We turn to the derivation of the weak–strong uniqueness principle in the case of different shear viscosities of the two fluids. In this regime, we cannot anymore ignore the viscous stress term \((\mu (\chi _v)-\mu (\chi _u))(\nabla v+\nabla v^T)\). The key idea is to construct a solenoidal vector field w which is small in the \(L^2\)-norm but whose gradient compensates for most of this problematic term, and then use the relative entropy inequality from Proposition 10 with this function. The precise definition as well as a list of all the relevant properties of this vector field are the content of Proposition 28.
A main ingredient for the construction of w are the local interface error heights as measured in orthogonal direction from the interface of the strong solution (see Fig. 3). For this reason, we first prove the relevant properties of the local heights of the interface error in Proposition 26. However, in order to control certain surface-tension terms in the relative entropy inequality, we actually need the vector field w to have bounded spatial derivatives. To this end, we perform an additional regularization of the height functions. This will be carried out in detail in Proposition 27 by a (time-dependent) mollification. After all these preparations, in Sections 6.4–6.8 we then further estimate the additional terms \(A_{visc}\), \(A_{dt}\), \(A_{adv}\), and \(A_{surTen}\) in the relative entropy inequality from Proposition 10. Based on these bounds, in Section 6.9 we finally provide the proof of the stability estimate and the weak–strong uniqueness principle for varifold solutions to the free boundary problem for the incompressible Navier–Stokes equation for two fluids (1a)–(1c) from Theorem 1.
The Evolution of the Local Height of the Interface Error
Consider a strong solution \((\chi _v,v)\) to the free boundary problem for the incompressible Navier–Stokes equation for two fluids (1a)–(1c) in the sense of Definition 6 on some time interval \([0,{T_{strong}})\). For the sake of better readability, let us recall some definitions and constructions related to the associated family of evolving interfaces \(I_v(t)\) of the strong solution.
For the family \((\Omega _t^+)_{t\in [0,{T_{strong}})}\) of smoothly evolving domains of the strong solution, the associated signed distance function is given by
$$\begin{aligned} {\text {dist}}^{\pm }(x,I_v(t)) = {\left\{ \begin{array}{ll} \mathrm {dist}(x,I_v(t)), &{} x\in \Omega ^+_t, \\ -\mathrm {dist}(x,I_v(t)), &{} x\notin \Omega ^+_t. \end{array}\right. } \end{aligned}$$
From Definition 5 of a family of smoothly evolving domains it follows that the family of maps \(\Phi _t:I_v(t)\times (-r_c,r_c) \rightarrow {\mathbb {R}^d}\) given by \(\Phi _t(x,y) := x+y\mathrm {n}_v(x,t)\) are \(C^2\)-diffeomorphisms onto their image \(\{x\in \mathbb {R}^d:{\text {dist}}(x,I_v(t))<r_c\}\). Here, \(\mathrm {n}_v(\cdot ,t)\) denotes the normal vector field of the interface \(I_v(t)\) pointing inwards \(\{x\in {\mathbb {R}^d}:\chi _v(x,t)=1\}\). The signed distance function (resp. its time derivative) to the interface \(I_v(t)\) of the strong solution is then of class \(C^0_tC^3_x\) (resp. \(C^0_tC^2_x\)) in the space-time tubular neighborhood \(\bigcup _{t\in [0,{T_{strong}})}\mathrm {im}(\Phi _t)\times \{t\}\) due to the regularity assumptions in Definition 5. Moreover, the projection \(P_{I_v(t)}x\) of a point x onto the nearest point on the manifold \(I_v(t)\) is well-defined and of class \(C^0_tC^2_x\) in the same tubular neighborhood. Observe that the inverse of \(\Phi _t\) is given by \(\Phi _t^{-1}(x) = (P_{I_v(t)}x,{\text {dist}}^{\pm }(x,I_v(t)))\) for all \(x\in {\mathbb {R}^d}\) such that \({\text {dist}}(x,I_v(t))<r_c\).
In Lemma 11, we computed the time evolution of the signed distance function to the interface \(I_v(t)\) of a strong solution. Recall also the various relations for the projected inner unit normal vector field \(\mathrm {n}_v(P_{I_v(t)}x,t)\) from Lemma 11, which will be of frequent use in subsequent computations. Finally, we remind the reader of the definition of the vector field \(\xi \) from Definition 13, which is a global extension of the inner unit normal vector field of the interface \(I_v(t)\). For an illustration of the vector field \(\xi \), we recall Fig. 2; for an illustration of \(h^+\), we refer to Fig. 3.
Proposition 26
Let \(\chi _v\in L^\infty ([0,{T_{strong}});{\text {BV}}(\mathbb {R}^d;\{0,1\}))\) be an indicator function such that \(\Omega ^+_t :=\{x\in \mathbb {R}^d:\chi _v(x,t)=1\}\) is a family of smoothly evolving domains and \(I_v(t) := \partial \Omega ^+_t\) is a family of smoothly evolving surfaces in the sense of Definition 5. Let \(\xi \) be the extension of the unit normal vector field \(\mathrm {n}_v\) from Definition 13.
Let \(\theta :[0,\infty )\rightarrow [0,1]\) be a smooth cutoff with \(\theta \equiv 0\) outside of \([0,\frac{1}{2}]\) and \(\theta \equiv 1\) in \([0,\frac{1}{4}]\). For an indicator function \(\chi _u\in L^\infty ([0,{T_{strong}}];{\text {BV}}(\mathbb {R}^d;\{0,1\}))\) and \(t\geqq 0\), we define the local height of the one-sided interface error \(h^+(\cdot ,t):I_v(t)\rightarrow \mathbb {R}_0^+\) as
$$\begin{aligned} h^+(x,t):= \int _0^\infty (1-\chi _u)(x+y\mathrm {n}_v(x,t),t) \, \theta \Big (\frac{y}{r_c}\Big ) \,{\mathrm {d}}y. \end{aligned}$$
(58)
Similarly, we introduce the local height of the interface error in the other direction
$$\begin{aligned} h^-(x,t):= \int _0^\infty \chi _u(x-y\mathrm {n}_v(x,t),t) \theta \Big (\frac{y}{r_c}\Big ) \,{\mathrm {d}}y. \end{aligned}$$
Then \(h^+\) and \(h^-\) have the following properties:
a) (\(L^2\)-bound) We have the estimates \(|h^\pm (x,t)|\leqq \frac{r_c}{2}\) and
$$\begin{aligned} \int _{I_v(t)} |h^\pm (x,t)|^2\,{\mathrm {d}}S(x) \leqq C\int _{\mathbb {R}^d}|\chi _u{-}\chi _v| \min \Big \{\frac{{\text {dist}}(x,I_v(t))}{r_c},1\Big \} \,{\mathrm {d}}x. \end{aligned}$$
(59a)
b) (\(H^1\)-bound) Moreover, the estimate holds
$$\begin{aligned}&\int _{I_v(t)} \min \{|\nabla ^{\tan } h^\pm (x,t)|^2,|\nabla ^{\tan } h^\pm (x,t)|\} \,{\mathrm {d}}S+ |D^s h^\pm |(I_v(t))\nonumber \\&\quad \leqq C \int _{\mathbb {R}^d}1-\xi \cdot \frac{\nabla \chi _u}{|\nabla \chi _u|} \,\mathrm{d}|\nabla \chi _u|+\frac{C}{r_c^2} \int _{\mathbb {R}^d}|\chi _u{-}\chi _v| \min \Big \{\frac{{\text {dist}}(x,I_v(t))}{r_c},1\Big \} \,{\mathrm {d}}x. \end{aligned}$$
(59b)
c) (Approximation property) The functions \(h^+\) and \(h^-\) provide an approximation of the set \(\{\chi _u=1\}\) in terms of a subgraph over the set \(I_v(t)\) by setting
$$\begin{aligned} \chi _{v,h^+,h^-}:=\chi _v - \chi _{0\leqq {\text {dist}}^{\pm }(x,I_v(t))\leqq h^+(P_{I_v(t)}x,t)} + \chi _{-h^-(P_{I_v(t)}x,t)\leqq {\text {dist}}^{\pm }(x,I_v(t))\leqq 0}, \end{aligned}$$
up to an error of
$$\begin{aligned}&\int _{\mathbb {R}^d}\big |\chi _u-\chi _{v,h^+,h^-}\big | \,{\mathrm {d}}x\nonumber \\&\quad \leqq C \int _{\mathbb {R}^d}1-\xi \cdot \frac{\nabla \chi _u}{|\nabla \chi _u|} \,\mathrm{d}|\nabla \chi _u| + C \int _{\mathbb {R}^d}|\chi _u-\chi _v| \min \Big \{\frac{{\text {dist}}(x,I_v(t))}{r_c},1\Big \} \,{\mathrm {d}}x. \end{aligned}$$
(59c)
d) (Time evolution) Let v be a solenoidal vector field
$$\begin{aligned} v\in L^2([0,{T_{strong}}];H^1({\mathbb {R}^d};{\mathbb {R}^d}))\cap L^\infty ([0,{T_{strong}}];W^{1,\infty }({\mathbb {R}^d};{\mathbb {R}^d})) \end{aligned}$$
such that in the domain \(\bigcup _{t\in [0,{T_{strong}})} (\Omega _t^+\cup \Omega _t^-) \times \{t\}\) the second spatial derivatives of the vector field v exist and satisfy \(\sup _{t\in [0,{T_{strong}})} \sup _{x\in \Omega _t^+\cup \Omega _t^-} |\nabla ^2 v(x,t)| <\infty \). Assume that \(\chi _v\) solves the equation \(\partial _t \chi _v = -\nabla \cdot (\chi _v v)\). If \(\chi _u\) solves the equation \(\partial _t \chi _u = -\nabla \cdot (\chi _u u)\) for another solenoidal vector field \(u\in L^2([0,{T_{strong}}];H^1(\mathbb {R}^d;\mathbb {R}^d))\), we have the following estimate on the time derivative of the local interface error heights \(h^\pm \):
$$\begin{aligned}&\bigg |\frac{\mathrm {d}}{\mathrm {d}t} \int _{I_v(t)} \eta (x) h^\pm (x,t) \,{\mathrm {d}}S(x) - \int _{I_v(t)} h^\pm (x,t) (\mathrm {Id}{-}\mathrm {n}_v\otimes \mathrm {n}_v)v(x,t) \cdot \nabla \eta (x) \,{\mathrm {d}}S(x)\bigg |\nonumber \\&\quad \quad \leqq \frac{C}{r_c^2}\Vert \eta \Vert _{W^{1,4}(I_v(t))}\Bigg (\int _{I_v(t)} |\bar{h}^\pm |^4 \,{\mathrm {d}}S\Bigg )^{1/4} \nonumber \\&\quad \qquad \quad \times \Bigg (\int _{I_v(t)} \sup _{y\in [-r_c,r_c]} |u-v|^2(x+y\mathrm {n}_v(x,t),t) \,{\mathrm {d}}S(x)\Bigg )^{1/2} \nonumber \\&\qquad +C\frac{1+\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}}{r_c^3}\Vert \eta \Vert _{L^2(I_v(t))} \nonumber \\&\quad \qquad \quad \times \Bigg (\int _{{\mathbb {R}^d}}|\chi _u(x,t)-\chi _v(x,t)| \,\min \Big \{\frac{{\text {dist}}(x,I_v(t))}{r_c},1\Big \} \,{\mathrm {d}}x\Bigg )^\frac{1}{2}\nonumber \\&\qquad +\frac{C(1+\Vert v\Vert _{W^{1,\infty }})}{r_c^2}\max _{p\in \{2,4\}}\Vert \eta \Vert _{W^{1,p}(I_v(t))} \int _{{\mathbb {R}^d}} 1-\xi \cdot \frac{\nabla \chi _u}{|\nabla \chi _u|} \,\mathrm{d}|\nabla \chi _u|\nonumber \\&\qquad +C\Vert \eta \Vert _{L^2(I_v(t))}\bigg (\int _{I_v(t)} |u-v|^2 \,{\mathrm {d}}S\bigg )^{1/2}, \end{aligned}$$
(59d)
for any test function \(\eta \in C_{cpt}^\infty ({\mathbb {R}^d})\) with \(\mathrm {n}_v\cdot \nabla \eta =0\) on the interface \(I_v(t)\), and where \(\bar{h}^\pm \) is defined as \(h^\pm \) but now with respect to the modified cut-off \(\bar{\theta }(\cdot )=\theta \big (\frac{\cdot }{2}\big )\).
Proof
Step 1: Proof of the estimate on the\(L^2\)-norm. The trivial estimate \(|h^\pm (x,t)|\leqq \frac{r_c}{2}\) follows directly from the definition of \(h^\pm \). To establish the \(L^2\)-estimate, let \(\ell ^+(x):=\int _0^{r_c}(1-\chi _u)(x+y\mathrm {n}_v(x,t),t)\,{\mathrm {d}}y\). A straighforward estimate then gives
$$\begin{aligned} |\ell ^+(x)|^2 = 2\int _0^{\ell ^+(x)}y\,{\mathrm {d}}y\leqq C\int _0^{r_c}|\chi _u(\Phi _t(x,y),t)-\chi _v(\Phi _t(x,y),t)|\frac{y}{r_c}\,{\mathrm {d}}y. \end{aligned}$$
(60)
Note that the term on the left hand side dominates \(|h^+|^2\) since we dropped the cutoff function. Hence, the desired estimate on the \(L^2\)-norm of \(h^+\) follows at once by a change of variables and recalling the fact that \({\text {dist}}(\Phi _t(x,y),I_v(t))=y\). The corresponding bound for \(h^-\) then follows along the same lines.
Step 2: Proof of the estimate on the spatial derivative (59b). The definition (58) is equivalent to
$$\begin{aligned} h^+(\Phi _t(x,0),t)=\int _0^\infty (1-\chi _u)(\Phi _t(x,y)) \, \theta \Big (\frac{y}{r_c}\Big ) \,{\mathrm {d}}y. \end{aligned}$$
We compute for any smooth vector field \(\eta \in C^\infty _{cpt}({\mathbb {R}^d};{\mathbb {R}^d})\) (recall that \(\Phi _t(x,0)=x\) and \({\text {dist}}(\Phi _t(x,y),I_v(t))=y\) for any \(x\in I_v(t)\) and any y with \(|y|\leqq r_c\)) that
$$\begin{aligned}&\int _{I_v(t)} \eta (x) \cdot \mathrm {d} (D_x^{\tan } h^+(\cdot ,t))(x)\\&\quad =-\int _{I_v(t)} h^+(x,t) \nabla ^{\tan } \cdot \eta (x) \,{\mathrm {d}}S(x) - \int _{I_v(t)} h^+(x,t) \eta (x) \cdot \mathrm {H}(x,t) \,{\mathrm {d}}S(x)\\&\quad =-\int _0^{r_c} \int _{I_v(t)} (1-\chi _u)(\Phi _t(x,y),t) \theta \Big (\frac{y}{r_c}\Big ) \nabla ^{\tan } \cdot \eta (x) \,{\mathrm {d}}S(x) \,{\mathrm {d}}y\\&\qquad -\int _0^{r_c} \int _{I_v(t)}(1-\chi _u)(\Phi _t(x,y),t) \theta \Big (\frac{y}{r_c}\Big ) \eta (x) \cdot \mathrm {H}(\Phi _t(x,0),t) \,{\mathrm {d}}S(x) \,{\mathrm {d}}y\\&\quad =-\int _{{\mathbb {R}^d}} (1-\chi _u)(x,t) \theta \Big (\frac{{\text {dist}}(x,I_v(t))}{r_c}\Big ) |\det \nabla \Phi _t^{-1}(x)|\\&\qquad \qquad \qquad \times ({\text {Id}}{-}\mathrm {n}_v(P_{I_v(t)}x)\otimes \mathrm {n}_v(P_{I_v(t)}x)):\nabla \eta (P_{I_v(t)}x) \,{\mathrm {d}}x\\&\qquad -\int _{{\mathbb {R}^d}} (1-\chi _u)(x,t) \theta \Big (\frac{{\text {dist}}(x,I_v(t))}{r_c}\Big ) \eta (P_{I_v(t)}x) \cdot \mathrm {H}(P_{I_v(t)}x) |\det \nabla \Phi _t^{-1}(x)| \,{\mathrm {d}}x\\&\quad =-\int _{{\mathbb {R}^d}} \theta \Big (\frac{{\text {dist}}(x,I_v(t))}{r_c}\Big ) |\det \nabla \Phi _t^{-1}(x)| \eta (P_{I_v(t)}x)\\&\qquad \qquad \qquad ({\text {Id}}{-}\mathrm {n}_v(P_{I_v(t)}x)\otimes \mathrm {n}_v(P_{I_v(t)}x))\cdot \,\mathrm{d} \nabla \chi _u\\&\qquad +\int _{{\mathbb {R}^d}} (1-\chi _u)(x,t) \theta \bigg (\frac{{\text {dist}}(x,I_v(t))}{r_c}\bigg ) \eta (P_{I_v(t)}x)\\&\qquad \qquad \quad \cdot \Big (\nabla \cdot \big ( ({\text {Id}}{-}\mathrm {n}_v(P_{I_v(t)}x)\otimes \mathrm {n}_v(P_{I_v(t)}x))|\det \nabla \Phi _t^{-1}|\big ) -\mathrm {H}(P_{I_v(t)}x) |\det \nabla \Phi _t^{-1}|\Big ) \,{\mathrm {d}}x, \end{aligned}$$
where in the last step we have used \(\nabla {\text {dist}}^{\pm }(x,I_v(t))=\mathrm {n}_v(P_{I_v(t)}x)\). This yields, by another change of variables in the second integral, the fact that \(\chi _v(\Phi _t(x,y),t)=1\) for any \(y>0\), (19), (20), \(|\det \nabla \Phi _t^{-1}|\leqq C\) as well as by abbreviating \(\mathrm {n}_u = \frac{\nabla \chi _u}{|\nabla \chi _u|}\)
$$\begin{aligned}&\int _{U\cap I_v(t)} 1 \,\mathrm{d}|D_x^{\tan } h^+(\cdot ,t)|\\&\quad \leqq C\int _{\{x+y\mathrm {n}_v(x,t):\,x\in U\cap I_v(t),y\in (-r_c,r_c)\}} \big |\mathrm {n}_v(P_{I_v(t)}x) - \mathrm {n}_u\big | \,\mathrm{d}|\nabla \chi _u(\cdot ,t)|\\&\qquad +\frac{C}{r_c} \int _{U\cap I_v(t)} \int _0^{r_c} |\chi _u(\Phi _t(x,y),t)-\chi _v(\Phi _t(x,y),t)| \,{\mathrm {d}}y\,{\mathrm {d}}S(x) \end{aligned}$$
for any Borel set \(U\subset {\mathbb {R}^d}\). Recall that the indicator function \(\chi _u(\cdot ,t)\) of the varifold solution is of bounded variation in \(I:=\{x\in {\mathbb {R}^d}:{\text {dist}}^{\pm }(x,I_v(t))\in (-r_c,r_c)\}\). In particular, \(E^+:=\{x\in {\mathbb {R}^d}:\chi _u>0\}\cap I\) is a set of finite perimeter in I. Applying Theorem 39 in local coordinates the sections
$$\begin{aligned} E^+_x = \{y\in (-r_c,r_c):\chi _u(x+y\mathrm {n}_v(x,t))>0\} \end{aligned}$$
