_t|_{{\mathbb {S}}^{d-1}}\,\mathrm {d}t. \end{aligned}$$
(65)
Furthermore, because of (44) we obtain
$$\begin{aligned}&\sigma \int _{0}^{T'} \int _{{\overline{\Omega }}} \xi \cdot ( \xi \cdot \nabla ) v \,\mathrm {d}|V_t|_{{\mathbb {S}}^{d-1}}\,\mathrm {d}t\nonumber \\ {}&=\sigma \int _{0}^{T'} \int _{\Omega }(1 {-} \theta _t) \xi \cdot ( \xi \cdot \nabla ) v \,\mathrm {d}|V_t|_{{\mathbb {S}}^{d-1}}\,\mathrm {d}t+ \sigma \int _{0}^{T'} \int _{\Omega }\theta _t \xi \cdot ( \xi \cdot \nabla ) v \,\mathrm {d}|V_t|_{{\mathbb {S}}^{d-1}}\,\mathrm {d}t\nonumber \\ {}&\quad + \sigma \int _{0}^{T'} \int _{\partial \Omega } \xi \cdot ( \xi \cdot \nabla ) v \,\mathrm {d}|V_t|_{{\mathbb {S}}^{d-1}}\,\mathrm {d}t\nonumber \\ {}&=\sigma \int _{0}^{T'} \int _{\Omega }(1 - \theta _t) \xi \cdot ( \xi \cdot \nabla ) v \,\mathrm {d}|V_t|_{{\mathbb {S}}^{d-1}}\,\mathrm {d}t+ \sigma \int _{0}^{T'} \int _{\Omega }\xi \cdot ( \xi \cdot \nabla ) v \,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t\nonumber \\ {}&\quad + \sigma \int _{0}^{T'} \int _{\partial \Omega } \xi \cdot ( \xi \cdot \nabla ) v \,\mathrm {d}|V_t|_{{\mathbb {S}}^{d-1}}\,\mathrm {d}t. \end{aligned}$$
(66)
The combination of (64), (65) and (66) together with \(\nabla \cdot v=0\) then implies
$$\begin{aligned} RHS_{surTen} =&- \sigma \int _{0}^{T'} \int _{{\overline{\Omega }} \times {\mathbb {S}}^{d-1}} (s - \xi ) \cdot ((s - \xi ) \cdot \nabla ) v \,\mathrm {d}V_t(x,s)\,\mathrm {d}t\nonumber \\ {}&+\sigma \int _{0}^{T'} \int _{\Omega }(1 - \theta _t) \xi \cdot ( \xi \cdot \nabla ) v \,\mathrm {d}|V_t|_{{\mathbb {S}}^{d-1}}\,\mathrm {d}t\nonumber \\ {}&+ \sigma \int _{0}^{T'} \int _{\partial \Omega } \xi \cdot ( \xi \cdot \nabla ) v \,\mathrm {d}|V_t|_{{\mathbb {S}}^{d-1}}\,\mathrm {d}t \nonumber \\&- \sigma \int _{0}^{T'} \int _{\Omega }\chi _v ((u-v)\cdot \nabla ) (\nabla \cdot \xi ) \,\mathrm {d}x\,\mathrm {d}t\nonumber \\ {}&- \sigma \int _{0}^{T'} \int _{\Omega }\xi \cdot ( (n_u - \xi )\cdot \nabla ) v \,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t\nonumber \\ {}&- \sigma \int _{0}^{T'} \int _{\Omega }(n_u - \xi ) \cdot ( \xi \cdot \nabla ) v \,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t\nonumber \\ {}&+ \sigma \int _{0}^{T'} \int _{\Omega }( {\text{ Id }}- \xi \otimes \xi ): \nabla v \,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t. \end{aligned}$$
(67)
In summary, plugging back (57)–(63) and (67) into (56), and then summing (55) to the resulting inequality yields in view of the definition (29) of the relative entropy
$$\begin{aligned}&E[\chi _u, u, V|\chi _v, v](T') + \int _{0}^{T'} \int _{\Omega }\frac{\mu }{2}|\nabla (u-v) + \nabla (u-v)^{\mathsf {T}} |^2 \,\mathrm {d}x\,\mathrm {d}t\nonumber \\&\le E[\chi _u, u, V|\chi _v, v](0) + R_{dt} + R_{adv} + R^{(1)}_{surTen} + R^{(2)}_{surTen} \end{aligned}$$
(68)
for almost every \(T' \in [0,T]\), where in addition to the notation of Proposition 5 we also defined the two auxiliary quantities
$$\begin{aligned} R^{(1)}_{surTen}&:= -\sigma \int _{0}^{T'} \int _{{\overline{\Omega }} \times {\mathbb {S}}^{d-1}} (s - \xi ) \cdot ((s - \xi ) \cdot \nabla ) v \,\mathrm {d}V_t(x,s)\,\mathrm {d}t\nonumber \\&\quad +\sigma \int _{0}^{T'} \int _{\Omega }(1 - \theta _t) \xi \cdot ( \xi \cdot \nabla ) v \,\mathrm {d}|V_t|_{{\mathbb {S}}^{d-1}}\,\mathrm {d}t\nonumber \\&\quad + \sigma \int _{0}^{T'} \int _{\partial \Omega } \xi \cdot ( \xi \cdot \nabla ) v \,\mathrm {d}|V_t|_{{\mathbb {S}}^{d-1}}\,\mathrm {d}t, \end{aligned}$$
(69)
$$\begin{aligned} R^{(2)}_{surTen}&:= \sigma \int _{0}^T \int _{\Omega }\chi _u (u \cdot \nabla ) (\nabla \cdot \xi ) \,\mathrm {d}x\,\mathrm {d}t\nonumber \\&\quad - \sigma \int _{0}^{T'} \int _{\Omega }\chi _v ((u{-}v)\cdot \nabla ) (\nabla \cdot \xi ) \,\mathrm {d}x\,\mathrm {d}t\nonumber \\&\quad - \sigma \int _{0}^{T'} \int _{\Omega }\xi \cdot ( (n_u - \xi )\cdot \nabla ) v \,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t\nonumber \\&\quad - \sigma \int _{0}^{T'} \int _{\Omega }(n_u - \xi ) \cdot ( \xi \cdot \nabla ) v \,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t\nonumber \\&\quad + \sigma \int _{0}^{T'} \int _{\Omega }( {\text {Id}}- \xi \otimes \xi ): \nabla v \,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t\nonumber \\&\quad - \sigma \int _{0}^{T'} \int _{\Omega }n_u \cdot \partial _t\xi \,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t. \end{aligned}$$
(70)
The remainder of the proof is concerned with the post-processing of the term \(R^{(2)}_{surTen}\).