are guaranteed to be one-dimensional Caccioppoli sets in \((-r_c,r_c)\) for \(\mathcal {H}^{d-1}\)-almost every \(x\in I_v(t)\). Note that whenever \(|\mathrm {n}_v\cdot \mathrm {n}_u|\leqq \frac{1}{2}\) then \(1-\mathrm {n}_v\cdot \mathrm {n}_u\geqq \frac{1}{2}\), and therefore, using as well the co-area formula for rectifiable sets (see [11, (2.72)])
$$\begin{aligned}&\int _{U\cap I_v(t)} 1 \,\mathrm{d}|D_x^{\tan } h^+(\cdot ,t)|\nonumber \\&\quad \leqq \frac{C}{r_c} \int _{U\cap I_v(t)} \int _0^{r_c} |\chi _u(\Phi _t(x,y),t)-\chi _v(\Phi _t(x,y),t)| \,{\mathrm {d}}y\,{\mathrm {d}}S(x)\nonumber \\&\qquad + C\int _{U\cap I_v(t)}\int _{\partial ^*E^+_x\cap \{\mathrm {n}_v(x)\cdot \mathrm {n}_u(x{+}y\mathrm {n}_v(x,t))\geqq \frac{1}{2}\}\cap (-r_c,r_c)} \frac{|\mathrm {n}_v(x)-\mathrm {n}_u|}{|\mathrm {n}_v(x)\cdot \mathrm {n}_u|} \,\mathrm{d}\mathcal {H}^0(y)\,\mathrm{d}S(x)\nonumber \\&\qquad + C\int _{\begin{array}{c} \{x{+}y\mathrm {n}_v(x,t):\,x\in U\cap I_v(t),y\in (-r_c,r_c),\\ \qquad \mathrm {n}_v(x)\cdot \mathrm {n}_u(x{+}y\mathrm {n}_v(x,t))\leqq \frac{1}{2}\} \end{array}} ~ \big (1-\mathrm {n}_v(P_{I_v(t)}x)\cdot \mathrm {n}_u\big ) \,\mathrm{d}|\nabla \chi _u(\cdot ,t)|. \end{aligned}$$
(61)
We now distinguish between different cases depending on \(x\in I_v(t)\) up to \(\mathcal {H}^{d-1}\)-measure zero. We start with the set of points \(x\in A_1\subset I_v(t)\) such that
$$\begin{aligned}&\int _0^{r_c} |\chi _u(\Phi _t(x,y),t)-\chi _v(\Phi _t(x,y),t)| \,{\mathrm {d}}y\nonumber \\&\quad +\int _{\partial ^*E^+_x\cap \{\mathrm {n}_v(x)\cdot \mathrm {n}_u(x{+}y\mathrm {n}_v(x,t))\geqq \frac{1}{2}\}\cap (-r_c,r_c)} \frac{|\mathrm {n}_v(x)-\mathrm {n}_u|}{|\mathrm {n}_v(x)\cdot \mathrm {n}_u|} \,\mathrm{d}\mathcal {H}^0(y)\nonumber \\&\quad +\sup _{y\in \{\tilde{y}\in (-r_c,r_c)\cap \partial ^*E^+_x:\mathrm {n}_v(x)\cdot \mathrm {n}_u(x{+}\tilde{y}\mathrm {n}_v(x,t))\leqq \frac{1}{2}\}} 1-\mathrm {n}_v(P_{I_v(t)}x)\cdot \mathrm {n}_u(x{+}y\mathrm {n}_v(x,t))\nonumber \\&\leqq \frac{1}{4}. \end{aligned}$$
(62)
By splitting the measure \(D_x^\mathrm {tan}h^+\) into a part which is absolutely continuous with respect to the surface measure on \(I_v(t)\), for which we denote the density by \(\nabla ^\mathrm {tan} h^+\), as well as a singular part \(D^sh^+\), we obtain from (61) (note that the third integral in (61) does not contribute to this estimate by the definition of the set \(A_1\subset I_v(t)\))
$$\begin{aligned}&\int _{U\cap I_v(t)\cap A_1} |\nabla ^\mathrm {tan}h^+|(x)\,{\mathrm {d}}S(x)\\&\quad \leqq \int _{U\cap I_v(t)\cap A_1} \frac{C}{r_c}\int _0^{r_c} |\chi _u(\Phi _t(x,y),t)-\chi _v(\Phi _t(x,y),t)| \,{\mathrm {d}}y\,{\mathrm {d}}S(x)\\&\qquad + \int _{U\cap I_v(t)\cap A_1}C\int _{\partial ^*E^+_x\cap \{\mathrm {n}_v(x)\cdot \mathrm {n}_u(x{+}y\mathrm {n}_v(x,t))\geqq \frac{1}{2}\}\cap (-r_c,r_c)}\frac{|\mathrm {n}_v(x)-\mathrm {n}_u|}{|\mathrm {n}_v(x)\cdot \mathrm {n}_u|} \,\mathrm{d}\mathcal {H}^0(y)\,\mathrm{d}S(x) \end{aligned}$$
for every Borel set \(U\subset {\mathbb {R}^d}\). Since U was arbitrary, we deduce that \(|\nabla ^\mathrm {tan}h^+|\) is bounded on \(A_1\) by the two integrands on the right hand side of the last inequality. Hence, we obtain
$$\begin{aligned}&\int _{A_1} |\nabla ^\mathrm {tan}h^+|^2(x)\,{\mathrm {d}}S(x) + |D^sh^+|(A_1)\\&\quad \leqq Cr_c^{-2}\int _{I_v(t)}\bigg |\int _0^{r_c} |\chi _u(\Phi _t(x,y),t)-\chi _v(\Phi _t(x,y),t)| \,{\mathrm {d}}y\bigg |^2 \,{\mathrm {d}}S(x)\\&\qquad + C\int _{I_v(t)\cap A_1}\bigg |\int _{\partial ^*E^+_x\cap \{\mathrm {n}_v(x)\cdot \mathrm {n}_u(x{+}y\mathrm {n}_v(x,t))\geqq \frac{1}{2}\}\cap (-r_c,r_c)} |\mathrm {n}_v-\mathrm {n}_u|\,\mathrm{d}\mathcal {H}^0(y) \bigg |^2\,{\mathrm {d}}S(x). \end{aligned}$$
The first term on the right hand side can be estimated as in the proof of the \(L^2\)-bound for \(h^\pm \). To bound the second term, we start by representing the one-dimensional Caccioppoli sets \(E^+_x\) as a finite union of disjoint intervals (see [11, Proposition 3.52]). It then follows from property iv) in Theorem 39 that \(\partial ^*E^+_x\cap (-r_c,r_c)\) can only contain at most one point. Indeed, otherwise we would find at least one point \(y\in \partial ^*E^+_x\cap (-r_c,r_c)\) such that \(\mathrm {n}_v(x)\cdot \mathrm {n}_u(x{+}y\mathrm {n}_v(x,t))<0\) which is a contradiction to the definition of \(A_1\). By another application of the co-area formula for rectifiable sets (see [11, (2.72)]) we therefore get
$$\begin{aligned}&\int _{A_1} |\nabla ^\mathrm {tan}h^+|^2(x)\,{\mathrm {d}}S(x) + |D^sh^+|(A_1)\nonumber \\&\quad \leqq \frac{C}{r_c^2} \int _{\mathbb {R}^d}|\chi _u-\chi _v| \min \Big \{\frac{{\text {dist}}(x,I_v(t))}{r_c},1\Big \} \,{\mathrm {d}}x\nonumber \\&\qquad +C\int _{\{{\text {dist}}(x,I_v(t))<r_c\}} 1-\mathrm {n}_v(P_{I_v(t)}x)\cdot \frac{\nabla \chi _u}{|\nabla \chi _u|} \,\mathrm{d}|\nabla \chi _u|(x). \end{aligned}$$
(63)
We now turn to the second case, namely the set of points \(A_2:=I_v(t){\setminus } A_1\). We begin with a preliminary computation. When splitting \(E^+_x\) into a finite family of disjoint open intervals as before, it again follows from property iv) in Theorem 39 that every second point \(y\in \partial ^*E^+_x\cap (-r_c,r_c)\) has to have the property that \(\mathrm {n}_v(x)\cdot \mathrm {n}_u(x{+}y\mathrm {n}_v(x,t))<0\), that is, \(|\mathrm {n}_v(x)-\mathrm {n}_u|\leqq 2\leqq 2(1-\mathrm {n}_v(x)\cdot \mathrm {n}_u)\). In particular, by another application of the co-area formula for rectifiable sets (see [11, (2.72)]) we obtain the bound
$$\begin{aligned}&\int _{A_2}\int _{\partial ^*E^+_x\cap \{\mathrm {n}_v(x)\cdot \mathrm {n}_u(x{+}y\mathrm {n}_v(x,t))\geqq \frac{1}{2}\}\cap (-r_c,r_c)} \frac{|\mathrm {n}_v(x)-\mathrm {n}_u|}{|\mathrm {n}_v(x)\cdot \mathrm {n}_u|} \,\mathrm{d}\mathcal {H}^0(y)\,\mathrm{d}S(x)\nonumber \\&\quad \leqq 8\int _{\{{\text {dist}}(x,I_v(t))<r_c\}} 1-\mathrm {n}_v(P_{I_v(t)}x)\cdot \frac{\nabla \chi _u}{|\nabla \chi _u|} \,\mathrm{d}|\nabla \chi _u|(x). \end{aligned}$$
(64)
Now, we proceed as follows. By definition of \(A_2\), either one of the three summands in (62) has to be \(\geqq \frac{1}{12}\). We distinguish between two cases. If the third one is not, then this actually means that the set \(\{\tilde{y}\in (-r_c,r_c)\cap \partial ^*E^+_x:\mathrm {n}_v(x)\cdot \mathrm {n}_u(x{+}\tilde{y}\mathrm {n}_v(x,t))\leqq \frac{1}{2}\}\) is empty, that is, the third summand has to vanish. Hence, either one of the first two summands in (62) has to be \(\geqq \frac{1}{8}\). If the first one is not, we use that \(\int _0^{r_c} |\chi _u(\Phi _t(x,y),t)-\chi _v(\Phi _t(x,y),t)| \,{\mathrm {d}}y\leqq r_c\) and bound this by the second term and then (64). If the second one is not, then
$$\begin{aligned} \ell ^+(x)&:=\int _0^{r_c} |\chi _u(\Phi _t(x,y),t)-\chi _v(\Phi _t(x,y),t)| \,{\mathrm {d}}y\leqq r_c\nonumber \\&\leqq \frac{C}{r_c}\int _0^{\ell ^+(x)}y\,{\mathrm {d}}y\leqq C\int _0^{r_c}|\chi _u(\Phi _t(x,y),t)-\chi _v(\Phi _t(x,y),t)|\frac{y}{r_c}\,{\mathrm {d}}y. \end{aligned}$$
(65)
Now, we move on with the remaining case, that is, that the third summand in (62) does not vanish. In other words, \(\{\tilde{y}\in (-r_c,r_c)\cap \partial ^*E^+_x:\mathrm {n}_v(x)\cdot \mathrm {n}_u(x{+}\tilde{y}\mathrm {n}_v(x,t))\leqq \frac{1}{2}\}\) is non-empty. We then estimate
$$\begin{aligned}&\int _0^{r_c} |\chi _u(\Phi _t(x,y),t)-\chi _v(\Phi _t(x,y),t)| \,{\mathrm {d}}y\nonumber \\&\leqq r_c \leqq 2r_c\int _{\partial ^*E^+_x\cap (-r_c,r_c)} 1-\mathrm {n}_v(x)\cdot \mathrm {n}_u(x{+}y\mathrm {n}_v(x,t)) \,\mathrm{d}\mathcal {H}^0(y). \end{aligned}$$
(66)
Taking finally \(U=A_2\) in (61), the conclusions of the above case study together with the three estimates (64), (65) and (66) followed by another application of the co-area formula for rectifiable sets (see [11, (2.72)]) to further estimate the latter, then imply that
$$\begin{aligned}&\int _{A_2} |\nabla ^\mathrm {tan}h^+|(x)\,{\mathrm {d}}S(x) + |D^sh^+|(A_2)\nonumber \\&\quad \leqq \frac{C}{r_c} \int _{\mathbb {R}^d}|\chi _u-\chi _v| \min \Big \{\frac{{\text {dist}}(x,I_v(t))}{r_c},1\Big \} \,{\mathrm {d}}x\nonumber \\&\qquad +C\int _{\{{\text {dist}}(x,I_v(t))<r_c\}} 1-\mathrm {n}_v(P_{I_v(t)}x)\cdot \frac{\nabla \chi _u}{|\nabla \chi _u|} \,\mathrm{d}|\nabla \chi _u|(x). \end{aligned}$$
(67)
The two estimates (63) and (67) thus entail the desired upper bound (59b) for the (tangential) gradient of \(h^\pm \) with \(\xi \) replaced by \(\mathrm {n}_v(P_{I_v(t)}x)\). However, one may replace \(\mathrm {n}_v(P_{I_v(t)}x)\) by \(\xi \) because of (38).
Step 3: Proof of the approximation property for the interface (59c). In order to establish (59c), we rewrite using the coordinate transform \(\Phi _t\) (recall that \({\text {dist}}^{\pm }(\Phi _t(x,y),I_v(t))=y\) and that \(|h^\pm |\leqq r_c\))
$$\begin{aligned} \int _{\mathbb {R}^d}&|\chi _u-\chi _{v,h^+,h^-}| \,{\mathrm {d}}x\nonumber \\&= \int _{I_v(t)} \int _{0}^{r_c} \det \nabla \Phi _t(x,y) |\chi _u(\Phi _t(x,y))-1+\chi _{\{y\leqq h^+(x)\}}| \,{\mathrm {d}}y\,{\mathrm {d}}S(x)\nonumber \\&\quad + \int _{I_v(t)} \int _{-r_c}^0 \det \nabla \Phi _t(x,y) |\chi _u(\Phi _t(x,y))-\chi _{\{y\geqq -h^-(x)\}}| \,{\mathrm {d}}y\,{\mathrm {d}}S(x)\nonumber \\&\quad + \int _{\{{\text {dist}}(x,I_v(t))\geqq r_c\}} |\chi _u-\chi _v| \,{\mathrm {d}}x. \end{aligned}$$
(68)
In order to derive a bound for the first term on the right-hand side of (68), we distinguish between different cases depending on \(x\in I_v(t)\) up to \(\mathcal {H}^{d-1}\)-measure zero. We first distinguish between \(h^+(x)\geqq \frac{r_c}{4}\) and \(h^+(x)<\frac{r_c}{4}\). In the former case, a straightforward estimate yields (recall (18))
$$\begin{aligned}&\bigg |\int _{0}^{r_c} \det \nabla \Phi _t(x,y) |\chi _u(\Phi _t(x,y))-1+\chi _{\{y\leqq h^+(x)\}}| \,{\mathrm {d}}y\bigg |\nonumber \\&\quad \leqq Cr_c \leqq \frac{C}{r_c}\int _0^{h^+(x)}y\,{\mathrm {d}}y\leqq C\int _0^{r_c}|\chi _u(\Phi _t(x,y))-\chi _v(\Phi _t(x,y))|\frac{y}{r_c}\,{\mathrm {d}}y, \end{aligned}$$
(69)
which is indeed of required order after a change of variables. We now consider the other case, that is, \(h^+(x)<\frac{r_c}{4}\). Recall that the indicator function \(\chi _u(\cdot ,t)\) of the varifold solution is of bounded variation in \(I^+:=\{x\in {\mathbb {R}^d}:{\text {dist}}^{\pm }(x,I_v(t))\in (0,r_c)\}\). In particular, \(E^+:=\{x\in {\mathbb {R}^d}:1-\chi _u>0\}\cap I_+\) is a set of finite perimeter in \(I^+\). Recall also that \(E^+=I^+\cap \{x\in {\mathbb {R}^d}:(\chi _v-\chi _u)_+>0\}\) since \(\chi _v\equiv 1\) in \(I^+\). Applying Theorem 39 in local coordinates, the sections
$$\begin{aligned} E^+_x = \{y\in (0,r_c):1-\chi _u(x+y\mathrm {n}_v(x,t))>0\} \end{aligned}$$
are guaranteed to be one-dimensional Caccioppoli sets in \((0,r_c)\) for \(\mathcal {H}^{d-1}\)-almost every \(x\in I_v(t)\). Hence, we may represent the one-dimensional section \(E^+_x\) for such \(x\in I_v(t)\) as a finite union of disjoint intervals (see [11, Proposition 3.52]):
$$\begin{aligned} E^+_x\cap (0,r_c) = \bigcup _{m=1}^{K(x)}(a_m,b_m). \end{aligned}$$
If \(K(x) = 0\) then \(h^+(x)=0\), and the inner integral in the first term on the right hand side of (68) vanishes for this x. If \(K(x) = 1\) and \(a_1=0\), then by definition of \(h^+(x)\) we have \((a_1,b_1)=(0,h^+(x))\) (recall that we now consider the case \(h^+(x)\leqq \frac{r_c}{4}\)). Thus, again the inner integral in the first term on the right hand side of (68) vanishes for this x. Hence, it remains to discuss the case that there is at least one non-empty interval in the decomposition of \(E^+_x\), say (a, b), such that \(a\in (0,r_c)\). From property iv) in Theorem 39 it then follows that
$$\begin{aligned} \mathrm {n}_v(x,t)\cdot \frac{-\nabla \chi _{E^+}}{|\nabla \chi _{E^+}|}(x+a\mathrm {n}_v(x,t)) \leqq 0. \end{aligned}$$
Hence, we may bound
$$\begin{aligned}&\bigg |\int _{0}^{r_c} \det \nabla \Phi _t(x,y) |\chi _u(\Phi _t(x,y))-1+\chi _{\{y\leqq h^+(x)\}}| \,{\mathrm {d}}y\bigg |\\&\quad \leqq Cr_c \leqq C\int _{(0,r_c)\cap (\partial ^*E^+)_x} 1-\mathrm {n}_v(x,t)\cdot \frac{-\nabla \chi _{E^+}}{|\nabla \chi _{E^+}|}(x+y\mathrm {n}_v(x,t))\, \mathrm {d}\mathcal {H}^0(y). \end{aligned}$$
Gathering the bounds from the different cases together with the estimate in (69), we therefore obtain by the co-area formula for rectifiable sets (see [11, (2.72)]) together with the change of variables \(\Phi _t(x,y)\):
$$\begin{aligned}&\bigg |\int _{I_v(t)}\int _{0}^{r_c} \det \nabla \Phi _t(x,y) |\chi _u(\Phi _t(x,y))-1+\chi _{\{y\leqq h^+(x)\}}| \,{\mathrm {d}}y\,{\mathrm {d}}S(x)\bigg | \\&\quad \leqq C \int _{I_v(t)}\int _{(0,r_c)\cap (\partial ^*E^+)_x} 1-\mathrm {n}_v(x,t)\cdot \frac{-\nabla \chi _{E^+}}{|\nabla \chi _{E^+}|}(x+y\mathrm {n}_v(x,t))\,\mathrm{d}\mathcal {H}^0(y)\,{\mathrm {d}}S(x)\\&\qquad + C \int _{\mathbb {R}^d}\int _{-r_c}^{r_c} |\chi _u(\Phi _t(x,y))-\chi _v(\Phi _t(x,y))| \frac{y}{r_c} \,\mathrm{d}y \,{\mathrm {d}}x\\&\quad \leqq C \int _{\{{\text {dist}}(x,I_v(t))<r_c\}} 1-\mathrm {n}_v(P_{I_v(t)}x)\cdot \frac{\nabla \chi _u}{|\nabla \chi _u|} \,\mathrm{d}|\nabla \chi _u|(x) \\&\qquad + C \int _{\mathbb {R}^d}|\chi _u-\chi _v| \min \Big \{\frac{{\text {dist}}(x,I_v(t))}{r_c},1\Big \} \,{\mathrm {d}}x, \end{aligned}$$
which is by (38) as well as (37) indeed a bound of desired order. Moreover, performing analogous estimates for the second term on the right-hand side of (68) and estimating the third term on the right-hand side of (68) trivially, we then get
$$\begin{aligned} \int _{\mathbb {R}^d}&|\chi _u-\chi _{v,h^+,h^-}| \,{\mathrm {d}}x\\&\leqq C \int _{\mathbb {R}^d}1-\xi \cdot \frac{\nabla \chi _u}{|\nabla \chi _u|} \,\mathrm{d}|\nabla \chi _u| + C \int _{\mathbb {R}^d}|\chi _u-\chi _v| \min \Big \{\frac{{\text {dist}}(x,I_v(t))}{r_c},1\Big \} \,{\mathrm {d}}x, \end{aligned}$$
which is precisely the desired estimate (59c).
Step 4: Proof of estimate on the time derivative (59d). To bound the time derivative, we compute using the weak formulation of the continuity equation \(\partial _t \chi _u = -\nabla \cdot (\chi _u u)\) and abbreviating \(I^+(t):=\{x\in {\mathbb {R}^d}:{\text {dist}}^{\pm }(x,I_v(t))\in [0,r_c)\}\) (recall that the boundary \(\partial I^+(t) = I_v(t)\) moves with normal speed \(\mathrm {n}_v \cdot v\)) to get
$$\begin{aligned}&\frac{\mathrm {d}}{{\mathrm {d}}t} \int _{I_v(t)} \eta (x) h^+(x,t) \,{\mathrm {d}}S(x)\\&\quad =\frac{\mathrm {d}}{{\mathrm {d}}t} \int _{I_v(t)} \int _0^\infty \eta (x) (1-\chi _u)(x+y\mathrm {n}_v(x,t),t) \, \theta \Big (\frac{y}{r_c}\Big ) \,{\mathrm {d}}y\,{\mathrm {d}}S(x)\\&\quad =\frac{\mathrm {d}}{{\mathrm {d}}t} \int _{I^+(t)} \eta (P_{I_v(t)}x) |\det \nabla \Phi _t^{-1}|(x) (1-\chi _u)(x,t) \, \theta \Big (\frac{{\text {dist}}(x,I_v(t))}{r_c}\Big ) \,{\mathrm {d}}x\\&\quad =\int _{I^+(t)} (1-\chi _u)(x,t) u \cdot \nabla \Big (\eta (P_{I_v(t)}x) |\det \nabla \Phi _t^{-1}|(x) \, \theta \Big (\frac{{\text {dist}}(x,I_v(t))}{r_c}\Big )\Big ) \,{\mathrm {d}}x\\&\qquad +\int _{I_v(t)} (\mathrm {n}_v\cdot u)(x,t) (1{-}\chi _u)(x,t) \eta (P_{I_v(t)}x) |\det \nabla \Phi _t^{-1}|(x) \theta \Big (\frac{{\text {dist}}(x,I_v(t))}{r_c}\Big ) \,{\mathrm {d}}S(x)\\&\qquad +\int _{I^+(t)} (1-\chi _u)(x,t) \, \frac{\mathrm {d}}{{\mathrm {d}}t}\Big (\eta (P_{I_v(t)}x) |\det \nabla \Phi _t^{-1}|(x) \theta \Big (\frac{{\text {dist}}(x,I_v(t))}{r_c}\Big )\Big ) \,{\mathrm {d}}x\\&\qquad -\int _{I_v(t)} (\mathrm {n}_v\cdot v)(x,t) (1{-}\chi _u)(x,t) \eta (P_{I_v(t)}x) |\det \nabla \Phi _t^{-1}|(x) \theta \Big (\frac{{\text {dist}}(x,I_v(t))}{r_c}\Big ) \,{\mathrm {d}}S(x). \end{aligned}$$
Recall from (27) the formula for the gradient of the projection onto the nearest point on the interface \(I_v(t)\). Recalling also the definitions of the extended normal velocity \(V_\mathrm {n}(x,t):= \big (v(x,t)\cdot \mathrm {n}_v(P_{I_v(t)}x,t)\big )\,\mathrm {n}_v(P_{I_v(t)}x,t)\) and its projection \(\bar{V}_{\mathrm {n}}(x,t):=V_{\mathrm {n}}(P_{I_v(t)}x,t)\) from (50) respectively (22), we also have
$$\begin{aligned}&-\int _{I^+} (1-\chi _u(x,t)) |\det \nabla \Phi _t^{-1}|(x) \theta \Big (\frac{{\text {dist}}(x,I_v(t))}{r_c}\Big ) (\nabla \eta )(P_{I_v(t)}x)\\&~~~~\qquad \cdot \big ((v(P_{I_v(t)}x,t)-\bar{V}_{\mathrm {n}}(x,t)) \cdot \nabla \big ) P_{I_v(t)}x \,{\mathrm {d}}x\\&\quad =-\int _{I_v(t)} \int _0^{r_c} (1-\chi _u(\Phi _t(x,y),t)) \theta \Big (\frac{y}{r_c}\Big ) \nabla \eta (x)\\&\qquad \qquad ~~~~~~~~~~~~~~~~ \cdot \big ((v(x,t)-V_{\mathrm {n}}(x,t)) \cdot \nabla \big ) P_{I_v(t)}(\Phi _t(x,y)) \,{\mathrm {d}}y\,{\mathrm {d}}S(x)\\&\quad =-\int _{I_v(t)} h^+(x,t) ({\text {Id}}{-}\mathrm {n}_v(x)\otimes \mathrm {n}_v(x))v(x,t) \cdot \nabla \eta (x) \,{\mathrm {d}}S(x) \\&\qquad +\int _{I^+(t)} (1-\chi _u(x,t)) |\det \nabla \Phi _t^{-1}|(x) \theta \Big (\frac{{\text {dist}}(x,I_v(t))}{r_c}\Big ){\text {dist}}(x,I_v(t))\\&\qquad ~~~~~~~~~~~~~~~~\times (\nabla \eta )(P_{I_v(t)}x)\cdot \big ((v(P_{I_v(t)}x,t)-\bar{V}_{\mathrm {n}}(x,t)) \cdot \nabla \big )\mathrm {n}_v(P_{I_v(t)}x) \,{\mathrm {d}}x. \end{aligned}$$
Adding this formula to the above formula for \(\frac{\mathrm {d}}{{\mathrm {d}}t} \int _{I_v(t)} \eta (x) h^+(x,t) \,{\mathrm {d}}S(x)\), introducing the abbreviation \(f:=|\det \nabla \Phi _t^{-1}|(x) \, \theta (\frac{{\text {dist}}(x,I_v(t))}{r_c})\), and using the fact that \(\chi _v=1\) in \(I^+(t)\), we obtain
$$\begin{aligned}&\frac{\mathrm {d}}{{\mathrm {d}}t} \int _{I_v(t)} \eta (x) h^+(x,t) \,{\mathrm {d}}x-\int _{I_v(t)} h^+(x,t) ({\text {Id}}{-}\mathrm {n}_v\otimes \mathrm {n}_v)v(x,t) \cdot \nabla \eta (x) \,{\mathrm {d}}S(x)\nonumber \\&\quad =\int _{I^+(t)} (\chi _u(x,t)-\chi _v(x,t))f(x){\text {dist}}(x,I_v(t)) (\nabla \eta )(P_{I_v(t)}x)\nonumber \\&\qquad ~~~~~~~~~~~\cdot \big ((v(P_{I_v(t)}x,t)-\bar{V}_{\mathrm {n}}(x,t)) \cdot \nabla \big )\mathrm {n}_v(P_{I_v(t)}x) \,{\mathrm {d}}x\nonumber \\&\qquad -\int _{I^+(t)} (\chi _u(x,t)-\chi _v(x,t)) \eta (P_{I_v(t)}x) (u-v) \cdot \nabla f \,{\mathrm {d}}x\nonumber \\&\qquad -\int _{I^+(t)} (\chi _u(x,t)-\chi _v(x,t)) f(x) (\nabla \eta )(P_{I_v(t)}x) \cdot ((u-v) \cdot \nabla )P_{I_v(t)}x \,{\mathrm {d}}x\nonumber \\&\qquad -\int _{I^+(t)} (\chi _u(x,t)-\chi _v(x,t)) f(x) (\nabla \eta )(P_{I_v(t)}x)\nonumber \\&\qquad ~~~~~~~~~~~~~~\cdot \big ((v(x,t)-(v(P_{I_v(t)}x,t)-\bar{V}_{\mathrm {n}}(x,t))) \cdot \nabla \big ) P_{I_v(t)}x \,{\mathrm {d}}x\nonumber \\&\qquad -\int _{I^+(t)} (\chi _u(x,t)-\chi _v(x,t)) f(x) (\nabla \eta )(P_{I_v(t)}x) \cdot \frac{\mathrm {d}}{{\mathrm {d}}t} P_{I_v(t)}x \,{\mathrm {d}}x\nonumber \\&\qquad -\int _{I^+(t)} (\chi _u(x,t)-\chi _v(x,t)) \, \eta (P_{I_v(t)}x) \Big (\frac{\mathrm {d}}{{\mathrm {d}}t}f+v \cdot \nabla f\Big ) \,{\mathrm {d}}x\nonumber \\&\qquad +\int _{I_v(t)} \mathrm {n}_v \cdot (u-v) (1-\chi _u) \eta \,{\mathrm {d}}S. \end{aligned}$$
(70)
Note that \(f(x)=|\det \nabla \Phi _t^{-1}|(x) \, \theta (\frac{{\text {dist}}(x,I_v(t))}{r_c})=1\) for any t and any \(x\in I_v(t)\). Thus, we have \(\frac{\mathrm {d}}{{\mathrm {d}}t} f + v\cdot \nabla f=0\) on \(I_v(t)\). Furthermore, we have \(|\nabla \bar{V}_{\mathrm {n}}| \leqq \frac{C}{r_c^2}\Vert v\Vert _{W^{1,\infty }}\) and \(|\nabla ^2\bar{V}_{\mathrm {n}}| \leqq \frac{C}{r_c^3}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}\) because of \(\bar{V}_{\mathrm {n}}(x)=V_{\mathrm {n}}(P_{I_v(t)}x)\), (19), the corresponding estimate (43) for the gradient of \(V_{\mathrm {n}}\) as well as the formula (27) for the gradient of \(P_{I_v(t)}\). Because of (23) and the equation (34) for the time evolution of the normal vector, we thus get the bounds \(|\frac{\mathrm {d}}{{\mathrm {d}}t}\nabla {\text {dist}}^{\pm }(\cdot ,I_v(t))|\leqq \frac{C}{r_c^2}\Vert v\Vert _{W^{1,\infty }}\) and \(|\nabla \frac{\mathrm {d}}{{\mathrm {d}}t}\nabla {\text {dist}}^{\pm }(\cdot ,I_v(t))|\leqq \frac{C}{r_c^3}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}\). Taking all of these bounds together, we obtain \(|f|\leqq \frac{C}{r_c}\), \(|\nabla f|\leqq \frac{C}{r_c^2}\) and \(|\nabla ^2 f|+|\nabla \frac{\mathrm {d}}{{\mathrm {d}}t} f|\leqq \frac{C}{r_c^3}(1+\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))})\). As a consequence, we get
$$\begin{aligned} \Big |\frac{\mathrm {d}}{{\mathrm {d}}t}f+v \cdot \nabla f\Big | \leqq \frac{C}{r_c^3}(1+\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}){\text {dist}}(\cdot ,I_v(t)). \end{aligned}$$
(71)
Moreover, we may compute
$$\begin{aligned} \frac{\mathrm {d}}{{\mathrm {d}}t} P_{I_v(t)}x = -\mathrm {n}_v(P_{I_v(t)}x) \frac{\mathrm {d}}{{\mathrm {d}}t}{\text {dist}}^{\pm }(x,I_v(t)) - {\text {dist}}^{\pm }(x,I_v(t)) \frac{\mathrm {d}}{{\mathrm {d}}t}(\mathrm {n}_v(P_{I_v(t)}x)). \end{aligned}$$
(72)
Since \(\mathrm {n}_v\cdot \nabla \eta = 0\) holds on the interface \(I_v(t)\) by assumption, we obtain from (72)
$$\begin{aligned}&-\int _{I^+(t)} (\chi _u(x,t)-\chi _v(x,t)) f(x) (\nabla \eta )(P_{I_v(t)}x) \cdot \frac{\mathrm {d}}{{\mathrm {d}}t} P_{I_v(t)}x \,{\mathrm {d}}x\\&\quad =\int _{I^+(t)} (\chi _u(x,t)-\chi _v(x,t)) {\text {dist}}^{\pm }(x,I_v(t)) f(x) (\nabla \eta )(P_{I_v(t)}x)\\&\qquad ~~~~~~~~~~~~\cdot \frac{\mathrm {d}}{dt}(\mathrm {n}_v(P_{I_v(t)}x)) \,{\mathrm {d}}x. \end{aligned}$$
In what follows, we will by slight abuse of notation use \(\nabla ^{\mathrm {tan}}g(x)\) as a shorthand for \((\mathrm {Id}-\mathrm {n}_v(P_{I_v(t)}x)\otimes \mathrm {n}_v(P_{I_v(t)}x))\nabla g(x)\) for scalar fields as well as \((\nabla ^{\mathrm {tan}}\cdot g)(x)\) instead of \((\mathrm {Id}-\mathrm {n}_v(P_{I_v(t)}x)\otimes \mathrm {n}_v(P_{I_v(t)}x)):\nabla g(x)\) for vector fields. Let us also abbreviate \(P^\mathrm {tan}x:=(\mathrm {Id}-\mathrm {n}_v(P_{I_v(t)}x)\otimes \mathrm {n}_v(P_{I_v(t)}x))\). Note that by assumption \((\nabla \eta )(P_{I_v(t)}x)=(\nabla ^{\mathrm {tan}}\eta )(P_{I_v(t)}x)\). Moreover, it follows from (24), (25) and (23) that \(\mathrm {n}_v(P_{I_v(t)}x)\cdot \frac{\mathrm {d}}{{\mathrm {d}}t}(\mathrm {n}_v(P_{I_v(t)}x))=0\). Hence, we may rewrite with an integration by parts (recall the notation \(P^\mathrm {tan}(x)=({\text {Id}}-\mathrm {n}_v\otimes \mathrm {n}_v)(P_{I_v(t)}x,t)\))
$$\begin{aligned}&\int _{I^+(t)} (\chi _u(x,t)-\chi _v(x,t)) {\text {dist}}^{\pm }(x,I_v(t)) f(x) (\nabla ^{\mathrm {tan}}\eta )(P_{I_v(t)}x)\cdot \frac{\mathrm {d}}{{\mathrm {d}}t} (\mathrm {n}_v(P_{I_v(t)}x)) \,{\mathrm {d}}x\nonumber \\&\quad =-\int _{I^+(t)} (\chi _u-\chi _v)(x,t) {\text {dist}}^{\pm }(x,I_v(t))\eta (P_{I_v(t)}x)\nonumber \\&\qquad ~~~~~~~~~~~~~~ \times \Big (\frac{\mathrm {d}}{{\mathrm {d}}t}(\mathrm {n}_v(P_{I_v(t)}x))\otimes \nabla \Big ) :f(x)P^\mathrm {tan}(x) \,{\mathrm {d}}x\nonumber \\&\qquad -\int _{I^+(t)} (\chi _u-\chi _v)(x,t) {\text {dist}}^{\pm }(x,I_v(t)) f(x)\eta (P_{I_v(t)}x)\nabla ^{\mathrm {tan}}\cdot \frac{\mathrm {d}}{{\mathrm {d}}t} (\mathrm {n}_v(P_{I_v(t)}x)) \,{\mathrm {d}}x\nonumber \\&\qquad -\int _{\mathbb {R}^d}{\text {dist}}^{\pm }(x,I_v(t)) f(x)\eta (P_{I_v(t)}x)\Big (\frac{\nabla \chi _u}{|\nabla \chi _u|}-\mathrm {n}_v(P_{I_v(t)}x)\Big ) \cdot \frac{\mathrm {d}}{{\mathrm {d}}t}(\mathrm {n}_v(P_{I_v(t)}x)) \,\mathrm{d}|\nabla \chi _u|. \end{aligned}$$
(73)
Using, from (25) and (23), that the spatial partial derivatives of the extended normal vector field are orthogonal to the gradient of the signed distance function, the same argument also shows that
$$\begin{aligned}&\int _{I^+(t)} (\chi _u(x,t)-\chi _v(x,t))f(x){\text {dist}}(x,I_v(t)) (\nabla ^{\mathrm {tan}} \eta )(P_{I_v(t)}x)\nonumber \\&\quad \quad ~~~~~~~~ \cdot \big ((v(P_{I_v(t)}x,t)-\bar{V}_{\mathrm {n}}(x,t)) \cdot \nabla \big )\mathrm {n}_v(P_{I_v(t)}x) \,{\mathrm {d}}x\nonumber \\&\quad =-\int _{I^+(t)} (\chi _u(x,t)-\chi _v(x,t)) {\text {dist}}(x,I_v(t)) \eta (P_{I_v(t)}x)\nonumber \\&\qquad ~~~~~~~~~~~~\times \big (((v(P_{I_v(t)}x,t)-\bar{V}_{\mathrm {n}}(x,t))\cdot \nabla ) \mathrm {n}_v(P_{I_v(t)}x)\otimes \nabla \big ):f(x)P^\mathrm {tan}(x) \,{\mathrm {d}}x\nonumber \\&\qquad -\int _{I^+(t)} (\chi _u(x,t)-\chi _v(x,t)) {\text {dist}}(x,I_v(t)) f(x)\eta (P_{I_v(t)}x)\nonumber \\&\qquad \qquad \qquad \times \nabla ^\mathrm {tan} \cdot \big (\big ((v(P_{I_v(t)}x,t)-\bar{V}_{\mathrm {n}}(x,t)) \cdot \nabla \big )\mathrm {n}_v(P_{I_v(t)}x) \big )\,{\mathrm {d}}x\nonumber \\&\qquad -\int _{\mathbb {R}^d}{\text {dist}}^{\pm }(x,I_v(t)) f(x)\eta (P_{I_v(t)}x)\Big (\frac{\nabla \chi _u}{|\nabla \chi _u|} -\mathrm {n}_v(P_{I_v(t)}x)\Big )\nonumber \\&\qquad ~~~~~~~~~~~~\cdot \big ((v(P_{I_v(t)}x,t)-\bar{V}_{\mathrm {n}}(x,t)) \cdot \nabla \big )\mathrm {n}_v(P_{I_v(t)}x)\,\mathrm{d}|\nabla \chi _u|. \end{aligned}$$
(74)
It follows from (27) as well as (25) and (23) that \((\mathrm {n}_v(P_{I_v(t)}x)\cdot \nabla )P_{I_v(t)}x = 0\). Hence, we obtain
$$\begin{aligned}&\int _{I^+(t)} (\chi _u(x,t)-\chi _v(x,t)) f(x) (\nabla \eta )(P_{I_v(t)}x)\nonumber \\&\qquad ~~~~~ \cdot \big ((v(x,t)-(v(P_{I_v(t)}x,t)-\bar{V}_{\mathrm {n}}(x,t))) \cdot \nabla \big ) P_{I_v(t)}(x) \,{\mathrm {d}}x\nonumber \\&\quad =\int _{I^+(t)} (\chi _u-\chi _v)(x,t) f(x) (\nabla \eta )(P_{I_v(t)}x) \nonumber \\&\qquad ~~~~~~~~~~\cdot \big ((v(x,t)-v(P_{I_v(t)}x,t)) \cdot \nabla \big )P_{I_v(t)}x \,{\mathrm {d}}x. \end{aligned}$$
(75)
Since the domain of integration is \(I^+(t)\), we may write
$$\begin{aligned}&v(x,t)-v(P_{I_v(t)}x,t) \\&\quad = {\text {dist}}^{\pm }(x,I_v(t))\int _{(0,1]} \nabla v\big (P_{I_v(t)}x+\lambda {\text {dist}}^{\pm }(x,I_v(t))\mathrm {n}_v(P_{I_v(t)}x)\big )\,\mathrm{d}\lambda \cdot \mathrm {n}_v(P_{I_v(t)}x). \end{aligned}$$
From this and the fact that \(\mathrm {n}_v(P_{I_v(t)})\cdot \nabla P_{I_v(t)}(x) = 0\), we deduce by another integration by parts that (where \(|F|\leqq r_c^{-1}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}\))