Step 3: By adding zeros, we can rewrite the last right hand side term of (70) as
$$\begin{aligned}&- \sigma \int _{0}^{T'} \int _{\Omega }n_u \cdot \partial _t\xi \,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t\nonumber \\ {}&= - \sigma \int _{0}^{T'} \int _{\Omega }(n_u {-} \xi ) \cdot (\partial _t\xi {+} (v \cdot \nabla )\xi {+} (\mathrm {Id}{-}\xi \otimes \xi )(\nabla v)^{\mathsf {T}} \xi ) \,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t\nonumber \\ {}&~~~~ - \sigma \int _0^{T'} \int _{\Omega }((n_u - \xi ) \cdot \xi ) (\xi \otimes \xi :\nabla v)\,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t\nonumber \\ {}&~~~~ - \sigma \int _{0}^{T'} \int _{\Omega }\Big (\partial _t\frac{1}{2}|\xi |^2 + (v \cdot \nabla )\frac{1}{2}|\xi |^2\Big ) \,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t\nonumber \\ {}&~~~~ + \sigma \int _{0}^{T'} \int _{\Omega }\xi \otimes (n_u - \xi ) : \nabla v \,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t\nonumber \\ {}&~~~~ + \sigma \int _{0}^{T'} \int _{\Omega }n_u \cdot ((v \cdot \nabla )\xi )\,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t. \end{aligned}$$
(71)
We proceed by manipulating the last term in the latter identity. To this end, we compute applying the product rule in the first step and then adding zero
$$\begin{aligned}&\sigma \int _{0}^{T'} \int _{\Omega }n_u \cdot ((v \cdot \nabla )\xi )\,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t\nonumber \\ {}&= \sigma \int _{0}^{T'} \int _{\Omega }n_u \cdot (\nabla \cdot (\xi \otimes v)) \,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t\nonumber \\ {}&~~~~+ \, \sigma \int _{0}^{T'} \int _{\Omega }(1 - n_u \cdot \xi ) (\nabla \cdot v) \,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t- \sigma \int _{0}^{T'} \int _{\Omega }{\text{ Id }}: \nabla v \,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t. \end{aligned}$$
(72)
Noting that for symmetry reasons \(\nabla \cdot (\nabla \cdot ( \xi \otimes v)) = \nabla \cdot (\nabla \cdot ( v \otimes \xi ))\), an integration by parts based on the boundary conditions (16b) and \((v\cdot n_{\partial \Omega })|_{\partial \Omega }=0\) entails
$$\begin{aligned} \sigma&\int _{0}^{T'} \int _{\Omega }n_u \cdot (\nabla \cdot (\xi \otimes v)) \,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t\\ =&- \sigma \int _{0}^{T'} \int _{\Omega }\chi _u \nabla \cdot (\nabla \cdot ( v \otimes \xi )) \,\mathrm {d}x\,\mathrm {d}t- \sigma \int _{0}^{T'} \int _{ \partial \Omega } \chi _u (n_{\partial \Omega }\otimes v : \nabla \xi )\,\mathrm {d}S\,\mathrm {d}t\\ =&\; \sigma \int _{0}^{T'} \int _{\Omega }n_u \cdot (\nabla \cdot (v \otimes \xi )) \,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t\\&+ \sigma \int _{0}^{T'} \int _{ \partial \Omega } \chi _u (n_{\partial \Omega }\cdot ((\xi \cdot \nabla )v- (v\cdot \nabla )\xi )) \,\mathrm {d}S\,\mathrm {d}t. \end{aligned}$$
We next observe that the last right hand side term of the previous display is zero. Indeed, note first that thanks to the boundary conditions (16b) and \((v\cdot n_{\partial \Omega })|_{\partial \Omega }=0\) the involved gradients are in fact tangential gradients along \(\partial \Omega \). Since the tangential gradient of a function only depends on its definition along the manifold, we are free to substitute \((\xi \cdot \tau _{\partial \Omega }) \tau _{\partial \Omega }\) for \(\xi \) resp. \((v \cdot \tau _{\partial \Omega }) \tau _{\partial \Omega }\) for v, obtaining in the process
$$\begin{aligned}&\int _{0}^{T'} \int _{\partial \Omega } \chi _u (n_{\partial \Omega }\cdot ((\xi \cdot \nabla ) v - (v \cdot \nabla ) \xi ))\,\mathrm {d}S\,\mathrm {d}t\\&= \int _{0}^{T'} \int _{\partial \Omega } \chi _u [ (\xi \cdot \nabla ) (v \cdot \tau _{\partial \Omega }) - (v \cdot \nabla ) (\xi \cdot \tau _{\partial \Omega })] (\tau _{\partial \Omega }\cdot n_{\partial \Omega }) \,\mathrm {d}S\,\mathrm {d}t\\&~~~ + \int _{0}^{T'} \int _{\partial \Omega } \chi _u [((v \cdot \tau _{\partial \Omega }) \xi - (\xi \cdot \tau _{\partial \Omega }) v)\cdot \nabla ) \tau _{\partial \Omega }] \cdot n_{\partial \Omega }\,\mathrm {d}S\,\mathrm {d}t=0 . \end{aligned}$$
The combination of the previous two displays together with an integration by parts and an application of the product rule thus yields
$$\begin{aligned}&\sigma \int _{0}^{T'} \int _{\Omega }n_u \cdot (\nabla \cdot (\xi \otimes v)) \,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t\\ {}&= \sigma \int _{0}^{T'} \int _{\Omega }(n_u \cdot v)(\nabla \cdot \xi ) \,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t+ \sigma \int _{0}^{T'} \int _{\Omega }n_u \otimes \xi : \nabla v \,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t. \end{aligned}$$