$$\begin{aligned}&\int _{I^+(t)} (\chi _u(x,t)-\chi _v(x,t)) f(x) (\nabla ^\mathrm {tan} \eta )(P_{I_v(t)}x)\nonumber \\&\qquad ~~~~ \cdot ((v(x,t)-v(P_{I_v(t)} x,t)) \cdot \nabla )P_{I_v(t)}x \,{\mathrm {d}}x\nonumber \\&\quad =-\int _{I^+(t)} (\chi _u(x,t)-\chi _v(x,t)) \eta (P_{I_v(t)}x)\nonumber \\&\qquad ~~~~~~~~~~~~ \times \big (((v(x,t)-v(P_{I_v(t)} x,t)) \cdot \nabla ) P_{I_v(t)}x\otimes \nabla \big ): f(x)P^\mathrm {tan}x \,{\mathrm {d}}x\nonumber \\&\qquad -\int _{I^+(t)} (\chi _u(x,t)-\chi _v(x,t)) f(x) \eta (P_{I_v(t)}x) ((v(x,t)-v(P_{I_v(t)} x,t))\nonumber \\&\qquad ~~~~~~~~~~~~\cdot \nabla (\nabla ^\mathrm {tan}\cdot P_{I_v(t)}x) \,{\mathrm {d}}x\nonumber \\&\qquad -\int _{I^+(t)} (\chi _u(x,t)-\chi _v(x,t)) {\text {dist}}(x,I_v(t)) f(x) \eta (P_{I_v(t)}x) F(x,t):\nabla P_{I_v(t)}x \,{\mathrm {d}}x\nonumber \\&\qquad -\int _{\mathbb {R}^d}f(x)\eta (P_{I_v(t)}x)\Big (\frac{\nabla \chi _u}{|\nabla \chi _u|}{-} \mathrm {n}_v(P_{I_v(t)}x)\Big )\nonumber \\&\qquad ~~~~~~~~~~~~\cdot \big ((v(x,t){-}v(P_{I_v(t)} x,t)) \cdot \nabla \big )P_{I_v(t)}x\,\mathrm{d}|\nabla \chi _u|. \end{aligned}$$
(76)
Hence, plugging in (75), (74) and (76), (73) into (70) and using the estimates \(\smash {|\nabla \bar{V}_{\mathrm {n}}| \leqq \frac{C}{r_c^2}\Vert v\Vert _{W^{1,\infty }}}\), \(\smash {|\frac{\mathrm {d}}{{\mathrm {d}}t}\mathrm {n}_v(P_{I_v(t)}x)|\leqq \frac{C}{r_c^2}\Vert v\Vert _{W^{1,\infty }}}\), \(|\nabla \frac{\mathrm {d}}{{\mathrm {d}}t}\mathrm {n}_v(P_{I_v(t)}x)|\leqq \frac{C}{r_c^3}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}\), and \(\smash {|\nabla f|\leqq \frac{C}{r_c^2}}\), we obtain
$$\begin{aligned}&\bigg |\frac{\mathrm {d}}{{\mathrm {d}}t} \int _{I_v(t)} \eta (x) h^+(x,t) \,{\mathrm {d}}x- \int _{I_v(t)} h^+(x,t) (\mathrm {Id}{-}\mathrm {n}_v\otimes \mathrm {n}_v)v(x,t) \cdot \nabla \eta (x) \,\mathrm{d}S(x)\bigg |\\&\quad \leqq \frac{C}{r_c^2} \int _{\{{\text {dist}}(x,I_v(t))\leqq r_c\}} |\chi _u(x,t)-\chi _v(x,t)| |u(x,t)-v(x,t)| |\eta (P_{I_v(t)}x)| \,{\mathrm {d}}x\\&\qquad +\frac{C}{r_c}\int _{\{{\text {dist}}(x,I_v(t))\leqq r_c\}} |\chi _u(x,t)-\chi _v(x,t)| |u(x,t)-v(x,t)| |\nabla \eta (P_{I_v(t)}x)| \,{\mathrm {d}}x\\&\qquad +\frac{C(1{+}\Vert v\Vert _{W^{1,\infty }})}{r_c}\int _{\{{\text {dist}}(x,I_v(t))\leqq r_c\}} \bigg |\frac{\nabla \chi _u}{|\nabla \chi _u|}-\mathrm {n}_v(P_{I_v(t)}x)\bigg |\\&\qquad ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\times \frac{|{\text {dist}}^{\pm }(x,I_v(t))|}{r_c} |\eta (P_{I_v(t)}x)| \,\mathrm{d}|\nabla \chi _u|(x)\\&\qquad +\frac{C(1{+}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))})}{r_c^3}\int _{\{{\text {dist}}(x,I_v(t))\leqq r_c\}} |\chi _u(x,t){-}\chi _v(x,t)|\\&\qquad \qquad \qquad \qquad \qquad \qquad \times \frac{|{\text {dist}}^{\pm }(x,I_v(t))|}{r_c} |\eta (P_{I_v(t)}x)| \,{\mathrm {d}}x\\&\qquad +C\int _{I_v(t)} |u-v| |\eta | \,{\mathrm {d}}S. \end{aligned}$$
This yields by the change of variables \(\Phi _t(x,y)\) and a straightforward estimate
$$\begin{aligned}&\bigg |\frac{\mathrm {d}}{{\mathrm {d}}t} \int _{I_v(t)} \eta (x) h^+(x,t) \,\mathrm{d}x - \int _{I_v(t)} h^+(x,t) (\mathrm {Id}{-}\mathrm {n}_v\otimes \mathrm {n}_v)v(x,t) \cdot \nabla \eta (x) \,{\mathrm {d}}S(x)\bigg |\\&\quad \leqq \frac{C}{r_c^2}\Vert \eta \Vert _{W^{1,4}(I_v(t))}\bigg (\int _{I_v(t)} \bigg (\int _{0}^{\frac{r_c}{2}} |\chi _u-\chi _v|(x+y\mathrm {n}_v(x,t),t) \,{\mathrm {d}}y\bigg )^4 \,{\mathrm {d}}S\bigg )^{1/4}\\&\qquad ~~~~~~ \times \bigg (\int _{I_v(t)} \sup _{y\in [-r_c,r_c]} |u-v|^2(x+y\mathrm {n}_v(x,t),t) \,{\mathrm {d}}S(x)\bigg )^{1/2}\\&\qquad +\frac{C(1{+}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))})}{r_c^3}\Vert \eta \Vert _{L^2(I_v(t))}\\&\qquad ~~~~~~ \times \bigg (\int _{{\mathbb {R}^d}}|\chi _u(x,t)-\chi _v(x,t)| \, \min \Big \{\frac{{\text {dist}}(x,I_v(t))}{r_c},1\Big \} \,{\mathrm {d}}x\bigg )^\frac{1}{2}\\&\qquad +\frac{C(1{+}\Vert v\Vert _{W^{1,\infty }})}{r_c}\Vert \eta \Vert _{L^\infty (I_v(t))} \bigg (\int _{\{{\text {dist}}(x,I_v(t))\leqq r_c\}} \bigg |\frac{\nabla \chi _u}{|\nabla \chi _u|}-\mathrm {n}_v(P_{I_v(t)}x)\bigg |^2 \,\mathrm{d}|\nabla \chi _u|\bigg )^\frac{1}{2}\\&\qquad ~~~~~~ \times \bigg (\int _{\{{\text {dist}}(x,I_v(t))\leqq r_c\}}\frac{|{\text {dist}}^{\pm }(x,I_v(t))|^2}{r_c^2} \,\mathrm{d}|\nabla \chi _u|\bigg )^\frac{1}{2}\\&\qquad +C\bigg (\int _{I_v(t)} |u-v|^2 \,\mathrm{d}S \bigg )^{1/2} \, \Vert \eta \Vert _{L^2(I_v(t))}. \end{aligned}$$
Using finally the Sobolev embedding to bound the \(L^\infty \)-norm of \(\eta \) on the interface (which is either one- or two-dimensional; note that the constant in the Sobolev embedding may be bounded by \(Cr_c^{-1}\) for our geometry), we infer from this estimate the desired bound (59d), using also (38) and (37). This concludes the proof. \(\quad \square \)
A Regularization of the Local Height of the Interface Error
In order to modify our relative entropy to compensate for the velocity gradient discontinuity at the interface, we need regularized versions of the local heights of the interface error \(h^+\) and \(h^-\) which in particular have Lipschitz regularity. To this end, we fix some function \(e(t)>0\) and basically apply a mollifier on scale e(t) to the local interface error heights \(\smash {h^+}\) and \(\smash {h^-}\) at each time. An illustration of \(\smash {h^+}\) and its mollification \(\smash {h^+_{e(t)}}\) is provided in Figs. 3 and 4. These regularized versions \(\smash {h^+_{e(t)}}\) and \(\smash {h^-_{e(t)}}\) of the local interface error heights then have the following properties:
Proposition 27
Let \(\chi _v\in L^\infty ([0,{T_{strong}});{\text {BV}}(\mathbb {R}^d;\{0,1\}))\) be an indicator function such that \(\Omega ^+_t :=\{x\in \mathbb {R}^d:\chi _v(x,t)=1\}\) is a family of smoothly evolving domains and \(I_v(t) := \partial \Omega ^+_t\) is a family of smoothly evolving surfaces in the sense of Definition 5. Let \(\xi \) be the extension of the unit normal vector field \(\mathrm {n}_v\) from Definition 13.
Let \(\chi _u\in L^\infty ([0,{T_{strong}});{\text {BV}}(\mathbb {R}^d;\{0,1\}))\) be another indicator function and let then \(h^+\) resp. \(h^-\) be as defined in Proposition 26. Let \(\theta :\mathbb {R}^+\rightarrow [0,1]\) be a smooth cutoff with \(\theta (s)=1\) for \(s\in [0,\frac{1}{4}]\) and \(\theta (s)=0\) for \(s\geqq \frac{1}{2}\). Let \(e:[0,{T_{strong}})\rightarrow (0,r_c]\) be a \(\smash {C^1}\)-function and define the regularized height of the local interface error
$$\begin{aligned} h^\pm _{e(t)}(x,t):= \frac{\int _{I_v(t)} \theta \big (\frac{|\tilde{x}-x|}{e(t)}\big ) h^\pm (\tilde{x},t) \,{\mathrm {d}}S(\tilde{x})}{\int _{I_v(t)} \theta \big (\frac{|\tilde{x}-x|}{e(t)}\big ) \,{\mathrm {d}}S(\tilde{x})}. \end{aligned}$$
(77)
Then \(h^+_{e(t)}\) and \(h^-_{e(t)}\) have the following properties:
a) (\(H^1\)-bound) If the interface error terms from the relative entropy are bounded by
$$\begin{aligned}&\int _{\mathbb {R}^d}1-\xi (\cdot ,t)\cdot \frac{\nabla \chi _u(\cdot ,t)}{|\nabla \chi _u(\cdot ,t)|} \,\mathrm{d}|\nabla \chi _u(\cdot ,t)|\\&\quad +\int _{\mathbb {R}^d}\big |\chi _u(\cdot ,t)-\chi _v(\cdot ,t)\big |\,\Big |\beta \Big (\frac{{\text {dist}}^{\pm }(\cdot ,I_v(t))}{r_c}\Big )\Big |\,{\mathrm {d}}x\leqq e(t)^2, \end{aligned}$$
we have the Lipschitz estimate \(\smash {|\nabla h^\pm _{e(t)}(\cdot ,t)|} \leqq C r_c^{-2}\), the global bound \(|\nabla ^2 \smash {h^\pm _{e(t)}}(\cdot ,t)| \leqq C e(t)^{-1} r_c^{-4}\), and the bound
$$\begin{aligned} \int _{I_v(t)} |\nabla h^\pm _{e(t)}|^2 + |h^\pm _{e(t)}|^2 \,{\mathrm {d}}S&\leqq \frac{C}{r_c^2} \int _{\mathbb {R}^d}1-\xi \cdot \frac{\nabla \chi _u}{|\nabla \chi _u|} \,\mathrm{d}|\nabla \chi _u|\nonumber \\&\quad + \frac{C}{r_c^4} \int _{\mathbb {R}^d}|\chi _u-\chi _v| \min \Big \{\frac{{\text {dist}}(x,I_v(t))}{r_c},1\Big \} \,{\mathrm {d}}x. \end{aligned}$$
(78a)
b) (Improved approximation property) The functions \(h^+_{e(t)}\) and \(h^-_{e(t)}\) provide an approximation for the interface of the weak solution
$$\begin{aligned} \chi _{v,h^+_{e(t)},h^-_{e(t)}}:=&\chi _v - \chi _{0\leqq {\text {dist}}^{\pm }(x,I_v(t))\leqq h^+_{e(t)}(P_{I_v(t)}x,t)}\nonumber \\&\quad + \chi _{-h^-_{e(t)}(P_{I_v(t)}x,t)\leqq {\text {dist}}^{\pm }(x,I_v(t))\leqq 0}, \end{aligned}$$
(78b)
up to an error of
$$\begin{aligned}&\int _{\mathbb {R}^d}\big |\chi _u-\chi _{v,h^+_{e(t)},h^-_{e(t)}}\big | \,{\mathrm {d}}x\nonumber \\&\quad \leqq C \int _{\mathbb {R}^d}1-\xi \cdot \frac{\nabla \chi _u}{|\nabla \chi _u|} \,\mathrm{d}|\nabla \chi _u| + C \int _{\mathbb {R}^d}|\chi _u-\chi _v| \min \Big \{\frac{{\text {dist}}(x,I_v(t))}{r_c},1\Big \} \,{\mathrm {d}}x\nonumber \\&\qquad +C e(t) \bigg (\int _{\mathbb {R}^d}1-\xi \cdot \frac{\nabla \chi _u}{|\nabla \chi _u|} \,\mathrm{d}|\nabla \chi _u|\bigg )^{1/2} \mathcal {H}^{d-1}(I_v(t))^{1/2}\nonumber \\&\qquad +C \frac{e(t)}{r_c} \bigg (\int _{\mathbb {R}^d}|\chi _u-\chi _v| \min \Big \{\frac{{\text {dist}}(x,I_v(t))}{r_c},1\Big \} \,{\mathrm {d}}x\bigg )^{1/2} \mathcal {H}^{d-1}(I_v(t))^{1/2}. \end{aligned}$$
(78c)
c) (Time evolution) Let v be a solenoidal vector field
$$\begin{aligned} v\in L^2([0,{T_{strong}}];H^1({\mathbb {R}^d};{\mathbb {R}^d}))\cap L^\infty ([0,{T_{strong}}];W^{1,\infty }({\mathbb {R}^d};{\mathbb {R}^d})) \end{aligned}$$
such that in the domain \(\bigcup _{t\in [0,{T_{strong}})} (\Omega _t^+\cup \Omega _t^-) \times \{t\}\) the second spatial derivatives of the vector field v exist and satisfy \(\sup _{t\in [0,{T_{strong}})} \sup _{x\in \Omega _t^+\cup \Omega _t^-} |\nabla ^2 v(x,t)| <\infty \). Assume that \(\chi _v\) solves the equation \(\partial _t \chi _v = -\nabla \cdot (\chi _v v)\). If \(\chi _u\) solves the equation \(\partial _t \chi _u = -\nabla \cdot (\chi _u u)\) for another solenoidal vector field \(u\in L^2([0,{T_{strong}}];H^1(\mathbb {R}^d;\mathbb {R}^d))\), we have the following estimate on the time derivative of \(h^\pm _{e(t)}\):
$$\begin{aligned}&\bigg |\frac{\mathrm {d}}{\mathrm {d}t} \int _{I_v(t)} \eta (x) h^\pm _{e(t)}(x,t) \,{\mathrm {d}}x- \int _{I_v(t)} h^\pm _{e(t)}(x,t) (\mathrm {Id}{-}\mathrm {n}_v\otimes \mathrm {n}_v)v(x,t) \cdot \nabla \eta (x) \,{\mathrm {d}}S(x) \bigg |\nonumber \\&\quad \leqq \frac{C}{e(t)r_c^2}\Vert \eta \Vert _{L^{4}(I_v(t))}\bigg (\int _{I_v(t)} |\bar{h}^\pm |^4 \,{\mathrm {d}}S\bigg )^{1/4}\nonumber \\&\qquad ~~~~ \times \bigg (\int _{I_v(t)} \sup _{y\in [-r_c,r_c]} |u-v|^2(x+y\mathrm {n}_v(x,t),t) \,{\mathrm {d}}S(x)\bigg )^{1/2}\nonumber \\&\qquad +C\frac{(1+\Vert v\Vert _{W^{1,\infty }})}{e(t) r_c}\max _{p\in \{2,4\}}\Vert \eta \Vert _{L^p(I_v(t))} \int _{{\mathbb {R}^d}} 1-\xi \cdot \frac{\nabla \chi _u}{|\nabla \chi _u|} \,\mathrm{d}|\nabla \chi _u|\nonumber \\&\qquad +Cr_c^{-4}\Vert v\Vert _{W^{1,\infty }}(1+e'(t)) \bigg (\int _{\mathbb {R}^d}1-\xi \cdot \frac{\nabla \chi _u}{|\nabla \chi _u|} \,\mathrm{d}|\nabla \chi _u|\bigg )^{1/2} ||\eta ||_{L^2(I_v(t))}\nonumber \\&\qquad +C\bigg (\frac{1+\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}}{r_c}+\frac{\Vert v\Vert _{W^{1,\infty }}}{r_c^6}(1+e'(t))\bigg ) \Vert \eta \Vert _{L^2(I_v(t))} \nonumber \\&\qquad ~~~~\times \bigg (\int _{{\mathbb {R}^d}}|\chi _u(x,t)-\chi _v(x,t)| \, \min \Big \{\frac{{\text {dist}}(x,I_v(t))}{r_c},1\Big \} \,{\mathrm {d}}x\bigg )^\frac{1}{2}\nonumber \\&\qquad +C\Vert \eta \Vert _{L^2(I_v(t))}\bigg (\int _{I_v(t)}|u-v|^2\,{\mathrm {d}}S\bigg )^\frac{1}{2} \end{aligned}$$
(78d)
for any smooth test function \(\eta \in C_{cpt}^\infty ({\mathbb {R}^d})\) with \(\mathrm {n}_v\cdot \nabla \eta = 0\) on the interface \(I_v(t)\), and where \(\bar{h}^\pm \) is defined as \(h^\pm \) but now with respect to the modified cut-off function \(\bar{\theta }(\cdot )=\theta \big (\frac{\cdot }{2}\big )\).