By another integration by parts, relying in the process also on \(\nabla \cdot v =0\) and \((v \cdot n_{\partial \Omega })|_{\partial \Omega } =0\), we may proceed computing
$$\begin{aligned} \sigma&\int _{0}^{T'} \int _{\Omega }n_u \cdot (\nabla \cdot (\xi \otimes v)) \,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t\nonumber \\ =&- \sigma \int _{0}^{T'} \int _{\Omega }\chi _u \nabla \cdot ( v(\nabla \cdot \xi )) \,\mathrm {d}x\,\mathrm {d}t+ \sigma \int _{0}^{T'} \int _{\Omega }n_u \otimes \xi : \nabla v \,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t\nonumber \\ =&- \sigma \int _{0}^{T'} \int _{\Omega }\chi _u (v \cdot \nabla )(\nabla \cdot \xi ) \,\mathrm {d}x\,\mathrm {d}t+ \sigma \int _{0}^{T'} \int _{\Omega }n_u \otimes \xi : \nabla v \,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t. \end{aligned}$$
(73)
In summary, taking together (71)–(73) and adding for a last time zero yields
$$\begin{aligned} -&\sigma \int _{0}^{T'} \int _{\Omega }n_u \cdot \partial _t\xi \,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t\nonumber \\ =&- \sigma \int _{0}^{T'} \int _{\Omega }\chi _u (v \cdot \nabla )(\nabla \cdot \xi ) \,\mathrm {d}x\,\mathrm {d}t\nonumber \\&- \sigma \int _{0}^{T'} \int _{\Omega }(n_u {-} \xi ) \cdot (\partial _t\xi {+} (v \cdot \nabla )\xi {+} (\mathrm {Id}{-}\xi \otimes \xi )(\nabla v)^{\mathsf {T}} \xi ) \,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t\nonumber \\&- \sigma \int _0^{T'} \int _{\Omega }((n_u - \xi ) \cdot \xi ) (\xi \otimes \xi :\nabla v)\,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t\nonumber \\&- \sigma \int _{0}^{T'} \int _{\Omega }\Big (\partial _t\frac{1}{2}|\xi |^2 + (v \cdot \nabla )\frac{1}{2}|\xi |^2\Big ) \,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t\nonumber \\&+ \sigma \int _{0}^{T'} \int _{\Omega }(1- n_u \cdot \xi ) (\nabla \cdot v )\,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t\nonumber \\&+ \sigma \int _{0}^{T'} \int _{\Omega }(n_u - \xi ) \otimes \xi : \nabla v \,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t+ \sigma \int _{0}^{T'} \int _{\Omega }\xi \otimes (n_u - \xi ) : \nabla v \,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t\nonumber \\&- \sigma \int _{0}^{T'} \int _{\Omega }({\text {Id}}- \xi \otimes \xi ): \nabla v \,\mathrm {d}|\nabla \chi _u|\,\mathrm {d}t. \end{aligned}$$
(74)
Inserting (74) into (70) then implies that \(R^{(1)}_{surTen} + R^{(2)}_{surTen}\) combines to the desired term \(R_{surTen}\). In particular, the estimate (68) upgrades to (30) as asserted. \(\square \)
Time Evolution of the Bulk Error: Proof of Lemma 6
Note that the sign conditions for the transported weight \(\vartheta \), see Definition 3, ensure that
$$\begin{aligned} E_{\mathrm {vol}}[\chi _u|\chi _v](t) = \int _\Omega \big (\chi _u(\cdot ,t) - \chi _v(\cdot ,t)\big ) \vartheta (\cdot ,t) \,\mathrm {d}x\end{aligned}$$
for all \(t\in [0,T]\). Hence, as a consequence of the transport equations for \(\chi _v\) and \(\chi _u\) (see Definitions 10 and 11, respectively) one obtains
$$\begin{aligned} E_{\mathrm {vol}}[\chi _u|\chi _v](T')&= E_{\mathrm {vol}}[\chi _u|\chi _v](0) \nonumber \\ {}&\quad + \int _0^{T'} \int _{\Omega } (\chi _u {-} \chi _v) \partial _t\vartheta \,\mathrm {d}x\,\mathrm {d}t+ \int _0^{T'} \int _{\Omega } (\chi _uu {-} \chi _vv) \cdot \nabla \vartheta \,\mathrm {d}x\,\mathrm {d}t \end{aligned}$$
(75)
for almost every \(T'\in [0,T]\). Note that for any sufficiently regular solenoidal vector field F with \((F\cdot n_{\partial \Omega })|_{\partial \Omega }=0\), since \(\vartheta =0\) along \(I_v\) (see Definition 3), an integration by parts yields
$$\begin{aligned} \int _\Omega \chi _v (F\cdot \nabla )\vartheta \,\mathrm {d}x= 0. \end{aligned}$$
(76)
Adding zero in (75) and making use of (76) with respect to the choices \(F=u\) and \(F=v\) in form of \(\int _\Omega \chi _v \big ((u{-}v)\cdot \nabla \big )\vartheta \,\mathrm {d}x= 0\) then updates (75) to (32). This concludes the proof of Lemma 6. \(\square \)
Conditional Weak-strong Uniqueness: Proof of Proposition 4
Starting point for a proof of the conditional weak-strong uniqueness principle is the following important coercivity estimate (cf. [12, Lemma 20]).
Lemma 12
Let the assumptions and notation of Proposition 4 be in place. Then there exists a constant \(C=C(\chi _v,v,T)>0\) such that for all \(\delta \in (0,1]\) it holds
$$\begin{aligned} \int _0^{T'} \int _{\Omega } |\chi _v{-}\chi _u||u{-}v| \,\mathrm {d}x\,\mathrm {d}t&\le \frac{C}{\delta } \int _0^{T'} E[\chi _u,u,V|\chi _v,v](t) + E_{\mathrm {vol}}[\chi _u|\chi _v](t) \,\mathrm {d}t\nonumber \\ {}&\quad + \delta \int _0^{T'} \int _{\Omega } |\nabla u - \nabla v|^2 \,\mathrm {d}x\,\mathrm {d}t \end{aligned}$$
(77)
for all \(T'\in [0,T]\).
Proof
It turns out to be convenient to introduce a decomposition of the interface \(I_v\) into its topological features: the connected components of \(I_v\cap \Omega \) and the connected components of \(I_v\cap \partial \Omega \). Let \(N\in {\mathbb {N}}\) denote the total number of such topological features of \(I_v\), and split
as follows. The subset \({\mathcal {I}}\) enumerates the space-time connected components of \(I_v\cap \Omega \) (being time-evolving connected interfaces), whereas the subset \({\mathcal {C}}\) enumerates the space-time connected components of \(I_v\cap \partial \Omega \) (being time-evolving contact points if \(d=2\), or time-evolving connected contact lines if \(d=3\)). If \(i\in {\mathcal {I}}\), we let \({\mathcal {T}}_i\) denote the space-time trajectory in \(\Omega \) of the corresponding connected interface. Furthermore, for every \(c\in {\mathcal {C}}\) we write \({\mathcal {T}}_c\) representing the space-time trajectory in \(\partial \Omega \) of the corresponding contact point (if \(d=2\)) or line (if \(d=3\)). Finally, let us write \(i\sim c\) for \(i\in {\mathcal {I}}\) and \(c\in {\mathcal {C}}\) if and only if \({\mathcal {T}}_i\) ends at \({\mathcal {T}}_c\). With this language and notation in place, the proof is now split into five steps.
Step 1: (Choice of a suitable localization scale) Denote by \(n_{\partial \Omega }\) the unit normal vector field of \(\partial \Omega \) pointing into \(\Omega \), and by \(n_{I_v}(\cdot ,t)\) the unit normal vector field of \(I_v(t)\) pointing into \(\Omega _v(t)\). Because of the uniform \(C^2_x\) regularity of the boundary \(\partial \Omega \) and the uniform \(C_tC^2_x\) regularity of the interface \(I_v(t)\), \(t\in [0,T]\), we may choose a scale \(r \in (0,\frac{1}{2}]\) such that for all \(t\in [0,T]\) and all \(i\in {\mathcal {I}}\) the maps
$$\begin{aligned} \Psi _{\partial \Omega }:\partial \Omega \times (-3r,3r) \rightarrow {\mathbb {R}}^d,&\quad (x,y) \mapsto x + yn_{\partial \Omega }(x), \end{aligned}$$
(78)
$$\begin{aligned} \Psi _{{\mathcal {T}}_i(t)}:{\mathcal {T}}_i(t) \times (-3r,3r) \rightarrow {\mathbb {R}}^d,&\quad (x,y) \mapsto x + yn_{I_v}(x,t) \end{aligned}$$
(79)
are \(C^1\) diffeomorphisms onto their image. By uniform regularity of \(\partial \Omega \) and \(I_v\) (the latter in space-time), we have bounds
$$\begin{aligned} \sup _{\partial \Omega {\times }[-r, r]} |\nabla \Psi _{\partial \Omega }| \le C,&\quad \sup _{\Psi _{\partial \Omega }(\partial \Omega {\times }[-r, r])} |\nabla \Psi _{\partial \Omega }^{-1}| \le C, \end{aligned}$$
(80)
$$\begin{aligned} \sup _{t\in [0,T]} \sup _{{\mathcal {T}}_i(t){\times }[-r,r]} |\nabla \Psi _{{\mathcal {T}}_i(t)}| \le C,&\quad \sup _{t\in [0,T]} \sup _{\Psi _{{\mathcal {T}}_i(t)}({\mathcal {T}}_i(t){\times }[-r,r])} |\nabla \Psi _{{\mathcal {T}}_i(t)}^{-1}| \le C \end{aligned}$$
(81)
for all \(i\in {\mathcal {I}}\). By possibly choosing \(r\in (0,\frac{1}{2}]\) even smaller, we may also guarantee that for all \(t\in [0,T]\) and all \(i\in {\mathcal {I}}\) it holds
$$\begin{aligned} \Psi _{{\mathcal {T}}_i(t)}({\mathcal {T}}_i(t){\times }[-r,r]) \cap \Psi _{{\mathcal {T}}_{i'}(t)}({\mathcal {T}}_{i'}(t){\times }[-r,r])&= \emptyset \text { for all } i'\in {\mathcal {I}},\,i'\ne i, \end{aligned}$$
(82)
$$\begin{aligned} \Psi _{{\mathcal {T}}_i(t)}({\mathcal {T}}_i(t){\times }[-r,r]) \cap \Psi _{\partial \Omega }(\partial \Omega {\times }[-r, r])&\ne \emptyset \,\Leftrightarrow \, \exists c \in {\mathcal {C}} :i \sim c, \end{aligned}$$
(83)
$$\begin{aligned} \Psi _{{\mathcal {T}}_i(t)}({\mathcal {T}}_i(t){\times }[-r,r]) \cap \Psi _{\partial \Omega }(\partial \Omega {\times }[-r, r])&\subset B_{2r}({\mathcal {T}}_c(t)) \text { if } \exists c \in {\mathcal {C}} :i \sim c \end{aligned}$$
(84)
$$\begin{aligned} B_{2r}({\mathcal {T}}_c(t)) \cap B_{2r}({\mathcal {T}}_{c'}(t))&= \emptyset \text { for all } c,c'\in {\mathcal {C}},\,c'\ne c. \end{aligned}$$
(85)
Note finally that because of the \(90^\circ \) contact angle condition and by possibly choosing \(r\in (0,\frac{1}{2}]\) even smaller, we can furthermore ensure that
$$\begin{aligned} \begin{aligned}&\Omega \setminus \Big (\Psi _{\partial \Omega }(\partial \Omega {\times }[-r, r]) \cup \bigcup _{i\in {\mathcal {I}}}\Psi _{{\mathcal {T}}_i(t)}({\mathcal {T}}_i(t){\times }[-r,r])\Big ) \\ {}&\subset \Omega \cap \{x\in {\mathbb {R}}^d:{\text {dist}}(x,\partial \Omega ) \wedge {\text {dist}}(x,I_v(t)) > r\} \end{aligned} \end{aligned}$$
(86)
for all \(t\in [0,T]\). Indeed, for \(x\in \Omega \setminus \big (\Psi _{\partial \Omega }(\partial \Omega {\times }[-r, r]) \cup \bigcup _{i\in {\mathcal {I}}}\Psi _{{\mathcal {T}}_i(t)}({\mathcal {T}}_i(t){\times }[-r,r])\big )\) it follows that \({\text {dist}}(x,\partial \Omega ) > r\). In case the interface \(I_v(t)\) intersects \(\partial \Omega \) it may not be immediately clear that also \({\text {dist}}(x,I_v(t)) > r\) holds true. Assume there exists a point \(x\in \Omega \setminus \big (\Psi _{\partial \Omega }(\partial \Omega {\times }[-r, r]) \cup \bigcup _{i\in {\mathcal {I}}}\Psi _{{\mathcal {T}}_i(t)}({\mathcal {T}}_i(t){\times }[-r,r])\big )\) such that \({\text {dist}}(x,I_v(t)) \le r\). Then necessarily \(x\in (\Omega \cap B_{r}(c(t)))\setminus \bigcup _{i\in {\mathcal {I}}}\Psi _{{\mathcal {T}}_i(t)}({\mathcal {T}}_i(t){\times }[-r,r])\) for some boundary point \(c(t)\in \partial \Omega \cap I_v(t)\). Hence, because of the uniform \(C^2_x\) regularity of \(\partial \Omega \) and \(I_v(t)\) intersecting \(\partial \Omega \) at an angle of \(90^\circ \), one may choose \(r\in (0,\frac{1}{2}]\) small enough such that \(x\in (\Omega \cap B_{r}(c(t)))\) implies \({\text {dist}}(x,\partial \Omega )\le r\). As we have already seen, this contradicts \(x\in \Omega \setminus \Psi _{\partial \Omega }(\partial \Omega {\times }[-r, r])\).