Proof
Proof of a). In order to estimate the spatial derivative \(\nabla {h^\pm _{e(t)}}\), we compute using the fact that \({\nabla _x \theta \big (\frac{|x-\tilde{x}|}{e(t)}\big )=-\nabla _{\tilde{x}} \theta \big (\frac{|x-\tilde{x}|}{e(t)}\big )}\) (note that all of the subsequent gradients are to be understood in the tangential sense on the manifold \(I_v(t)\))
$$\begin{aligned} \nabla h^\pm _{e(t)} (x,t)&= -\frac{\int _{I_v(t)} \nabla _{\tilde{x}} \theta \big (\frac{|\tilde{x}-x|}{e(t)}\big ) h^\pm (\tilde{x},t) \,{\mathrm {d}}S(\tilde{x})}{\int _{I_v(t)} \theta \big (\frac{|\tilde{x}-x|}{e(t)}\big ) \,{\mathrm {d}}S(\tilde{x})}\\&\quad +\frac{\int _{I_v(t)} \theta \big (\frac{|\tilde{x}-x|}{e(t)}\big ) h^\pm (\tilde{x},t) \,{\mathrm {d}}S(\tilde{x}) \int _{I_v(t)} \nabla _{\tilde{x}} \theta \big (\frac{|\tilde{x}-x|}{e(t)}\big ) \,{\mathrm {d}}S(\tilde{x})}{\big (\int _{I_v(t)} \theta \big (\frac{|\tilde{x}-x|}{e(t)}\big ) \,{\mathrm {d}}S(\tilde{x})\big )^2}\\&= \frac{\int _{I_v(t)} \theta \big (\frac{|\tilde{x}-x|}{e(t)}\big ) \nabla h^\pm (\tilde{x},t) \,{\mathrm {d}}S(\tilde{x})}{\int _{I_v(t)} \theta \big (\frac{|\tilde{x}-x|}{e(t)}\big ) \,{\mathrm {d}}S(\tilde{x})} +\frac{\int _{I_v(t)} \theta \big (\frac{|\tilde{x}-x|}{e(t)}\big ) \,\mathrm{d}D^s h^\pm (\tilde{x})}{\int _{I_v(t)} \theta \big (\frac{|\tilde{x}-x|}{e(t)}\big ) \,{\mathrm {d}}S(\tilde{x})}\\&\quad +\frac{\int _{I_v(t)} \theta \big (\frac{|\tilde{x}-x|}{e(t)}\big ) h^\pm (\tilde{x},t) \mathrm {H}(\tilde{x},t) \,{\mathrm {d}}S(\tilde{x})}{\int _{I_v(t)} \theta \big (\frac{|\tilde{x}-x|}{e(t)}\big ) \,{\mathrm {d}}S(\tilde{x})}\\&\quad -\frac{\int _{I_v(t)} \theta \big (\frac{|\tilde{x}-x|}{e(t)}\big ) h^\pm (\tilde{x},t) \,{\mathrm {d}}S(\tilde{x}) \int _{I_v(t)} \theta \big (\frac{|\tilde{x}-x|}{e(t)}\big ) \mathrm {H}(\tilde{x}, t) \,{\mathrm {d}}S(\tilde{x})}{\big (\int _{I_v(t)} \theta \big (\frac{|\tilde{x}-x|}{e(t)}\big ) \,{\mathrm {d}}S(\tilde{x})\big )^2}. \end{aligned}$$
Introduce the convex function
$$\begin{aligned} G(p):= {\left\{ \begin{array}{ll} |p|^2&{}\text {for }|p|\leqq 1,\\ 2|p|-1&{}\text {for }|p|\geqq 1. \end{array}\right. } \end{aligned}$$
(79)
Using the estimate (20), the obvious bounds \(G(p+\tilde{p})\leqq CG(p)+CG(\tilde{p})\) and \(G(\lambda p)\leqq C(\lambda +\lambda ^2) G(p)\) for any p, \(\tilde{p}\), and \(\lambda >0\), and Jensen’s inequality, we obtain (as the recession function of G is given by 2|p|)
$$\begin{aligned} G(|\nabla h^\pm _{e(t)} (x,t)|)&\leqq C \frac{\int _{I_v(t)} \theta \big (\frac{|\tilde{x}-x|}{e(t)}\big ) \big (G(|\nabla h^\pm (\tilde{x},t)|) + G(r_c^{-1} |h^\pm (\tilde{x},t)|)\big ) \,{\mathrm {d}}S(\tilde{x})}{\int _{I_v(t)} \theta \big (\frac{|\tilde{x}-x|}{e(t)}\big ) \,{\mathrm {d}}S(\tilde{x})}\nonumber \\&\quad +C\frac{\int _{I_v(t)} \theta \big (\frac{|\tilde{x}-x|}{e(t)}\big ) \,\mathrm{d}|D^s h^\pm |(\tilde{x},t)}{\int _{I_v(t)} \theta \big (\frac{|\tilde{x}-x|}{e(t)}\big ) \,{\mathrm {d}}S(\tilde{x})}. \end{aligned}$$
(80)
Consider \(x\in I_v(t)\). By the assumption from Definition 5, there is a \(C^3\)-function \(g:B_1(0)\subset \mathbb {R}^{d-1}\rightarrow \mathbb {R}\) with \(\Vert \nabla g\Vert _{L^\infty }\leqq 1\), \(g(0)=0\), and \(\nabla g(0)=0\), and such that \(I_v(t)\cap B_{2r_c}(x)\) is after rotation and translation given as the graph \(\{(x,g(x)):x\in \mathbb {R}^{d-1}\}\). Using the fact that \(\theta \equiv 0\) on \(\mathbb {R}{\setminus }[0,\frac{1}{2}]\) and \(e(t)<r_c\leqq 1\), that is, the map \(\smash {I_v(t)\ni \tilde{x}\mapsto \theta (\frac{|\tilde{x} - x|}{e(t)})}\) is supported in a coordinate patch given by the graph of g, we then may bound
$$\begin{aligned} \int _{I_v(t)} \theta \Big (\frac{|\tilde{x}-x|}{e(t)}\Big ) \,{\mathrm {d}}S(\tilde{x})&\leqq \int _{I_v(t)\cap B_{\frac{e(t)}{2}}(x)} 1\,{\mathrm {d}}S(\tilde{x}) \leqq C \int _{\{\tilde{x}\in \mathbb {R}^{d-1} :|\tilde{x}|<\frac{e(t)}{2}\}} 1\,\mathrm{d}\tilde{x}\\&\leqq Ce(t)^{d-1}. \end{aligned}$$
We also obtain a lower bound using that \(\theta \equiv 1\) on \([0,\frac{1}{4}]\) and again \(e(t)<r_c\leqq 1\)
$$\begin{aligned} \int _{I_v(t)} \theta \Big (\frac{|\tilde{x}-x|}{e(t)}\Big ) \,{\mathrm {d}}S(\tilde{x})&\geqq \int _{I_v(t)\cap B_{\frac{e(t)}{4}}(x)} 1\,{\mathrm {d}}S(\tilde{x}) \geqq c \int _{\{\tilde{x}\in \mathbb {R}^{d-1} :|\tilde{x}|<c e(t)\}} 1\,\mathrm{d}\tilde{x}\\&\geqq ce(t)^{d-1}. \end{aligned}$$
In summary, we infer that
$$\begin{aligned} c e(t)^{d-1} \leqq \int _{I_v(t)} \theta \Big (\frac{|\tilde{x}-x|}{e(t)}\Big ) \,{\mathrm {d}}S(\tilde{x}) \leqq C e(t)^{d-1}. \end{aligned}$$
(81)
Making use of (81), the assumptions \(\int _{\mathbb {R}^d}1-\xi \cdot \mathrm {n}_u \,\mathrm{d}|\nabla \chi _u|\leqq e(t)^2\leqq r_c^2 <1\) and \(d\leqq 3\), the upper bounds \(|\theta |\leqq 1\) and \(G(\lambda p)\leqq C(\lambda +\lambda ^2) G(p)\), as well as the already established \(L^2\)- resp. \(H^1\)-bound for the local interface error heights \(h^\pm \) from (59a) resp. (59b) we deduce
$$\begin{aligned} G(|\nabla h^\pm _{e(t)} (x,t)|) \leqq C r_c^{-2}, \end{aligned}$$
which is precisely the first assertion in a). Similarly, one derives the other desired estimate \(G(e(t) |\nabla ^2 h^\pm _{e(t)}(x,t)|) \leqq C r_c^{-4}\).
Integrating (80) over \(I_v(t)\) and employing the global upper bound \(|\nabla h^\pm _{e(t)}(\cdot ,t)|\leqq C r_c^{-2}\), which in turn entails \(\smash {G(|\nabla h^\pm _{e(t)}(\cdot ,t)|)\geqq c r_c^2 |\nabla h^\pm _{e(t)}(\cdot ,t)|^2}\), we get
$$\begin{aligned}&\int _{I_v(t)} |\nabla h^\pm _{e(t)} (x,t)|^2 \,{\mathrm {d}}S(x)\nonumber \\&\quad \leqq C r_c^{-2} \int _{I_v(t)} \frac{\int _{I_v(t)} \theta \big (\frac{|\tilde{x}-x|}{e(t)}\big ) G(|\nabla h^\pm (\tilde{x},t)|) + G(r_c^{-1} |h^\pm (\tilde{x},t)|) \,{\mathrm {d}}S(\tilde{x})}{\int _{I_v(t)} \theta \big (\frac{|\tilde{x}-x|}{e(t)}\big ) \,{\mathrm {d}}S(\tilde{x})} \,{\mathrm {d}}S(x)\nonumber \\&\qquad +C r_c^{-2} \int _{I_v(t)} \frac{\int _{I_v(t)} \theta \big (\frac{|\tilde{x}-x|}{e(t)}\big ) \,\mathrm{d}|D^s h^\pm |(\tilde{x},t)}{\int _{I_v(t)} \theta \big (\frac{|\tilde{x}-x|}{e(t)}\big ) \,{\mathrm {d}}S(\tilde{x})} \,{\mathrm {d}}S(x). \end{aligned}$$
(82)
Applying Fubini’s theorem and using the bounds (81), \(G(\lambda p)\leqq C(\lambda +\lambda ^2) G(p)\), as well as (59a) and (59b) we deduce the estimate on \(\smash {\int _{I_v(t)} |\nabla h^\pm _{e(t)}|^2 \,{\mathrm {d}}S}\) stated in a). The estimate on \(\smash {\int _{I_v(t)} |h^\pm _{e(t)}|^2 \,{\mathrm {d}}S}\) follows by an analogous argument, first squaring (77) and applying Jensen’s inequality, then integrating over \(I_v(t)\), and finally using (81), Fubini as well as (59a) and (59b).
Proof of b). We start with a change of variables to estimate (recall (18))
$$\begin{aligned}&\int _{\mathbb {R}^d}|\chi _{v,h^+_{e(t)},h^-_{e(t)}}-\chi _{v,h^+,h^-}| \,{\mathrm {d}}x\\&\quad \leqq C\int _{I_v(t)}\int _{0}^{r_c}|\chi _{0\leqq {\text {dist}}^{\pm }(x,I_v(t))\leqq h^+_{e(t)}(P_{I_v(t)}x,t)} -\chi _{0\leqq {\text {dist}}^{\pm }(x,I_v(t))\leqq h^+(P_{I_v(t)}x,t)}|\,{\mathrm {d}}y\,{\mathrm {d}}S\\&\qquad ~~~~~~~~~~~~ +C\int _{I_v(t)}\int _{0}^{r_c}| \chi _{-h^-_{e(t)}(P_{I_v(t)}x,t)\leqq {\text {dist}}^{\pm }(x,I_v(t))\leqq 0} - \chi _{-h^-(P_{I_v(t)}x,t)\leqq {\text {dist}}^{\pm }(x,I_v(t))\leqq 0}|\,{\mathrm {d}}y\,{\mathrm {d}}S\\&\quad \leqq C\int _{I_v(t)} |h^+_{e(t)}(x,t){-}h^+(x,t)| + |h^-_{e(t)}(x,t)-h^-(x,t)| \,{\mathrm {d}}S(x). \end{aligned}$$
By adding zero and using (59c) we therefore obtain
$$\begin{aligned} \int _{\mathbb {R}^d}&|\chi _u-\chi _{v,h^+_{e(t)},h^-_{e(t)}}| \,{\mathrm {d}}x\\&\leqq \int _{\mathbb {R}^d}|\chi _u-\chi _{v,h^+,h^-}| \,{\mathrm {d}}x+\int _{\mathbb {R}^d}|\chi _{v,h^+_{e(t)},h^-_{e(t)}}-\chi _{v,h^+,h^-}| \,{\mathrm {d}}x\\&\leqq C \int _{\mathbb {R}^d}1-\xi \cdot \frac{\nabla \chi _u}{|\nabla \chi _u|} \,\mathrm{d}|\nabla \chi _u|\\&\qquad + C \int _{{\mathbb {R}^d}} |\chi _u-\chi _v| \min \Big \{\frac{{\text {dist}}(x,I_v(t))}{r_c},1\Big \}\,{\mathrm {d}}x\\&\qquad +C\int _{I_v(t)} |h^+_{e(t)}(x,t)-h^+(x,t)| + |h^-_{e(t)}(x,t)-h^-(x,t)| \,{\mathrm {d}}S(x). \end{aligned}$$
Observe that one can decompose
$$\begin{aligned} h^\pm (x,t)=h^\pm _{e(t)}(x,t) + \sum _{k=0}^\infty \big (h^\pm _{2^{-k-1}e(t)}(x,t)- h^\pm _{2^{-k}e(t)}(x,t)\big ). \end{aligned}$$
A straightforward estimate in local coordinates then yields
$$\begin{aligned} \int _{I_v(t)}&\big |h_{2^{-k}e(t)}^\pm -h_{2^{-k-1}e(t)}^\pm \big | \,{\mathrm {d}}S\\&\leqq C 2^{-k} e(t) \int _{I_v(t)} 1 \,\mathrm{d}|D^{{\text {tan}}}h^\pm |\\&\leqq C 2^{-k} e(t) \int _{I_v(t)} 1 \,\mathrm{d}|D^s h^\pm | + C 2^{-k} e(t) \int _{I_v(t)} |\nabla h^+| \chi _{\{|\nabla h^+|\geqq 1\}} \,{\mathrm {d}}S\\&\quad + C 2^{-k} e(t) \bigg (\int _{I_v(t)} |\nabla h^+|^2 \chi _{\{|\nabla h^+|\leqq 1\}} \,{\mathrm {d}}S\bigg )^{1/2} \mathcal {H}^{d-1}(I_v(t))^{1/2}. \end{aligned}$$
Using (59b) and summing with respect to \(k\in \mathbb {N}\), we get the desired estimate (78c).
Proof of c). Note that
$$\begin{aligned} \int _{I_v(t)} \eta (x) h^\pm _{e(t)}(x,t) \,{\mathrm {d}}S= \int _{I_v(t)} h^\pm (\tilde{x},t) \int _{I_v(t)} \frac{\theta \big (\frac{|\tilde{x}-x|}{e(t)}\big ) \eta (x)}{\int _{I_v(t)}\theta \big (\frac{|\hat{x}-x|}{e(t)}\big ) \,{\mathrm {d}}S(\hat{x})} \,{\mathrm {d}}S(x) \,{\mathrm {d}}S(\tilde{x}). \end{aligned}$$
Abbreviating
$$\begin{aligned} \eta _e(\tilde{x},t):=\int _{I_v(t)}\frac{\theta \big (\frac{|\tilde{x}-x|}{e(t)}\big ) \eta (x)}{\int _{I_v(t)}\theta \big (\frac{|\hat{x}-x|}{e(t)}\big )\,{\mathrm {d}}S(\hat{x})} \,{\mathrm {d}}S(x), \end{aligned}$$
we compute
$$\begin{aligned} |\nabla _{\tilde{x}}^\mathrm {tan}\eta _e(\tilde{x},t)|&=\bigg |\int _{I_v(t)}\frac{\nabla _{\tilde{x}}^\mathrm {tan} \theta \big (\frac{|\tilde{x}-x|}{e(t)}\big ) \eta (x)}{\int _{I_v(t)}\theta \big (\frac{|\hat{x}-x|}{e(t)}\big )\, {\mathrm {d}}S(\hat{x})} \,{\mathrm {d}}S(x)\bigg |\\&\leqq \int _{I_v(t)}\bigg (\frac{\big |\theta '\big | \big (\frac{|\tilde{x}-x|}{e(t)}\big )}{\int _{I_v(t)}\theta \big (\frac{|\hat{x}-x|}{e(t)}\big )\, {\mathrm {d}}S(\hat{x})}\bigg )\frac{\eta (x)}{e(t)} \,{\mathrm {d}}S(x). \end{aligned}$$
As in the argument for (81), one checks that \(\int _{I_v(t)}|\theta '|(\frac{|\tilde{x} - x|}{e(t)})\,{\mathrm {d}}S(x)\leqq Ce(t)^{d-1}\). Using the lower bound from (81), the proof for the standard \(L^p\)-inequality for convolutions carries over and we obtain \(\Vert \eta _e\Vert _{L^p(I_v(t))}\leqq C\Vert \eta \Vert _{L^p(I_v(t))}\) as well as
$$\begin{aligned} \int _{I_v(t)} |\nabla \eta _e(x,t)|^{p} \,{\mathrm {d}}S(x)&\leqq \frac{C}{e(t)^p} \int _{I_v(t)} |\eta (x,t)|^p \,{\mathrm {d}}S(x) \end{aligned}$$
for any \(p \geqq 1\). As a consequence of (59d) and these considerations, we deduce that
$$\begin{aligned}&\bigg |\frac{\mathrm {d}}{{\mathrm {d}}t} \int _{I_v(t)} \eta (x) h^\pm _{e(t)}(x,t) \,{\mathrm {d}}x-\int _{I_v(t)} h^\pm (\tilde{x},t) \frac{\mathrm {d}}{{\mathrm {d}}t} \eta _e(\tilde{x},t) \,{\mathrm {d}}S(\tilde{x})\nonumber \\&\qquad -\int _{I_v(t)} h^\pm (\tilde{x},t) (\mathrm {Id}{-}\mathrm {n}_v\otimes \mathrm {n}_v) v(\tilde{x},t) \cdot \nabla _{\tilde{x}} \eta _e(\tilde{x},t)\,{\mathrm {d}}S(\tilde{x})\bigg |\nonumber \\&\quad \leqq \frac{C}{e(t)r_c^2}\Vert \eta \Vert _{L^{4}(I_v(t))}\Bigg (\int _{I_v(t)} |\bar{h}^\pm |^4 \,{\mathrm {d}}S\Bigg )^{1/4}\nonumber \\&\qquad ~~~~\times \Bigg (\int _{I_v(t)} \sup _{y\in [-r_c,r_c]} |u-v|^2(x+y\mathrm {n}_v(x,t),t) \,{\mathrm {d}}S(x)\Bigg )^{1/2}\nonumber \\&\qquad +C\frac{1+\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}}{r_c}\Vert \eta \Vert _{L^2(I_v(t))}\nonumber \\&\qquad ~~~~\times \Bigg (\int _{{\mathbb {R}^d}}|\chi _u(x,t)-\chi _v(x,t)| \,\min \Big \{\frac{{\text {dist}}(x,I_v(t))}{r_c},1\Big \} \,{\mathrm {d}}x\Bigg )^\frac{1}{2}\nonumber \\&\qquad +C\frac{(1+\Vert v\Vert _{W^{1,\infty }})}{r_c e(t)}\max _{p\in \{2,4\}}\Vert \eta \Vert _{L^p(I_v(t))} \int _{{\mathbb {R}^d}} 1-\xi \cdot \frac{\nabla \chi _u}{|\nabla \chi _u|} \,\mathrm{d}|\nabla \chi _u|\nonumber \\&\qquad +C\Vert \eta \Vert _{L^2(I_v(t))}\bigg (\int _{I_v(t)} |u-v|^2 \,{\mathrm {d}}S\bigg )^{1/2}. \end{aligned}$$
(83)
Using the estimate \(|v(x,t)-v(\tilde{x},t)|\leqq C |x-\tilde{x}| \Vert \nabla v\Vert _{L^\infty }\), we infer
$$\begin{aligned}&\Bigg |\int _{I_v(t)} h^\pm (\tilde{x},t) v(\tilde{x},t) \cdot \nabla _{\tilde{x}} \int _{I_v(t)} \frac{\theta \big (\frac{|\tilde{x}-x|}{e(t)}\big ) \eta (x)}{\int _{I_v(t)} \theta \big (\frac{|\hat{x}-x|}{e(t)}\big )\,{\mathrm {d}}S(\hat{x})} \,{\mathrm {d}}S(x) \,{\mathrm {d}}S(\tilde{x})\nonumber \\&\qquad +\int _{I_v(t)} \eta (x)(v(x,t)\cdot \nabla ) h^\pm _{e(t)}(x,t) \,{\mathrm {d}}S(x)\Bigg |\nonumber \\&\quad =\Bigg |\int _{I_v(t)} \int _{I_v(t)} \eta (x)h^\pm (\tilde{x},t) v(\tilde{x},t) \cdot \nabla _{\tilde{x}} \frac{\theta \big (\frac{|\tilde{x}-x|}{e(t)}\big )}{\int _{I_v(t)} \theta \big (\frac{|\hat{x}-x|}{e(t)}\big )\,{\mathrm {d}}S(\hat{x})} \,{\mathrm {d}}S(x) \,{\mathrm {d}}S(\tilde{x})\nonumber \\&\qquad +\int _{I_v(t)} \int _{I_v(t)} \eta (x) h^\pm (\tilde{x},t) v(x,t) \cdot \nabla _x \frac{\theta \big (\frac{|\tilde{x}-x|}{e(t)}\big )}{\int _{I_v(t)}\theta \big (\frac{|\hat{x}-x|}{e(t)}\big ) \,{\mathrm {d}}S(\hat{x})} \,{\mathrm {d}}S(\tilde{x}) \,{\mathrm {d}}S(x)\Bigg |\nonumber \\&\quad \leqq \int _{I_v(t)} \int _{I_v(t)} h^\pm (\tilde{x},t) \Vert \nabla v\Vert _{L^\infty } \frac{|\theta '|\big (\frac{|\tilde{x}-x|}{e(t)}\big ) |\tilde{x}-x| |\eta (x)|}{e(t) \int _{I_v(t)}\theta \big (\frac{|\hat{x}-x|}{e(t)}\big )\,{\mathrm {d}}S(\hat{x})} \,{\mathrm {d}}S(x) \,{\mathrm {d}}S(\tilde{x})\nonumber \\&\qquad +\int _{I_v(t)} \int _{I_v(t)} h^\pm (\tilde{x},t) \Vert v\Vert _{L^\infty } \frac{\theta \big (\frac{|\tilde{x}-x|}{e(t)}\big ) |\eta (x)|\Big |\nabla _x \int _{I_v(t)}\theta \big (\frac{|\hat{x}-x|}{e(t)}\big )\,{\mathrm {d}}S(\hat{x})\Big |}{\big (\int _{I_v(t)} \theta \big (\frac{|\hat{x}-x|}{e(t)}\big )\,{\mathrm {d}}S(\hat{x})\big )^2}\nonumber \\&\qquad ~~~~~~~~~~~~~~~~~~~~~~~\,{\mathrm {d}}S(x) \,{\mathrm {d}}S(\tilde{x})\nonumber \\&\quad \leqq Cr_c^{-1}\Vert v\Vert _{W^{1,\infty }} \bigg (\int _{I_v(t)} |h^\pm (x,t)|^2 \,{\mathrm {d}}S(x) \bigg )^{1/2} \bigg (\int _{I_v(t)} |\eta (x)|^2 \,{\mathrm {d}}S(x) \bigg )^{1/2}, \end{aligned}$$