Step 2: (A reduction argument) We may estimate by a union bound and (86)
$$\begin{aligned}&\int _0^{T'} \int _{\Omega } |\chi _v{-}\chi _u||u{-}v| \,\mathrm {d}x\,\mathrm {d}t\nonumber \\&\le \int _0^{T'} \int _{\Omega \cap \Psi _{\partial \Omega }(\partial \Omega {\times }[-r, r]) \setminus \bigcup _{c\in {\mathcal {C}}} B_{2r}({\mathcal {T}}_c(t))} |\chi _v{-}\chi _u||u{-}v| \,\mathrm {d}x\,\mathrm {d}t\nonumber \\&\quad + \sum _{i\in {\mathcal {I}}}\int _0^{T'} \int _{\Omega \cap \Psi _{{\mathcal {T}}_i(t)}({\mathcal {T}}_i(t){\times }[-r,r]) \setminus \bigcup _{c\in {\mathcal {C}}} B_{2r}({\mathcal {T}}_c(t))} |\chi _v{-}\chi _u||u{-}v| \,\mathrm {d}x\,\mathrm {d}t\nonumber \\&\quad + C \sum _{c\in {\mathcal {C}}}\int _0^{T'} \int _{\Omega \cap B_{2r}({\mathcal {T}}_c(t))} |\chi _v{-}\chi _u||u{-}v| \,\mathrm {d}x\,\mathrm {d}t\nonumber \\&\quad + \int _0^{T'} \int _{\Omega \cap \{{\text {dist}}(\cdot ,\partial \Omega )\wedge {\text {dist}}(\cdot ,I_v(t)) > r\}} |\chi _v{-}\chi _u||u{-}v| \,\mathrm {d}x\,\mathrm {d}t. \end{aligned}$$
(87)
An application of Hölder’s inequality and Young’s inequality, the definition (29) of the relative entropy functional, the coercivity estimate (27) for the transported weight, and the definition (31) of the bulk error functional further imply
$$\begin{aligned}&\int _0^{T'} \int _{\Omega \cap \{{\text {dist}}(\cdot ,\partial \Omega )\wedge {\text {dist}}(\cdot ,I_v(t))> r\}} |\chi _v{-}\chi _u||u{-}v| \,\mathrm {d}x\,\mathrm {d}t\\ {}&\le C\int _0^{T'} \int _{\Omega \cap \{{\text {dist}}(\cdot ,\partial \Omega ) \wedge {\text {dist}}(\cdot ,I_v(t)) > r\}} |\chi _v{-}\chi _u| \,\mathrm {d}x\,\mathrm {d}t+ C \int _0^{T'} E[\chi _u,u,V|\chi _v,v](t) \,\mathrm {d}t\\ {}&\le C\int _0^{T'} E[\chi _u,u,V|\chi _v,v](t) + E_{\mathrm {vol}}[\chi _u,\chi _v](t) \,\mathrm {d}t. \end{aligned}$$
Hence, it remains to estimate the first three terms on the right hand side of (87).
Step 3: (Estimate near the interface but away from contact points) First of all, because of the localization properties (82)–(84) it holds for all \(i\in {\mathcal {I}}\)
$$\begin{aligned} {\text {dist}}(\cdot ,{\mathcal {T}}_i) = {\text {dist}}(\cdot ,\partial \Omega ) \wedge {\text {dist}}(\cdot ,I_v(t)) \end{aligned}$$
(88)
in \(\Omega \cap \Psi _{{\mathcal {T}}_i(t)}({\mathcal {T}}_i(t){\times }[-r,r]) \setminus \bigcup _{c\in {\mathcal {C}}} B_{2r}({\mathcal {T}}_c(t))\). Hence, the local interface error height as measured in the direction of \(n_{I_v}\) on \({\mathcal {T}}_i\)
$$\begin{aligned} h_{{\mathcal {T}}_i}(x,t) := \int _{-r}^r |\chi _u - \chi _v|(\Psi _{{\mathcal {T}}_i(t)}(x,y),t) \,\mathrm {d}y, \quad x\in {\mathcal {T}}_i(t),\,t\in [0,T], \end{aligned}$$
is, because of (88) and the coercivity estimate (27) of the transported weight \(\vartheta \), subject to the estimate
$$\begin{aligned} h_{{\mathcal {T}}_i}^2(x,t)&\le C\int _{-r}^r |\chi _u - \chi _v|(\Psi _{{\mathcal {T}}_i(t)}(x,y),t) y \,\mathrm {d}y\nonumber \\&\le C \int _{-r}^r |\chi _u - \chi _v|(\Psi _{{\mathcal {T}}_i(t)}(x,y),t) |\vartheta |(\Psi _{{\mathcal {T}}_i(t)}(x,y),t) \,\mathrm {d}y\end{aligned}$$
(89)
for all \(x\in {\mathcal {T}}_i(t)\setminus \bigcup _{c\in {\mathcal {C}}}B_{2r}({\mathcal {T}}_c(t))\), all \(t\in [0,T]\) and all \(i\in {\mathcal {I}}\). Carrying out the slicing argument of the proof of [12, Lemma 20] in \(\Omega \cap \Psi _{{\mathcal {T}}_i(t)}({\mathcal {T}}_i(t){\times }[-r,r]) \setminus \bigcup _{c\in {\mathcal {C}}} B_{2r}({\mathcal {T}}_c(t))\) by means of \( \Psi _{{\mathcal {T}}_i(t)}\), which is indeed admissible thanks to (79), (81) and (89), shows that one obtains an estimate of required form
$$\begin{aligned}&\sum _{i\in {\mathcal {I}}}\int _0^{T'} \int _{\Omega \cap \Psi _{{\mathcal {T}}_i(t)}({\mathcal {T}}_i(t){\times }[-r,r]) \setminus \bigcup _{c\in {\mathcal {C}}} B_{2r}({\mathcal {T}}_c(t))} |\chi _v{-}\chi _u||u{-}v| \,\mathrm {d}x\,\mathrm {d}t\\ {}&\le \frac{C}{\delta } \int _0^{T'} E[\chi _u,u,V|\chi _v,v](t) + E_{\mathrm {vol}}[\chi _u|\chi _v](t) \,\mathrm {d}t+ \delta \int _0^{T'} \int _{\Omega } |\nabla u - \nabla v|^2 \,\mathrm {d}x\,\mathrm {d}t. \end{aligned}$$
Step 4: (Estimate near the boundary of the domain but away from contact points) The argument is similar to the one of the previous step, with the only major difference being that the slicing argument of the proof of [12, Lemma 20] is now carried out in \(\Omega \cap \Psi _{\partial \Omega }(\partial \Omega {\times }[-r, r]) \setminus \bigcup _{c\in {\mathcal {C}}} B_{2r}({\mathcal {T}}_c(t))\) by means of \(\Psi _{\partial \Omega }\). This in turn is facilitated by the following facts. First, the localization properties (82)–(84) ensure
$$\begin{aligned} {\text {dist}}(\cdot ,\partial \Omega ) = {\text {dist}}(\cdot ,\partial \Omega ) \wedge {\text {dist}}(\cdot ,I_v(t)) \end{aligned}$$
(90)
in \(\Omega \cap \Psi _{\partial \Omega }(\partial \Omega {\times }[-r, r]) \setminus \bigcup _{c\in {\mathcal {C}}} B_{2r}({\mathcal {T}}_c(t))\). Second, as a consequence of (90) and the coercivity estimate (27) of the transported weight \(\vartheta \), the local interface error height as measured in the direction of \(n_{\partial \Omega }\)
$$\begin{aligned} h_{\partial \Omega }(x,t) := \int _{-r}^r |\chi _u - \chi _v|(\Psi _{\partial \Omega }(x,y),t) \,\mathrm {d}y, \quad x\in \partial \Omega ,\,t\in [0,T], \end{aligned}$$
satisfies the estimate
$$\begin{aligned} h_{\partial \Omega }^2(x,t)&\le C\int _{-r}^r |\chi _u - \chi _v|(\Psi _{\partial \Omega }(x,y),t) y \,\mathrm {d}y\nonumber \\&\le C \int _{-r}^r |\chi _u - \chi _v|(\Psi _{\partial \Omega }(x,y),t) |\vartheta |(\Psi _{\partial \Omega }(x,y),t) \,\mathrm {d}y. \end{aligned}$$
(91)
Hence, we obtain
$$\begin{aligned}&\int _0^{T'} \int _{\Omega \cap \Psi _{\partial \Omega }(\partial \Omega {\times }[-r, r]) \setminus \bigcup _{c\in {\mathcal {C}}} B_{2r}({\mathcal {T}}_c(t))} |\chi _v{-}\chi _u||u{-}v| \,\mathrm {d}x\,\mathrm {d}t\\ {}&\le \frac{C}{\delta } \int _0^{T'} E[\chi _u,u,V|\chi _v,v](t) + E_{\mathrm {vol}}[\chi _u|\chi _v](t) \,\mathrm {d}t+ \delta \int _0^{T'} \int _{\Omega } |\nabla u - \nabla v|^2 \,\mathrm {d}x\,\mathrm {d}t. \end{aligned}$$