(84)
where in the last step we have used the simple equality
$$\begin{aligned} \nabla _x^\mathrm {tan} \int _{I_v(t)}\theta \Big (\frac{|\hat{x}-x|}{e(t)}\Big )\,{\mathrm {d}}S(\hat{x})&=-\int _{I_v(t)}\nabla _{\hat{x}}^\mathrm {tan} \theta \Big (\frac{|\hat{x}-x|}{e(t)}\Big )\,{\mathrm {d}}S(\hat{x})\nonumber \\&=\int _{I_v(t)} \theta \Big (\frac{|\hat{x}-x|}{e(t)}\Big ) \mathrm {H}(\hat{x}) \,{\mathrm {d}}S(\hat{x}) \end{aligned}$$
(85)
and the bounds (20) and (81). Recall from the transport theorem for moving hypersurfaces (see [85]) that we have for any \(f\in C^1({\mathbb {R}^d}\times [0,{T_{strong}}))\) that
$$\begin{aligned} \frac{\mathrm {d}}{{\mathrm {d}}t}\int _{I_v(t)}f(x,t)\,{\mathrm {d}}S(x)&= \int _{I_v(t)}\partial _t f(x,t)\,{\mathrm {d}}S(x) +\int _{I_v(t)} V_{\mathrm {n}}\cdot \nabla f(x,t)\,{\mathrm {d}}S(x)\nonumber \\&\quad +\int _{I_v(t)} f(x,t)\,\mathrm {H}\cdot V_{\mathrm {n}}\,{\mathrm {d}}S(x), \end{aligned}$$
(86)
with the normal velocity \(V_{\mathrm {n}}(x,t)=(v(x,t)\cdot \mathrm {n}_v(P_{I_v(t)}x,t))\mathrm {n}_v(P_{I_v(t)}x,t)\). Making use of (86) and \(\frac{\mathrm {d}}{{\mathrm {d}}t}P_{I_v(t)}\tilde{x} = -V_{\mathrm {n}}(\tilde{x},t)\) for \(\tilde{x}\in I_v(t)\) (see (72)), we then compute for every \(\tilde{x} \in I_v(t)\)
$$\begin{aligned}&\frac{\mathrm {d}}{{\mathrm {d}}t} \int _{I_v(t)} \theta \Big (\frac{|\hat{x}-x|}{e(t)}\Big ) \,{\mathrm {d}}S(\hat{x}) =\frac{\mathrm {d}}{{\mathrm {d}}t} \int _{I_v(t)} \theta \Big (\frac{|P_{I_v(t)}\hat{x}-P_{I_v(t)}x|}{e(t)}\Big ) \,{\mathrm {d}}S(\hat{x})\\&\quad =-\frac{e'(t)}{e(t)^2}\int _{I_v(t)} \theta '\Big (\frac{| \hat{x}- x|}{e(t)}\Big ) |\hat{x}- x| \,{\mathrm {d}}S(\hat{x})\\&\qquad +\frac{1}{e(t)}\int _{I_v(t)} \theta '\Big (\frac{| \hat{x}- x|}{e(t)}\Big ) \frac{( \hat{x}- x)\cdot (V_{\mathrm {n}}(\hat{x},t)-V_{\mathrm {n}}(x,t))}{e(t) |\hat{x}-x|} \,{\mathrm {d}}S(\hat{x})\\&\qquad +\int _{I_v(t)} \theta \Big (\frac{|\hat{x}-x|}{e(t)}\Big ) V_{\mathrm {n}}(\hat{x}) \cdot \mathrm {H}(\hat{x}) \,{\mathrm {d}}S(\hat{x}). \end{aligned}$$
This, together with another application of (86) and the fact that \(\mathrm {n}_v\cdot \nabla \eta =0\) on the interface \(I_v(t)\), implies for \(\tilde{x}\in I_v(t)\), that
$$\begin{aligned}&\frac{\mathrm {d}}{{\mathrm {d}}t}\eta _e(\tilde{x},t) =\frac{\mathrm {d}}{{\mathrm {d}}t}\int _{I_v(t)} \frac{\theta \big (\frac{|P_{I_v(t)}\tilde{x}-P_{I_v(t)}x|}{e(t)}\big )\eta (x)}{\int _{I_v(t)}\theta \big (\frac{|P_{I_v(t)}\hat{x}-P_{I_v(t)}x|}{e(t)}\big )\,{\mathrm {d}}S(\hat{x})} \,{\mathrm {d}}S(x)\nonumber \\&\quad =\int _{I_v(t)}\bigg (\frac{\theta \big (\frac{|\tilde{x} - x|}{e(t)}\big )\eta (x)}{\int _{I_v(t)}\theta \big (\frac{|\hat{x} -x|}{e(t)}\big )\,{\mathrm {d}}S(\hat{x})}\bigg ) V_{\mathrm {n}}(x)\cdot \mathrm {H}(x)\,{\mathrm {d}}S(x)\nonumber \\&\qquad -\int _{I_v(t)}\frac{\theta \big (\frac{|\tilde{x} - x|}{e(t)}\big )\eta (x) \big (\int _{I_v(t)}\theta \big (\frac{|\hat{x} -x|}{e(t)}\big )V_{\mathrm {n}}(\hat{x})\cdot \mathrm {H}(\hat{x})\,{\mathrm {d}}S(\hat{x})\big )}{\big (\int _{I_v(t)}\theta \big (\frac{|\hat{x} -x|}{e(t)}\big )\,{\mathrm {d}}S(\hat{x})\big )^2}\,{\mathrm {d}}S(x)\nonumber \\&\qquad +\int _{I_v(t)}\frac{\eta (x)\theta '\big (\frac{|\tilde{x} - x|}{e(t)}\big ) \frac{(\tilde{x}-x)\cdot (V_{\mathrm {n}}(\tilde{x})-V_{\mathrm {n}}( x))}{e(t)|\tilde{x}-x|}}{\int _{I_v(t)}\theta \big (\frac{|\hat{x} -x|}{e(t)}\big )\,{\mathrm {d}}S(\hat{x})}\,{\mathrm {d}}S(x)\nonumber \\&\qquad -\int _{I_v(t)}\frac{\theta \big (\frac{|\tilde{x} - x|}{e(t)}\big )\eta (x) \int _{I_v(t)}\theta '\big (\frac{|\hat{x} -x|}{e(t)}\big )\frac{(\hat{x}-x)\cdot (V_{\mathrm n}(\hat{x},t)-V_{\mathrm n}(x,t))}{e(t)|\hat{x}-x|}\,{\mathrm {d}}S(\hat{x})}{\big (\int _{I_v(t)}\theta \big (\frac{|\hat{x} -x|}{e(t)}\big )\,{\mathrm {d}}S(\hat{x})\big )^2}\,{\mathrm {d}}S(x) \nonumber \\&\qquad -\frac{e'(t)}{e(t)}\int _{I_v(t)}\frac{F'_{e,\theta }(\tilde{x},x)\eta (x)}{\int _{I_v(t)}\theta \big (\frac{|\hat{x} -x|}{e(t)}\big )\,{\mathrm {d}}S(\hat{x})}\,{\mathrm {d}}S(x), \end{aligned}$$
(87)
where \(F'_{e,\theta }(t):I_v(t)\times I_v(t)\rightarrow \mathbb {R}\) is the kernel
$$\begin{aligned}&F'_{e,\theta }(t)(\tilde{x},x) := \theta '\Big (\frac{|\tilde{x} - x|}{e(t)}\Big ) \frac{|P_{I_v(t)} \tilde{x} - P_{I_v(t)} x|}{e(t)} \nonumber \\&\qquad -\theta \Big (\frac{|\tilde{x} {-} x|}{e(t)}\Big ) \frac{\int _{I_v(t)}\theta '\big (\frac{|\hat{x} - x|}{e(t)}\big ) \frac{|P_{I_v(t)}\hat{x}-P_{I_v(t)}x|}{e(t)} \,{\mathrm {d}}S(\hat{x})}{\int _{I_v(t)}\theta \big (\frac{|\hat{x} - x|}{e(t)}\big )\,{\mathrm {d}}S(\hat{x})}. \end{aligned}$$
(88)
Observe that we have
$$\begin{aligned} \int _{I_v(t)}F'_{e,\theta }(t)(\tilde{x},x)\,{\mathrm {d}}S(\tilde{x}) = 0. \end{aligned}$$
(89)
By the choice of the cutoff \(\theta \), we see that for every given \(x\in I_v(t)\) the kernel \(F'_{e,\theta }(t)\) is supported in \(B_{e(t)/2}(x)\cap I_v(t)\). Moreover, the exact same argumentation which led to the upper bound in (81) (we only used the support and upper bound for \(\theta \) as well as \(e(t)\leqq r_c\)) shows that the kernel \(F'_{e,\theta }\) satisfies the upper bound
$$\begin{aligned} \int _{I_v(t)} |F'_{e,\theta }(\tilde{x},x)|^p\,{\mathrm {d}}S(\tilde{x}) \leqq C(p) e(t)^{d-1} \end{aligned}$$
(90)
for any \(1\leqq p<\infty \). We next intend to rewrite the function \(F'_{e,\theta }(\tilde{x},x)\) for fixed x as the divergence of a vector field. By the property (89), we may consider Neumann problem for the (tangential) Laplacian with right hand side \(F'_{e,\theta }(\cdot ,x)\) in some neighborhood (of scale e(t)) of the point x. To do this we first rescale the setup, that is, we consider the kernel \(F'_1(\tilde{x},x):=F'_{e,\theta }(e(t)\tilde{x},e(t)x)\) for \(\tilde{x},x\in e(t)^{-1}I_v(t)\). By scaling and the fact that \(F'_{e,\theta }\) is supported on scale e(t)/2, it follows that \(F'_1(\cdot ,x)\) has zero average on \(e(t)^{-1}I_v(t)\cap B_1(x)\) for every point \(x\in e(t)^{-1}I_v(t)\) and that
$$\begin{aligned} \int _{e(t)^{-1}I_v(t)} |F'_{1}(\tilde{x},x)|^p\,{\mathrm {d}}S(\tilde{x}) \leqq C(p). \end{aligned}$$
(91)
We fix \(x\in e(t)^{-1}I_v(t)\) and solve on \(e(t)^{-1}I_v(t)\cap B_1(x)\) the weak formulation of the equation \(-\Delta ^\mathrm {tan}_{\tilde{x}}\hat{F}_{1}(\cdot ,x) = F'_{1}(\cdot ,x)\) with vanishing Neumann boundary condition. More precisely, we require \(\hat{F}_1(\cdot , x)\) to have vanishing average on \(e(t)^{-1}I_v(t)\cap B_1(x)\) (note that in the weak formulation the curvature term does not appear because it gets contracted with the tangential derivative of the test function). By elliptic regularity and (91), it follows that
$$\begin{aligned} ||\nabla ^\mathrm {tan}\hat{F}_1(\tilde{x}, x)||_{L^\infty } \leqq C. \end{aligned}$$
(92)
We now rescale back to \(I_v(t)\) and define \(\hat{F}_{e,\theta }(\tilde{x},x):=e(t)^2\hat{F}_1(e(t)^{-1}\tilde{x},e(t)^{-1}x)\) for \(x\in I_v(t)\) and \(\tilde{x}\in I_v(t)\cap B_{e(t)}(x)\). For fixed \(x\in I_v(t)\), \(\hat{F}_{e,\theta }(\cdot , x)\) has vanishing average on \(I_v(t)\cap B_{e(t)}(x)\) and solves \(-\Delta ^\mathrm {tan}_{\tilde{x}}\hat{F}_{e,\theta }(\cdot ,x) =F'_{e,\theta }(\cdot ,x)\) on \(I_v(t)\cap B_{e(t)}(x)\) with vanishing Neumann boundary condition. We finally introduce \(F_{e,\theta }(\tilde{x},x):=\nabla ^\mathrm {tan}_{\tilde{x}}\hat{F}_{e,\theta }(\tilde{x},x)\) for \(x\in I_v(t)\) and \(\tilde{x}\in I_v(t)\cap B_{e(t)}(x)\). It then follows from scaling, (92) as well as \(e(t)<r_c\) that \(\nabla _{\tilde{x}} \cdot F_{e,\theta }(\tilde{x},x)=F'_{e,\theta }\) and
$$\begin{aligned}&||e^{-1}(t)F_{e,\theta }(\tilde{x},x)||_{L^\infty } \leqq C. \end{aligned}$$
(93)
We now have everything in place to proceed with estimating the term
$$\begin{aligned} \bigg |\int _{I_v(t)} h^\pm (\tilde{x},t) \frac{\mathrm {d}}{{\mathrm {d}}t} \eta _e(\tilde{x},t) \,{\mathrm {d}}S(\tilde{x})\bigg |. \end{aligned}$$
To this end, we will make use of (87) and estimate term by term. Because of (20), (81), \(\Vert \eta _e\Vert _{L^p(I_v(t))}\leqq C\Vert \eta \Vert _{L^p(I_v(t))}\), the estimate
$$\begin{aligned} \int _{I_v(t)} |\theta '|\Big (\frac{|\tilde{x}-x|}{e(t)}\Big ) \,{\mathrm {d}}S(\tilde{x}) \leqq C e(t)^{d-1}, \end{aligned}$$
the Lipschitz property \(|V_{\mathrm n}(x)-V_{\mathrm n}(\tilde{x})|\leqq ||\nabla v||_{L^\infty } |x-\tilde{x}|\), and the fact that \(\theta (s)=0\) for \(s\geqq 1\), the first four terms on the right-hand side of (87) are straightforward to estimate and result in the bound
$$\begin{aligned} Cr_c^{-1}\Vert v\Vert _{W^{1,\infty }}\Vert h^\pm (\cdot ,t)\Vert _{L^2(I_v(t))}\Vert \eta \Vert _{L^2(I_v(t))}. \end{aligned}$$
(94)
To estimate the fifth term, we first apply Fubini’s theorem and then perform an integration by parts (recall that we imposed vanishing Neumann boundary conditions) which entails, because of the above considerations,
$$\begin{aligned}&\frac{1}{e(t)}\int _{I_v(t)} h^\pm (\tilde{x},t) \int _{I_v(t)}\frac{F'_{e,\theta }(\tilde{x},x)}{\int _{I_v(t)}\theta \big (\frac{|\hat{x} - x|}{e(t)}\big )\,{\mathrm {d}}S(\hat{x})}\eta (x)\,{\mathrm {d}}S(x) \,{\mathrm {d}}S(\tilde{x}) \\&\quad =\int _{I_v(t)}\bigg (\int _{I_v(t)\cap B_{\frac{3}{4}e(t)}(x)} h^\pm (\tilde{x},t) \frac{e(t)^{-1}F'_{e,\theta }(\tilde{x},x)}{\int _{I_v(t)}\theta \big (\frac{|\hat{x} - x|}{e(t)}\big )\,{\mathrm {d}}S(\hat{x})}\,{\mathrm {d}}S(\tilde{x})\bigg ) \eta (x)\,{\mathrm {d}}S(x) \\&\quad =-\int _{I_v(t)}\bigg (\int _{I_v(t)\cap B_{\frac{3}{4}e(t)}} \nabla _{\tilde{x}}h^\pm (\tilde{x},t)\cdot \frac{e(t)^{-1}F_{e,\theta }(\tilde{x},x)}{\int _{I_v(t)}\theta \big (\frac{|\hat{x} - x|}{e(t)}\big )\,{\mathrm {d}}S(\hat{x})}\,{\mathrm {d}}S(\tilde{x})\bigg )\eta (x)\,{\mathrm {d}}S(x)\\&\qquad -\int _{I_v(t)}h^\pm (\tilde{x},t)\mathrm {H}(\tilde{x}, t)\cdot \bigg (\int _{I_v(t)}\frac{e(t)^{-1}F_{e,\theta }(\tilde{x},x)}{\int _{I_v(t)}\theta \big (\frac{|\hat{x} - x|}{e(t)}\big )\,{\mathrm {d}}S(\hat{x})}\eta (x)\,{\mathrm {d}}S(x)\bigg )\,{\mathrm {d}}S(\tilde{x}). \end{aligned}$$
Using (93) as well as the lower bound from (81) we see that the second term can be estimated by a term of the form (94). For the first term, note that by the properties of \(F_{e,\theta }\) we may interpret the integral in brackets as the mollification of \(\nabla h^\pm \) on scale e(t). Applying the argument which led to (82) (for this, we only need the upper bound (93) for \(F_{e,\theta }\), a lower bound as in (81) is only required for \(\theta \)) we observe that one can bound this term similar to \(\smash {\Vert \nabla h^\pm _{e(t)}(\cdot ,t)\Vert _{L^2(I_v(t))}}\). We therefore obtain the bound
$$\begin{aligned}&\bigg |\int _{I_v(t)} h^\pm (\tilde{x},t) \frac{\mathrm {d}}{{\mathrm {d}}t} \eta _e(\tilde{x},t) \,{\mathrm {d}}S(\tilde{x})\bigg | \\&\quad \leqq Cr_c^{-4}\Vert v\Vert _{W^{1,\infty }}(1+e'(t)) \bigg (\int _{\mathbb {R}^d}1-\xi \cdot \frac{\nabla \chi _u}{|\nabla \chi _u|} \,\mathrm{d}|\nabla \chi _u|\bigg )^{1/2} ||\eta ||_{L^2(I_v(t))}\\&\qquad +Cr_c^{-6}\Vert v\Vert _{W^{1,\infty }}(1+e'(t)) \bigg (\int _{\mathbb {R}^d}|\chi _u-\chi _v| \min \Big \{\frac{{\text {dist}}(x,I_v(t))}{r_c},1\Big \} \,{\mathrm {d}}x\bigg )^{1/2}\\&\qquad ~~~~~~~~~~~~~~~~~~~~ \times ||\eta ||_{L^2(I_v(t))}. \end{aligned}$$
Hence, combining (83) with these estimates for the fourth term from (87) as well as (94) and (84), we obtain the desired estimate on the time derivative. This concludes the proof. \(\quad \square \)
Construction of the Compensation Function w for the Velocity Gradient Discontinuity
We turn to the construction of a compensating vector field, which shall be small in the \(L^2\)-norm but whose associated viscous stress \(\mu (\chi _u){D^{{\text {sym}}}}w\) shall compensate for (most of) the problematic viscous term \((\mu (\chi _u)-\mu (\chi _v)){D^{{\text {sym}}}}v\) appearing on the right hand side of the relative entropy inequality from Proposition 10 in the case of different shear viscosities.
Before we state the main result of this section, we introduce some further notation. Let \(\smash {h^+_{e(t)}}\) be defined as in Proposition 27. We then denote by \(\smash {P_{h^+_{e(t)}}}\) the downward projection onto the graph of \(\smash {h^+_{e(t)}}\), that is,
$$\begin{aligned} P_{h^+_{e(t)}}(x,t) := P_{I_v(t)}x + h^+_{e(t)}(P_{I_v(t)}x,t)\mathrm {n}_v(P_{I_v(t)}x,t), \end{aligned}$$
for all (x, t) such that \({\text {dist}}(x,I_v(t))<r_c\). Note that this map does not define an orthogonal projection. Analogously, one introduces the projection \(\smash {P_{h^-_{e(t)}}}\) onto the graph of \(\smash {h^-_{e(t)}}\).
Proposition 28
Let \((\chi _u,u,V)\) be a varifold solution to the free boundary problem for the incompressible Navier–Stokes equation for two fluids (1a)–(1c) in the sense of Definition 2 on some time interval \([0,{T_{vari}})\). Let \((\chi _v,v)\) be a strong solution to (1a)–(1c) in the sense of Definition 6 on some time interval \([0,{T_{strong}})\) with \({T_{strong}}\leqq {T_{vari}}\). Let \(\xi \) be the extension of the inner unit normal vector field \(\mathrm {n}_v\) of the interface \(I_v(t)\) from Definition 13. Let \(e:[0,{T_{strong}})\rightarrow (0,r_c]\) be a \(C^1\)-function and assume that the relative entropy is bounded by \(E[\chi _u,u,V|\chi _v,v](t)\leqq e(t)^2\). Let the regularized local interface error heights \(h^+_{e(t)}\) and \(h^-_{e(t)}\) be defined as in Proposition 27.