Step 5: (Estimate near contact points) Fix \(c\in {\mathcal {C}}\), and let \(i\in {\mathcal {I}}\) denote the unique connected interface \({\mathcal {T}}_i\) such that \(i\sim c\). Because of the regularity of \(\partial \Omega \), the regularity of \({\mathcal {T}}_i\), and the \(90^\circ \) contact angle condition we may decompose the neighborhood \(\Omega \cap B_{2r}({\mathcal {T}}_c(t))\)—by possibly reducing the localization scale \(r\in (0,\frac{1}{2}]\) even further—into three pairwise disjoint open sets \(W_{\partial \Omega }(t)\), \(W_{{\mathcal {T}}_i}(t)\) and \(W_{\partial \Omega \sim {\mathcal {T}}_i}(t)\) such that \(\Omega \cap B_{2r}({\mathcal {T}}_c(t)) \setminus \big (W_{\partial \Omega }(t) \cup W_{{\mathcal {T}}_i}(t) \cup W_{\partial \Omega \sim {\mathcal {T}}_i}(t)\big )\) is an \({\mathcal {H}}^d\) null set and
$$\begin{aligned} {\text {dist}}(\cdot ,\partial \Omega )&= {\text {dist}}(\cdot ,\partial \Omega ) \wedge {\text {dist}}(\cdot ,I_v(t))&\text { in } W_{\partial \Omega }(t), \end{aligned}$$
(92)
$$\begin{aligned} {\text {dist}}(\cdot ,{\mathcal {T}}_i(t))&= {\text {dist}}(\cdot ,\partial \Omega ) \wedge {\text {dist}}(\cdot ,I_v(t))&\text { in } W_{{\mathcal {T}}_i}(t), \end{aligned}$$
(93)
$$\begin{aligned} {\text {dist}}(\cdot ,\partial \Omega )&\sim {\text {dist}}(\cdot ,{\mathcal {T}}_i(t)) \sim {\text {dist}}(\cdot ,I_v(t))&\text { in } W_{\partial \Omega \sim {\mathcal {T}}_i}(t), \end{aligned}$$
(94)
as well as
$$\begin{aligned} W_{\partial \Omega }(t)&\subset \Psi _{\partial \Omega }(\partial \Omega {\times }(-3r,3r)), \end{aligned}$$
(95)
$$\begin{aligned} W_{{\mathcal {T}}_i}(t)&\subset \Psi _{{\mathcal {T}}_i(t)}({\mathcal {T}}_i(t){\times }(-3r,3r)), \end{aligned}$$
(96)
$$\begin{aligned} W_{\partial \Omega \sim {\mathcal {T}}_i}(t)&\subset \Psi _{\partial \Omega }(\partial \Omega {\times }(-3r,3r)) \cap \Psi _{{\mathcal {T}}_i(t)}({\mathcal {T}}_i(t){\times }(-3r,3r)). \end{aligned}$$
(97)
(Up to a rigid motion, these sets can in fact be defined independent of \(t\in [0,T]\).) Hence, applying the argument of Step 3 based on (93) and (96) with respect to \(\Omega \cap B_{2r}({\mathcal {T}}_c(t))\cap W_{{\mathcal {T}}_i}(t)\), the argument of Step 4 based on (92) and (95) with respect to \(\Omega \cap B_{2r}({\mathcal {T}}_c(t))\cap W_{\partial \Omega }(t)\), and either the argument of Step 3 or Step 4 based on (94) and (97) with respect to \(\Omega \cap B_{2r}({\mathcal {T}}_c(t))\cap W_{\partial \Omega \sim {\mathcal {T}}_i}(t)\) entails
$$\begin{aligned}&\sum _{c\in {\mathcal {C}}}\int _0^{T'} \int _{\Omega \cap B_{2r}({\mathcal {T}}_c(t))} |\chi _v{-}\chi _u||u{-}v| \,\mathrm {d}x\,\mathrm {d}t\\ {}&\le \frac{C}{\delta } \int _0^{T'} E[\chi _u,u,V|\chi _v,v](t) + E_{\mathrm {vol}}[\chi _u|\chi _v](t) \,\mathrm {d}t+ \delta \int _0^{T'} \int _{\Omega } |\nabla u - \nabla v|^2 \,\mathrm {d}x\,\mathrm {d}t. \end{aligned}$$
This in turn concludes the proof of Lemma 12. \(\square \)
Proof of Proposition 4
The proof proceeds in three steps.
Step 1: (Post-processing the relative entropy inequality (30)) It follows immediately from the \(L^\infty _{x,t}\)-bound for \(\partial _t v\) and \(\rho (\chi _v)-\rho (\chi _u)=(\rho ^+{-}\rho ^-)(\chi _v{-}\chi _u)\) that
$$\begin{aligned} |R_{dt}| \le C \int _0^{T'} \int _{\Omega } |\chi _v{-}\chi _u||u{-}v| \,\mathrm {d}x\,\mathrm {d}t\end{aligned}$$
(98)
for almost every \(T'\in [0,T]\). Furthermore, the \(L^\infty _tW^{1,\infty }_x\)-bound for v, the definition (29) of the relative entropy functional, and again the identity \(\rho (\chi _v)-\rho (\chi _u)=(\rho ^+{-}\rho ^-)(\chi _v{-}\chi _u)\) imply that
$$\begin{aligned} |R_{adv}| \le C \int _0^{T'} \int _{\Omega } |\chi _v{-}\chi _u||u{-}v| \,\mathrm {d}x\,\mathrm {d}t+ C \int _0^{T'} E[\chi _u,u,V|\chi _v,v](t) \,\mathrm {d}t\end{aligned}$$
(99)
for almost every \(T'\in [0,T]\). For a bound on the interface contribution \(R_{surTen}\), we rely on the \(L^\infty _tW^{1,\infty }_x\)-bound for v, the \(L^\infty _tW^{2,\infty }_x\)-bound for \(\xi \), the \(L^\infty _tW^{1,\infty }_x\)-bound for B, the definition (29) of the relative entropy functional, as well as the estimates (16d) and (16e) of a boundary adapted extension \(\xi \) of \(n_{I_v}\) to the effect that
$$\begin{aligned} |R_{surTen}|&\le C \int _0^{T'} \int _{\Omega } |\chi _v{-}\chi _u||u{-}v| \,\mathrm {d}x\,\mathrm {d}t\nonumber \\&\quad + C \int _0^{T'} \int _{{\overline{\Omega }}\times {\mathbb {S}}^{d-1}} |s-\xi |^2 \,\mathrm {d}V_t(x,s) \,\mathrm {d}t\nonumber \\&\quad + C \int _{0}^{T'} \int _{\Omega } 1- \theta _t \,\mathrm {d} |V_t|_{{\mathbb {S}}^{d-1}} \,\mathrm {d}t\nonumber \\&\quad + C \int _{0}^{T'} \int _{\partial \Omega } 1 \,\mathrm {d} |V_t|_{{\mathbb {S}}^{d-1}} \,\mathrm {d}t\nonumber \\&\quad + C \int _0^{T'} \int _{\Omega } |n_u - \xi |^2 \,\mathrm {d}|\nabla \chi _u| \,\mathrm {d}t\nonumber \\&\quad + C \int _0^{T'} \int _{\Omega } {\text {dist}}^2(\cdot ,I_v) \wedge 1 \,\mathrm {d}|\nabla \chi _u| \,\mathrm {d}t\nonumber \\&\quad + C \int _0^{T'} \int _{\Omega } |\xi \cdot (\xi - n_u)| \,\mathrm {d}|\nabla \chi _u| \,\mathrm {d}t\nonumber \\&\quad + C \int _0^{T'} E[\chi _u,u,V|\chi _v,v](t) \,\mathrm {d}t\end{aligned}$$