Then there exists a solenoidal vector field \(w\in L^2([0,{T_{strong}}];H^1({\mathbb {R}^d}))\) such that w is subject to the estimates
$$\begin{aligned}&\int _{\mathbb {R}^d}|w|^2 \,{\mathrm {d}}x\leqq C (r_c^{-4}R^2\Vert v\Vert ^2_{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}+1)\nonumber \\&\qquad \times \int _{I_v(t)} |h^+_{e(t)}|^2 {+} |\nabla h^+_{e(t)}|^2 + |h^-_{e(t)}|^2 {+} |\nabla h^-_{e(t)}|^2 \,{\mathrm {d}}S, \end{aligned}$$
(95)
where \(R>0\) is such that \(I_v(t)+B_{r_c}\subset B_R(0)\), and
$$\begin{aligned}&\int _{\{{\text {dist}}^{\pm }(x,I_v(t))\geqq 0\}} \big |\nabla w - \chi _{0\leqq {\text {dist}}^{\pm }(x,I_v(t))\leqq h^+_{e(t)}(P_{I_v(t)} x)} W \otimes \mathrm {n}_v(P_{I_v(t)}x,t)\big |^2 \,{\mathrm {d}}x\nonumber \\&\qquad +\int _{\{{\text {dist}}^{\pm }(x,I_v(t))\leqq 0\}} \big |\nabla w - \chi _{-h^-_{e(t)}(P_{I_v(t)} x)\leqq {\text {dist}}^{\pm }(x,I_v(t))\leqq 0} W \otimes \mathrm {n}_v(P_{I_v(t)}x,t)\big |^2 \,{\mathrm {d}}x\nonumber \\&\qquad +\int _{\mathbb {R}^d}\chi _{{\text {dist}}^{\pm }(x,I_v(t))\notin [-h^-_{e(t)}(P_{I_v(t)}x),h^+_{e(t)}(P_{I_v(t)} x)]} |\nabla w|^2 \,{\mathrm {d}}x\nonumber \\&\quad \leqq C r_c^{-4}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}^2 \int _{I_v(t)} |h^+_{e(t)}|^2 + |\nabla h^+_{e(t)}|^2 + |h^-_{e(t)}|^2 + |\nabla h^-_{e(t)}|^2 \,{\mathrm {d}}S, \end{aligned}$$
(96)
where the vector field W is given by
$$\begin{aligned} W(x,t)&:= \frac{2(\mu _+-\mu _-)}{\mu _+\,(1{-}\chi _v)+\mu _-\chi _v}\big ({\text {Id}}-\mathrm {n}_v \otimes \mathrm {n}_v\big )(P_{I_v(t)}x)\big ({D^{{\text {sym}}}}v\cdot \mathrm {n}_v(P_{I_v(t)}x)\big ), \end{aligned}$$
(97)
with the symmetric gradient defined by \({D^{{\text {sym}}}}v := \frac{1}{2}(\nabla v + \nabla v^T)\), as well as the estimates
$$\begin{aligned}&\int _{I_v(t)} \sup _{y\in (-r_c,r_c)} |w(x + y\mathrm {n}_v(x,t))|^2 \,{\mathrm {d}}S(x)\nonumber \\&\quad \leqq C r_c^{-4}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}^2 \int _{I_v(t)} |h^+_{e(t)}|^2 + |\nabla h^+_{e(t)}|^2 + |h^-_{e(t)}|^2 + |\nabla h^-_{e(t)}|^2 \,{\mathrm {d}}S, \end{aligned}$$
(98)
$$\begin{aligned}&\Vert \nabla w\Vert _{L^\infty } \leqq Cr_c^{-4}|\log e(t)|\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))} + Cr_c^{-3}\Vert \nabla ^3 v\Vert _{L^\infty ({\mathbb {R}^d}{\setminus } I_v(t))}\nonumber \\&\quad ~~~~~~~~~~~~~~~~~~ + C r_c^{-9} \big (1{+}\mathcal {H}^{d-1}(I_v(t))\big )\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))} , \end{aligned}$$
(99)
$$\begin{aligned}&\bigg (\int _{I_v(t)}\sup _{y\in [-r_c,r_c]}|(\nabla w)^T(x+y\mathrm {n}_v(x,t)) \mathrm {n}_v(x,t)|^2\,{\mathrm {d}}S(x)\bigg )^\frac{1}{2}\nonumber \\&\quad \leqq Cr_c^{-9}(1+\mathcal {H}^{d-1}(I_v(t)))\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}e(t) +Cr_c^{-2}\Vert v\Vert _{W^{3,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}e(t)\nonumber \\&\qquad +Cr_c^{-1}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}|\log e(t)|^\frac{1}{2}e(t) \end{aligned}$$
(100)
and
$$\begin{aligned} \partial _t w(\cdot ,t) = - \big (v(\cdot ,t)\cdot \nabla \big )w(\cdot ,t) + g + \hat{g}, \end{aligned}$$
(101)
where the vector fields g and \(\hat{g}\) are subject to the bounds
$$\begin{aligned}&\Vert \hat{g}\Vert _{L^\frac{4}{3}({\mathbb {R}^d})} \nonumber \\&\quad \leqq C\frac{\Vert v\Vert _{W^{1,\infty }}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}}{e(t)r_c^3}\bigg (\int _{I_v(t)} |\bar{h}^\pm |^4 \,{\mathrm {d}}S\bigg )^\frac{1}{4}\nonumber \\&\qquad ~~~~~ \times \bigg (\int _{I_v(t)} |h^+_{e(t)}|^2 + |\nabla h^+_{e(t)}|^2 + |h^-_{e(t)}|^2 + |\nabla h^-_{e(t)}|^2 \,{\mathrm {d}}S\bigg )^\frac{1}{2}\nonumber \\&\qquad + C\frac{\Vert v\Vert _{W^{1,\infty }}}{e(t)r_c^2}\bigg (\int _{I_v(t)} |\bar{h}^\pm |^4 \,{\mathrm {d}}S\bigg )^\frac{1}{4} \nonumber \\&\qquad ~~~~~ \times (\Vert u{-}v{-}w\Vert _{L^2}^\frac{1}{2}\Vert \nabla (u{-}v{-}w)\Vert _{L^2}^\frac{1}{2}+\Vert u{-}v{-}w\Vert _{L^2})\nonumber \\&\qquad +C\frac{\Vert v\Vert _{W^{1,\infty }}(1+\Vert v\Vert _{W^{1,\infty }})}{e(t)} \int _{{\mathbb {R}^d}} 1-\xi \cdot \frac{\nabla \chi _u}{|\nabla \chi _u|} \,\mathrm{d}|\nabla \chi _u|, \end{aligned}$$
(102)
and
$$\begin{aligned}&\Vert g\Vert _{L^2({\mathbb {R}^d})} \nonumber \\&\quad ~~~~ \leqq C \frac{1{+}\Vert v\Vert _{W^{1,\infty }}}{r_c^2} (\Vert \partial _t \nabla v\Vert _{L^\infty ({\mathbb {R}^d}{\setminus } I_v(t))}{+}(R^2{+}1) \Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))})\nonumber \\&\qquad ~~~~~ \times \bigg (\int _{I_v(t)} |h^+_{e(t)}|^2 + |\nabla h^+_{e(t)}|^2 + |h^-_{e(t)}|^2 + |\nabla h^-_{e(t)}|^2 \,{\mathrm {d}}S\bigg )^\frac{1}{2}\nonumber \\&\qquad +C\frac{\Vert v\Vert _{W^{1,\infty }}(1+\Vert v\Vert _{W^{1,\infty }})}{e(t)r_c} \int _{{\mathbb {R}^d}} 1-\xi \cdot \frac{\nabla \chi _u}{|\nabla \chi _u|} \, \mathrm {d}|\nabla \chi _u|\nonumber \\&\qquad +Cr_c^{-2}(1+e'(t))\Vert v\Vert _{W^{1,\infty }}^2 \bigg (\int _{I_v(t)}|h^\pm |^2\,{\mathrm {d}}S\bigg )^\frac{1}{2}\nonumber \\&\qquad +C\frac{\Vert v\Vert _{W^{1,\infty }}(1{+}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))})}{r_c} \bigg (\int _{{\mathbb {R}^d}}|\chi _u{-}\chi _v| \min \Big \{\frac{{\text {dist}}(x,I_v(t))}{r_c},1\Big \} \,{\mathrm {d}}x\bigg )^\frac{1}{2}\nonumber \\&\qquad +C\Vert v\Vert _{W^{1,\infty }}(\Vert u{-}v{-}w\Vert _{L^2}^\frac{1}{2}\Vert \nabla (u{-}v{-}w) \Vert _{L^2}^\frac{1}{2}+\Vert u{-}v{-}w\Vert _{L^2}), \end{aligned}$$
(103)
where \(\bar{h}^\pm \) is defined as \(h^\pm \) but now with respect to the modified cut-off function \(\bar{\theta }(\cdot )=\theta \big (\frac{\cdot }{2}\big )\), see Proposition 26. Furthermore, w may be taken to have the regularity \(\nabla w(\cdot ,t)\in W^{1,\infty }({\mathbb {R}^d}{\setminus } (I_v(t)\cup I_{h^+_e}(t)\cup I_{h^+_e}(t)))\) for almost every t, where \(I_{h^\pm _e}(t)\) denotes the \(C^3\)-manifold \(\{x\pm h^\pm _{e(t)}(x) \mathrm {n}_v(x):x\in I_v(t)\}\).
Proof
Step 1: Definition ofw. Let \(\eta \) be a cutoff supported at each \(t\in [0,{T_{strong}})\) in the set \(I_v(t)+B_{r_c/2}\) with \(\eta \equiv 1\) in \(I_v(t)+B_{r_c/4}\) and \(|\nabla \eta |\leqq C r_c^{-1}\), \(|\nabla ^2 \eta |\leqq C r_c^{-2}\) as well as \(|\partial _t \eta |\leqq C r_c^{-1} \Vert v\Vert _{L^\infty }\) and \(|\partial _t \nabla \eta |\leqq C r_c^{-2} \Vert v\Vert _{W^{1,\infty }}\). For example, one may choose \(\eta (x,t):=\theta (\frac{{\text {dist}}(x,I_v(t))}{r_c})\) where \(\theta :\mathbb {R}^+\rightarrow [0,1]\) is the smooth cutoff already used in the definition of the regularized local interface error heights in Proposition 27.
Define the vector field W as given in (97) and set (making use of the notation \(a\wedge b = \min \{a,b\}\) and \(a\vee b = \max \{a,b\}\))
$$\begin{aligned} w^+(x,t):= \eta \int _0^{({\text {dist}}^{\pm }(x,I_v(t)) \vee 0) \wedge h^+_{e(t)}(P_{I_v(t)} x)} W (P_{I_v(t)} x + y\mathrm {n}_v(P_{I_v(t)}x,t)) \,{\mathrm {d}}y\end{aligned}$$
(104)
as well as
$$\begin{aligned} w^-(x,t):= \eta \int _{({\text {dist}}^{\pm }(x,I_v(t)) \wedge 0) \vee -h^-_{e(t)}(P_{I_v(t)} x)}^0 W (P_{I_v(t)} x + y\mathrm {n}_v(P_{I_v(t)}x,t)) \,{\mathrm {d}}y. \end{aligned}$$
(105)
For this choice, we have
$$\begin{aligned}&\nabla w^+ (x,t)\nonumber \\&\quad = \chi _{0\leqq {\text {dist}}^{\pm }(x,I_v(t))\leqq h^+_{e(t)}(P_{I_v(t)} x)} W(x) \otimes \mathrm {n}_v(P_{I_v(t)}x)\nonumber \\&\qquad + \eta \, \chi _{{\text {dist}}^{\pm }(x,I_v(t))>h^+_{e(t)}(P_{I_v(t)} x)} W(P_{h^+_{e(t)}} x) \otimes \nabla h^+_{e(t)}(P_{I_v(t)}x) \nabla P_{I_v(t)}(x)\nonumber \\&\qquad +\eta \int _0^{({\text {dist}}^{\pm }(x,I_v(t)) \vee 0) \wedge h^+_{e(t)}(P_{I_v(t)} x)} \nabla W (P_{I_v(t)} x {+} y \mathrm {n}_v(P_{I_v(t)}x)) \nonumber \\&\qquad \quad \qquad \qquad (\nabla P_{I_v(t)}x{+}y\nabla \mathrm {n}_v(P_{I_v(t)}x)) \,{\mathrm {d}}y\nonumber \\&\qquad +\nabla \eta \int _0^{({\text {dist}}^{\pm }(x,I_v(t)) \vee 0) \wedge h^+_{e(t)}(P_{I_v(t)} x)} W (P_{I_v(t)} x {+} y \mathrm {n}_v(P_{I_v(t)}x)) \,{\mathrm {d}}y\end{aligned}$$
(106)
(note that this directly implies the last claim about the regularity of w, namely \(\nabla w(\cdot ,t)\in W^{1,\infty }({\mathbb {R}^d}{\setminus } (I_v(t)\cup I_{h^+_e}(t)\cup I_{h^+_e}(t)))\) for almost every t) as well as
$$\begin{aligned}&\partial _t w^+(x,t)\nonumber \\&\quad =\chi _{0\leqq {\text {dist}}^{\pm }(x,I_v(t))\leqq h^+_{e(t)}(P_{I_v(t)} x)} W (x) \partial _t {\text {dist}}^{\pm }(x,I_v(t))\nonumber \\&\qquad +\eta \, \chi _{{\text {dist}}^{\pm }(x,I_v(t))> h^+_{e(t)}(P_{I_v(t)} x)} W (P_{h^+_{e(t)}} x) \nonumber \\&\qquad \qquad \times \big (\partial _t h^+_{e(t)}(P_{I_v(t)}x)+\partial _t P_{I_v(t)}x\cdot \nabla h^+_{e(t)}(P_{I_v(t)}x)\big )\nonumber \\&\qquad +\eta \int _0^{({\text {dist}}^{\pm }(x,I_v(t)) \vee 0) \wedge h^+_{e(t)}(P_{I_v(t)} x)} \partial _t W (P_{I_v(t)} x {+} y \mathrm {n}_v(P_{I_v(t)}x)) \,{\mathrm {d}}y\nonumber \\&\qquad +\eta \int _0^{({\text {dist}}^{\pm }(x,I_v(t)) \vee 0) \wedge h^+_{e(t)}(P_{I_v(t)} x)} \nabla W (P_{I_v(t)} x {+} y \mathrm {n}_v(P_{I_v(t)}x))\nonumber \\&\qquad \qquad \qquad (\partial _t P_{I_v(t)}x{+}y\partial _t \mathrm {n}_v(P_{I_v(t)}x)) \,{\mathrm {d}}y\nonumber \\&\qquad +\partial _t \eta \int _0^{({\text {dist}}^{\pm }(x,I_v(t)) \vee 0) \wedge h^+_{e(t)}(P_{I_v(t)} x)} W (P_{I_v(t)} x + y \mathrm {n}_v(P_{I_v(t)}x)) \,{\mathrm {d}}y. \end{aligned}$$
(107)
Moreover, note that (106) entails by the definition of the vector field W
$$\begin{aligned}&\nabla \cdot w^+ (x,t)\nonumber \\&\quad =\eta \, \chi _{{\text {dist}}^{\pm }(x,I_v(t))>h^+_{e(t)}(P_{I_v(t)} x)} W(P_{h^+_{e(t)}} x) \cdot \nabla h^+_{e(t)}(P_{I_v(t)}x) \nabla P_{I_v(t)}(x)\nonumber \\&\quad +\eta \int _0^{({\text {dist}}^{\pm }(x,I_v(t)) \vee 0) \wedge h^+_{e(t)}(P_{I_v(t)} x)} {\text {tr}} \nabla W (P_{I_v(t)} x {+} y \mathrm {n}_v(P_{I_v(t)}x))\nonumber \\&\qquad ~~~~~~~~(\nabla P_{I_v(t)}x{+}y\nabla \mathrm {n}_v(P_{I_v(t)}x)) \,{\mathrm {d}}y\nonumber \\&\quad +\nabla \eta \cdot \int _0^{({\text {dist}}^{\pm }(x,I_v(t)) \vee 0) \wedge h^+_{e(t)}(P_{I_v(t)} x)} W (P_{I_v(t)} x {+} y \mathrm {n}_v(P_{I_v(t)}x)) \,{\mathrm {d}}y. \end{aligned}$$
(108)
Analogous formulas and properties can be derived for \(w^-\). The function \(w^+ + w^-\) would then satisfy our conditions, with the exception of the solenoidality \(\nabla \cdot w=0\). For this reason, we introduce the (usual) kernel
$$\begin{aligned} \theta (x):=\frac{1}{\mathcal {H}^{d-1}(\mathbb {S}^{d-1})}\frac{x}{|x|^d} \end{aligned}$$
and set
$$\begin{aligned} w(x,t)&:= w^+(x,t)-(\theta *\nabla \cdot w^+)(x,t) +w^-(x,t)-(\theta *\nabla \cdot w^-)(x,t). \end{aligned}$$
(109)
It is immediately apparent that \(\nabla \cdot w=0\).
Step 2: Estimates onwand\(\nabla w\). From (106), \(|\nabla \eta |\leqq Cr_c^{-1}\) as well as the bounds (19) and (28) we deduce the pointwise bound
$$\begin{aligned}&\big |\nabla w^+ - \chi _{0\leqq {\text {dist}}^{\pm }(x,I_v(t))\leqq h^+_{e(t)}(P_{I_v(t)} x)} W(x) \otimes \mathrm {n}_v(P_{I_v(t)}x)\big |\nonumber \\&\quad \leqq C \chi _{\mathrm{supp}\,\eta } r_c^{-1} \Vert \nabla v\Vert _{L^\infty } |\nabla h^+_{e(t)}(P_{I_v(t)} x)|\nonumber \\&\qquad +C \chi _{\mathrm{supp}\,\eta } \big (r_c^{-2} \Vert \nabla v\Vert _{L^\infty } + r_c^{-1} \Vert \nabla ^2 v\Vert _{L^\infty ({\mathbb {R}^d}{\setminus } I_v(t))}\big ) |h^+_{e(t)}(P_{I_v(t)} x)|\nonumber \\&\qquad +C r_c^{-1} \chi _{\mathrm{supp}\,\eta } \Vert \nabla v\Vert _{L^\infty } |h^+_{e(t)}(P_{I_v(t)} x)|, \end{aligned}$$
(110)
and therefore, by integration and a change of variables \(\Phi _t\),
$$\begin{aligned}&\int _{\mathbb {R}^d}\Big |\nabla w^+ - \chi _{0\leqq {\text {dist}}^{\pm }(x,I_v(t))\leqq h^+_{e(t)}(P_{I_v(t)} x)} W(x) \otimes \mathrm {n}_v(P_{I_v(t)}x)\Big |^2 \,{\mathrm {d}}x\nonumber \\&\quad \leqq C (r_c^{-4}\Vert \nabla v\Vert _{L^\infty }^2+r_c^{-2}\Vert \nabla ^2 v\Vert _{L^\infty ({\mathbb {R}^d}{\setminus } I_v(t))}^2)\nonumber \\&\qquad \times \int _{\mathbb {R}^d}\chi _{\mathrm {supp}\,\eta }(|h^+_{e(t)}|^2 + |\nabla h^+_{e(t)}|^2)(P_{I_v(t)}x)\,{\mathrm {d}}x\nonumber \\&\quad \leqq C r_c^{-4}\Vert v\Vert ^2_{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))} \int _{I_v(t)} |h^+_{e(t)}|^2 + |\nabla h^+_{e(t)}|^2 \,{\mathrm {d}}S. \end{aligned}$$
(111)
Observe that this also implies, by (97), that
$$\begin{aligned}&\int _{\mathbb {R}^d}|\nabla \cdot w^+|^2\,{\mathrm {d}}x\leqq C r_c^{-4}\Vert v\Vert ^2_{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))} \int _{I_v(t)} |h^+_{e(t)}|^2 + |\nabla h^+_{e(t)}|^2 \,{\mathrm {d}}S. \end{aligned}$$
(112)
From this, Theorem 38, and the fact that \(\nabla \theta \) is a singular integral kernel subject to the assumptions of Theorem 38, we deduce
$$\begin{aligned} \int _{\mathbb {R}^d}\big |\nabla (\theta *(\nabla \cdot w^+)) \big |^2 \,{\mathrm {d}}x\leqq C r_c^{-4}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}^2 \int _{I_v(t)} |h^+_{e(t)}|^2 + |\nabla h^+_{e(t)}|^2 \,{\mathrm {d}}S. \end{aligned}$$
(113)
Combining the estimates (111) and (113) with the corresponding inequalities for \(w^-\) and \(\theta *\nabla \cdot w^-\), we deduce our estimate (96).