(100)
for almost every \(T'\in [0,T]\). It follows from property (16a) of a boundary adapted extension \(\xi \) and the trivial estimates \(|\xi \cdot (\xi - n_u)| \le (1{-}|\xi |^2) + (1 {-} n_u \cdot \xi ) \le 2(1{-}|\xi |) + (1 {-} n_u \cdot \xi )\) and \(1-|\xi | \le 1 - n_u\cdot \xi \) that
$$\begin{aligned}&\int _0^{T'} \int _{\Omega } {\text {dist}}^2(\cdot ,I_v) \wedge 1 \,\mathrm {d}|\nabla \chi _u| \,\mathrm {d}t+ \int _0^{T'} \int _{\Omega } |\xi \cdot (\xi - n_u)| \,\mathrm {d}|\nabla \chi _u| \,\mathrm {d}t\nonumber \\&\le C \int _0^{T'} E[\chi _u,u,V|\chi _v,v](t) \,\mathrm {d}t. \end{aligned}$$
(101)
Moreover, the trivial estimate \(|n_u-\xi |^2 \le 2(1-n_u\cdot \xi )\) implies
$$\begin{aligned} \int _0^{T'} \int _{\Omega } |n_u - \xi |^2 \,\mathrm {d}|\nabla \chi _u| \,\mathrm {d}t\le C \int _0^{T'} E[\chi _u,u,V|\chi _v,v](t) \,\mathrm {d}t. \end{aligned}$$
(102)
Recall finally from (24) and (20) that
$$\begin{aligned} \int _0^{T'} \int _{{\overline{\Omega }}\times {\mathbb {S}}^{d-1}} |s-\xi |^2 \,\mathrm {d}V_t(x,s) \,\mathrm {d}t&\le C \int _0^{T'} E[\chi _u,u,V|\chi _v,v](t) \,\mathrm {d}t, \nonumber \\ \int _{0}^{T'} \int _{\Omega } 1- \theta _t \,\mathrm {d} |V_t|_{{\mathbb {S}}^{d-1}} \,\mathrm {d}t+ \int _{0}^{T'} \int _{\partial \Omega } 1 \,\mathrm {d} |V_t|_{{\mathbb {S}}^{d-1}} \,\mathrm {d}t&\le C \int _0^{T'} E[\chi _u,u,V|\chi _v,v](t) \,\mathrm {d}t. \end{aligned}$$
(103)
By inserting back the estimates (98)–(103) into the relative entropy inequality (30), then making use of the coercivity estimate (77) and Korn’s inequality, and finally carrying out an absorption argument, it follows that there exist two constants \(c=c(\chi _v,v,T)>0\) and \(C=C(\chi _v,v,T)>0\) such that for almost every \(T'\in [0,T]\)
$$\begin{aligned}&E[\chi _u,u,V|\chi _v,v](T') + c\int _{0}^{T'} \int _{\Omega } |\nabla (u{-}v) + \nabla (u{-}v)^{\mathsf {T}} |^2 \,\mathrm {d}x\,\mathrm {d}t\nonumber \\&\le E[\chi _u,u,V|\chi _v,v](0) + C\int _0^{T'} E[\chi _u,u,V|\chi _v,v](t) + E_{\mathrm {vol}}[\chi _u|\chi _v](t) \,\mathrm {d}t. \end{aligned}$$
(104)
Step 2: (Post-processing the identity (32)) By the \(L^\infty _tW^{1,\infty }_x\)-bound for the transported weight \(\vartheta \), the estimate (28) on the advective derivative of the transported weight \(\vartheta \), and the definition (31) of the bulk error functional we infer that
$$\begin{aligned} E_{\mathrm {vol}}[\chi _u|\chi _v](T')&\le E_{\mathrm {vol}}[\chi _u|\chi _v](0) + C\int _0^{T'} E_{\mathrm {vol}}[\chi _u|\chi _v](t) \,\mathrm {d}t\\&\quad + C\int _0^{T'} \int _{\Omega } |\chi _v{-}\chi _u||u{-}v| \,\mathrm {d}x\,\mathrm {d}t\end{aligned}$$
for almost every \(T'\in [0,T]\). Adding (104) to the previous display, and making use of the coercivity estimate (77) in combination with Korn’s inequality and an absorption argument thus implies that for almost every \(T'\in [0,T]\)
$$\begin{aligned}&E[\chi _u,u,V|\chi _v,v](T') + E_{\mathrm {vol}}[\chi _u|\chi _v](T') + c\int _{0}^{T'} \int _{\Omega } |\nabla (u{-}v) + \nabla (u{-}v)^{\mathsf {T}} |^2 \,\mathrm {d}x\,\mathrm {d}t\nonumber \\ {}&\le E[\chi _u,u,V|\chi _v,v](0) + E_{\mathrm {vol}}[\chi _u|\chi _v](0) \nonumber \\&\quad + C\int _0^{T'} E[\chi _u,u,V|\chi _v,v](t) + E_{\mathrm {vol}}[\chi _u|\chi _v](t) \,\mathrm {d}t. \end{aligned}$$
(105)
Step 3: (Conclusion) The stability estimates (11) and (12) are an immediate consequence of the estimate (105) by an application of Gronwall’s lemma. In case of coinciding initial conditions, it follows that \(E_{\mathrm {vol}}[\chi _u|\chi _v](t) = 0\) for almost every \(t\in [0,T]\). This in turn implies that \(\chi _u(\cdot ,t)=\chi _v(\cdot ,t)\) almost everywhere in \(\Omega \) for almost every \(t\in [0,T]\). The asserted representation of the varifold follows from the fact that \(E[\chi _u,u,V|\chi _v,v](t) = 0\) for almost every \(t\in [0,T]\). This concludes the proof of the conditional weak-strong uniqueness principle. \(\square \)
Proof of Theorem 1
This is now an immediate consequence of Proposition 4 and the existence results of Proposition 7 and Lemma 8, respectively. \(\square \)