The trivial estimate \(|w^+(x,t)|\leqq \chi _{\mathrm{supp}\,\eta }(x,t) \Vert \nabla v\Vert _{L^\infty } h^+_{e(t)}(P_{I_v(t)}x)\) gives by the change of variables \(\Phi _t \)
$$\begin{aligned}&\int _{\mathbb {R}^d}|w^+|^2 \,{\mathrm {d}}x\leqq C r_c \int _{I_v(t)} |h^+_{e(t)}|^2 \,{\mathrm {d}}S. \end{aligned}$$
(114)
Now, let \(R>1\) be big enough such that \(I_v(t)+B_{r_c}\subset B_R(0)\) for all \(t\in [0,{T_{strong}})\). We then estimate with an integration by parts and Theorem 38 applied to the singular integral operator \(\nabla \theta \)
$$\begin{aligned} \int _{{\mathbb {R}^d}{\setminus } B_{3R}(0)} \big | \theta *(\nabla \cdot w^+)\big |^2 \,{\mathrm {d}}x&= \int _{{\mathbb {R}^d}{\setminus } B_{3R}(0)}\bigg |\int _{B_R(0)}\theta (x-\tilde{x}) (\nabla \cdot w^+(\tilde{x}))\,\mathrm{d}\tilde{x}\bigg |^2\,{\mathrm {d}}x\nonumber \\&\leqq \int _{{\mathbb {R}^d}}\bigg |\int _{B_R(0)} \nabla \theta (x-\tilde{x})w^+(\tilde{x}) \,\mathrm{d}\tilde{x}\bigg |^2\,{\mathrm {d}}x\nonumber \\&\leqq C\int _{B_R(0)}|w^+|^2\,{\mathrm {d}}x. \end{aligned}$$
(115)
By Young’s inequality for convolutions, (112), (114) and (115), we then obtain
$$\begin{aligned}&\int _{\mathbb {R}^d}\big | \theta *(\nabla \cdot w^+)\big |^2 \,{\mathrm {d}}x\nonumber \\&\quad =\int _{B_{3R}(0)} \big | \theta *(\nabla \cdot w^+)\big |^2 \,{\mathrm {d}}x+\int _{{\mathbb {R}^d}{\setminus } B_{3R}(0)} \big | \theta *(\nabla \cdot w^+)\big |^2 \,{\mathrm {d}}x\nonumber \\&\quad \leqq C\bigg (\int _{B_{3R}(0)}\frac{1}{|x|^{d-1}}\,{\mathrm {d}}x\bigg )^2 \int _{\mathbb {R}^d}|\nabla \cdot w^+ |^2 \,{\mathrm {d}}x+C\int _{{\mathbb {R}^d}}|w^+|^2\,{\mathrm {d}}x\nonumber \\&\quad \leqq C (r_c^{-4}R^2\Vert v\Vert ^2_{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}+1) \int _{I_v(t)} |h^+_{e(t)}|^2 + |\nabla h^+_{e(t)}|^2 \,{\mathrm {d}}S. \end{aligned}$$
(116)
Together with the respective estimates for \(w^-\) and \(\theta *(\nabla \cdot w^-)\), this implies (95). The estimate (98) follows directly from (104) and the estimates (113) and (116) on the \(H^1\)-norm of \(\theta *(\nabla \cdot w^+)\) as well as the definition of \(w^-\) and the analogous estimates for \(\theta *(\nabla \cdot w^-)\).
Step 3:\(L^\infty \)-estimates for\(\nabla w\). Regarding the estimate (99) on \(\Vert \nabla w\Vert _{L^\infty }\) we have by (110) and the estimates \(|\nabla h^+_{e(t)}|\leqq Cr_c^{-2}\) and \(|h^+_{e(t)}|\leqq r_c\leqq 1\) from Proposition 27
$$\begin{aligned} \Vert \nabla w^+\Vert _{L^\infty }\leqq Cr_c^{-4}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}. \end{aligned}$$
(117)
To estimate \(|\nabla (\theta *(\nabla \cdot w^+))|\), we first compute, starting with (108),
$$\begin{aligned}&\nabla (\nabla \cdot w^+) (x,t)\nonumber \\&=\eta \, \chi _{{\text {dist}}^{\pm }(x,I_v)>h^+_{e(t)}(P_{I_v(t)} x)} W(P_{h^+_{e(t)}} x) \cdot \nabla ^2 h^+_{e(t)}(P_{I_v(t)}x) \nabla P_{I_v(t)}(x)\nabla P_{I_v(t)}(x)\nonumber \\&\quad +\big (W(P_{h^+_{e(t)}} x) \cdot \nabla h^+_{e(t)}(P_{I_v(t)}x) \nabla P_{I_v(t)}(x)\big ) \, \nabla \chi _{{\text {dist}}^{\pm }(x,I_v)>h^+_{e(t)}(P_{I_v(t)} x)}\nonumber \\&\quad +F(x,t), \end{aligned}$$
(118)
where F(x, t) is subject to a bound of the form \(|F(x,t)|\leqq C r_c^{-5}\Vert v\Vert _{W^{3,\infty }({\mathbb {R}^d}{\setminus } I_v(t))})\) and supported in \(I_v(t)+B_{r_c}\). Next, we decompose the kernel \(\theta \) as \(\theta = \sum _{k=-\infty }^\infty \theta _k\) with smooth functions \(\theta _k\) with \(\mathrm{supp}\,\theta _k \subset B_{2^{k+1}}{\setminus } B_{2^{k-1}}\). More precisely, we first choose a smooth function \(\varphi :\mathbb {R}_+\rightarrow [0,1]\) such that \(\varphi (s)=0\) whenever \(s\notin [-1/2,2]\) and such that \(\sum _{k\in \mathbb {Z}}\varphi (2^ks)=1\) for all \(s>0\). Such a function indeed exists, see for instance [16]. We then let \(\theta _k(x):=\varphi (2^k|x|)\theta (x)\). Note that \(\Vert \theta _k\Vert _{L^1({\mathbb {R}^d})}\leqq C 2^k\), \(\Vert \nabla \theta _k\Vert _{L^1({\mathbb {R}^d})}\leqq C\) as well as \(|\nabla \theta _k|\leqq C (2^k)^{-d}\). We estimate
$$\begin{aligned} |\nabla (\theta *(\nabla \cdot w^+))|&\leqq \sum _{k=\lfloor \log e^2(t)\rfloor }^0 |\nabla (\theta _k *(\nabla \cdot w^+))| +\sum _{k=1}^\infty |\nabla (\theta _k *(\nabla \cdot w^+))|\nonumber \\&\quad +\sum _{k=-\infty }^{\lfloor \log e^2(t)\rfloor -1} |\theta _k *\nabla (\nabla \cdot w^+)|. \end{aligned}$$
(119)
Using Young’s inequality for convolutions as well as the estimate \(\Vert \nabla \theta _k\Vert _{L^1({\mathbb {R}^d})}\leqq C\) we obtain
$$\begin{aligned} \sum _{k=\lfloor \log e^2(t)\rfloor }^0 |\nabla (\theta _k *(\nabla \cdot w^+))| \leqq 2C|\log e(t)|\Vert \nabla \cdot w^+\Vert _{L^\infty }. \end{aligned}$$
(120)
Moreover, it follows from \(|\nabla \theta _k|\leqq C (2^k)^{-d}\), the precise formula for \(\nabla \cdot w^+\) in (108), (19), (28), a change of variables and Hölder’s inequality that
$$\begin{aligned}&\sum _{k=1}^\infty |\nabla (\theta _k *(\nabla \cdot w^+))|\nonumber \\&\quad \leqq Cr_c^{-2}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}\nonumber \\&\qquad \times \sum _{k=1}^\infty (2^k)^{-d} \int _{I_v(t)+B_{r_c/2}} |\nabla h^+_{e(t)}(P_{I_v(t)}x)| + |h^+_{e(t)}(P_{I_v(t)}x)| \,{\mathrm {d}}x\nonumber \\&\quad \leqq Cr_c^{-2}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}\sqrt{\mathcal {H}^{d-1}(I_v(t))} \bigg (\int _{I_v(t)} |\nabla h^+_{e(t)}|^2+|h^+_{e(t)}|^2\,{\mathrm {d}}S\bigg )^\frac{1}{2}. \end{aligned}$$
(121)
Using (118), the estimate \(|\nabla ^2 h^\pm _{e(t)}(\cdot ,t)| \leqq C r_c^{-4} e(t)^{-1}\) from Proposition 27, (19), (28) and again Young’s inequality for convolutions (recall that \(\Vert \theta _k\Vert _{L^1({\mathbb {R}^d})}\leqq C 2^k\)), we get
$$\begin{aligned}&\sum _{k=-\infty }^{\lfloor \log e^2(t)\rfloor -1} |\theta _k *\nabla (\nabla \cdot w^+)|(\tilde{x},t) \leqq I + II + III \end{aligned}$$
(122)
where the three terms on the right hand side are given by
$$\begin{aligned} I := \sum _{k=-\infty }^{\lfloor \log e^2(t)\rfloor -1} 2^{k} Cr_c^{-5}\Vert v\Vert _{W^{3,\infty }({\mathbb {R}^d}{\setminus } I_v(t))} \leqq Cr_c^{-5}\Vert v\Vert _{W^{3,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}e^2(t) \end{aligned}$$
(123)
and
$$\begin{aligned} II := Cr_c^{-5}\Vert v\Vert _{W^{1,\infty }} e(t)^{-1} \sum _{k=-\infty }^{\lfloor \log e^2(t)\rfloor -1} 2^{k} \leqq Cr_c^{-5}\Vert v\Vert _{W^{1,\infty }} e(t) \end{aligned}$$
(124)
as well as
$$\begin{aligned}&III := \sum _{k=-\infty }^{\lfloor \log e^2(t) \rfloor -1} \bigg |\int _{\mathbb {R}^d}\theta _k(x{-}\tilde{x}) \otimes \big (W(P_{h^+_{e(t)}} x) \cdot (\nabla P_{I_v(t)})^T(x) \nabla h^+_{e(t)}(P_{I_v(t)} x)\big )\nonumber \\&\qquad ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\,\mathrm{d}\nabla \chi _{{\text {dist}}^{\pm }(x,I_v(t))>h^+_{e(t)}(P_{I_v(t)}x)}(x)\bigg |. \end{aligned}$$
(125)
To estimate the latter term, we proceed first by noting the definition of \(\smash {h^+_{e(t)}}\) in (77), as well as the trivial bound \(|h^+|\leqq r_c\) it holds \(\smash {|h^+_{e(t)}|}\leqq r_c\). Then for all \(\tilde{x}\in I_v(t)+\{|x|> r_c+2^{\lfloor \log e^2(t) \rfloor }\}\) and all \(k\leqq \lfloor \log e^2(t) \rfloor - 1\) we observe that \(\chi _{\{{\text {dist}}^{\pm }(x,I_v(t))>h^+_{e(t)}(P_{I_v(t)}x)\}}(x)=1\) for all \(x\in {\mathbb {R}^d}\) such that \(|x-\tilde{x}|\leqq 2^{k+1}\). In particular, for such \(\tilde{x}\) the third term on the right hand side of (122) vanishes since the corresponding second term in the formula for \(\nabla (\nabla \cdot w^+)\) (see (118)) does not appear anymore.
Hence, let \(\tilde{x}\in I_v(t)+\{|x|\leqq r_c+2^{\lfloor \log e^2(t) \rfloor }\}\) and denote by F the tangent plane to the manifold \(\{{\text {dist}}^{\pm }(x,I_v(t))=h^+_{e(t)}(P_{I_v(t)}x)\}\) at the nearest point to \(\tilde{x}\). We then have for any \(\psi \in C^\infty _{cpt}({\mathbb {R}^d})\) that
$$\begin{aligned}&\int _{\mathbb {R}^d}\psi (x) \,\mathrm{d}\nabla \chi _{\{{\text {dist}}^{\pm }(x,I_v(t))>h^+_{e(t)}(P_{I_v(t)}x)\}}(x) -\int _{\mathbb {R}^d}\psi (x) \,\mathrm{d}\nabla \chi _{\{{\text {dist}}^{\pm }(x,F)>0\}}(x)\\&\quad =\int _{\{{\text {dist}}^{\pm }(x,I_v(t))>h^+_{e(t)}(P_{I_v(t)}x)\}} \nabla \psi (x) \,\mathrm{d}x -\int _{\{{\text {dist}}^{\pm }(x,F)>0\}} \nabla \psi (x) \,\mathrm{d}x, \end{aligned}$$
and as a consequence,
$$\begin{aligned}&\int _{\mathbb {R}^d}\theta _k(x-\tilde{x}) \otimes \big (W(P_{h^+_{e(t)}} x) \cdot (\nabla P_{I_v(t)})^T(x) \nabla h^+_{e(t)}(P_{I_v(t)} x)\big ) \,\\&\qquad ~~~~~\mathrm{d}\nabla \chi _{\{{\text {dist}}^{\pm }(x,I_v(t))>h^+_{e(t)}(P_{I_v(t)}x)\}}(x)\\&\quad =\int _F \theta _k(x-\tilde{x}) \otimes \big (W(P_{h^+_{e(t)}} x) \cdot (\nabla P_{I_v(t)})^T(x) \nabla h^+_{e(t)}(P_{I_v(t)} x)\big ) \mathrm {n}_F \,{\mathrm {d}}S(x)\\&\qquad +\int _{\mathbb {R}^d}(\chi _{\{{\text {dist}}^{\pm }(x,I_v(t))>h^+_{e(t)}(P_{I_v(t)}x)\}}-\chi _{\{{\text {dist}}^{\pm }(x,F)>0\}})\\&\qquad ~~~~~~~~~~~~ \nabla \big (\theta _k(x-\tilde{x}) \otimes \big (W(P_{h^+_{e(t)}} x) \cdot (\nabla P_{I_v(t)})^T(x) \nabla h^+_{e(t)}(P_{I_v(t)} x)\big )\big ) \,{\mathrm {d}}x. \end{aligned}$$
Recall that we defined \(\theta _k(x):=\varphi (2^k|x|)\theta (x)\) where \(\varphi :\mathbb {R}_+\rightarrow [0,1]\) is a smooth function such that \(\varphi (s)=0\) whenever \(s\notin [-1/2,2]\) and such that \(\sum _{k\in \mathbb {Z}}\varphi (2^ks)=1\) for all \(s>0\). Hence, \(|\mathrm {n}_F \cdot \theta _k(x-\tilde{x})|\leqq C\frac{|\mathrm {n}_F\cdot (x-\cdot \tilde{x})|}{|x-\tilde{x}|^d} \leqq C\frac{{\text {dist}}(\tilde{x},F)}{|x-\tilde{x}|^d}\) for all \(x\in F\). It also follows from the definition of \(\theta \) that \(\int _F ({\text {Id}}-\mathrm {n}_F\otimes \mathrm {n}_F) \theta _k(x-\tilde{x})\,{\mathrm {d}}S(x)=0\). Hence we may solve \(({\text {Id}}-\mathrm {n}_F\otimes \mathrm {n}_F) \theta _k(\cdot \,-\tilde{x}) =\Delta _x^{\mathrm {tan}}\tilde{\theta }_k(\cdot ,\tilde{x})\) on \(B_{2^{k+2}}(\tilde{x})\cap F\) with vanishing Neumann boundary conditions. In particular, for \(\hat{\theta }_k(x,\tilde{x}):= \nabla ^{\mathrm {tan}}_x\tilde{\theta }_k(x,\tilde{x})\) we obtain \(({\text {Id}}-\mathrm {n}_F\otimes \mathrm {n}_F) \theta _k(x-\tilde{x})=\nabla ^{\mathrm {tan}}_x\cdot \nabla _x\hat{\theta }_k(x,\tilde{x})\). It follows from elliptic regularity that \(\hat{\theta }(\cdot ,\tilde{x})\) is \(C^\infty \). Moreover, since we could have rescaled \(\theta _k\) first to unit scale, then solved the associated problem on that scale, and finally rescaled the solution back to the dyadic scale k we see that \(|\hat{\theta }_k(x,\tilde{x})|\leqq C (2^k)^{2-d}\). We then have by an integration by parts
$$\begin{aligned} \bigg |\int _F({\text {Id}}-\mathrm {n}_F\otimes \mathrm {n}_F) \theta _k(x-\tilde{x})\otimes \psi \,{\mathrm {d}}S(x)\bigg |&\leqq \int _{F\cap B_{2^{k+1}}(\tilde{x})}|\hat{\theta }_k(x,\tilde{x})||\nabla ^{\mathrm {tan}}\psi |\,{\mathrm {d}}S(x) \\&\leqq C(2^k)^{2-d}\int _{F\cap B_{2^{k+1}}(\tilde{x})}|\nabla ^{\mathrm {tan}}\psi |\,{\mathrm {d}}S(x) \end{aligned}$$
for any \(\psi \in C^1_{cpt}({\mathbb {R}^d};{\mathbb {R}^d})\). Furthermore, it holds that
$$\begin{aligned} \int _{B_{2^k}(\tilde{x})} |\chi _{\{{\text {dist}}^{\pm }(x,I_v(t))>h^+_{e(t)}(P_{I_v(t)}x)\}}-\chi _{\{{\text {dist}}^{\pm }(x,F)>0\}}| \,{\mathrm {d}}x&\leqq C \Vert \nabla ^2 h^+_{e(t)}\Vert _{L^\infty } (2^k)^{d+1}. \end{aligned}$$
Using these considerations in the previous formula, we obtain
$$\begin{aligned}&\bigg |\int _{\mathbb {R}^d}\theta _k(x-\tilde{x}) \otimes \big (W(P_{h^+_{e(t)}} x) \cdot (\nabla P_{I_v(t)})^T(x) \nabla h^+_{e(t)}(P_{I_v(t)} x)\big ) \nonumber \\&\qquad ~~~\mathrm{d}\nabla \chi _{\{{\text {dist}}^{\pm }(x,I_v(t))>h^+_{e(t)}(P_{I_v(t)}x)\}}(x)\bigg |\nonumber \\&\quad \leqq \int _{F\cap B_{2^{k+1}}(\tilde{x}){\setminus } B_{2^{k-1}}(\tilde{x})} \frac{{\text {dist}}(\tilde{x},F)}{|\tilde{x}-x|^d} \nonumber \\&\qquad \qquad \times |W(P_{h^+_{e(t)}} x) \cdot (\nabla P_{I_v(t)})^T(x)\nabla h^+_{e(t)}(P_{I_v(t)} x)| \,{\mathrm {d}}S(x)\nonumber \\&\qquad +\int _{F\cap B_{2^{k+1}}(\tilde{x})} C (2^k)^{2-d} |\nabla (W(P_{h^+_{e(t)}} x) \cdot (\nabla P_{I_v(t)})^T(x) \nabla h^+_{e(t)}(P_{I_v(t)}x))| \,{\mathrm {d}}S(x)\nonumber \\&\qquad +C \Vert \nabla ^2 h^+_{e(t)}\Vert _{L^\infty } (2^k)^{d+1} \nonumber \\&\qquad ~~~~\times \big \Vert \nabla \big (\theta _k(x-\tilde{x}) \otimes \big (W(P_{h^+_{e(t)}} x)\cdot (\nabla P_{I_v(t)})^T(x) \nabla h^+_{e(t)}(P_{I_v(t)} x)\big )\big )\big \Vert _{L^\infty }. \end{aligned}$$
(126)
Making use of the fact that the integral vanishes for \({\text {dist}}(\tilde{x},F)\geqq 2^{k+1}\) and the bounds (19) and (28), we obtain
$$\begin{aligned}&\int _{F\cap B_{2^{k+1}}(\tilde{x}){\setminus } B_{2^{k-1}}(\tilde{x})} \frac{{\text {dist}}(\tilde{x},F)}{|\tilde{x}-x|^d} |W(P_{h^+_{e(t)}} x) \cdot (\nabla P_{I_v(t)})^T(x) \nabla h^+_{e(t)}(P_{I_v(t)} x)| \,{\mathrm {d}}S(x)\nonumber \\&\quad \leqq \chi _{\{{\text {dist}}(\tilde{x},F)< 2^{k}\}}Cr_c^{-3}\Vert v\Vert _{W^{1,\infty }}\frac{{\text {dist}}(\tilde{x}, F)}{2^k}\nonumber \\&\qquad \times \int _{F\cap B_{2^{k+1}}(\tilde{x}){\setminus } B_{2^{k-1}}(\tilde{x})} \frac{|\nabla h^+_{e(t)}(P_{I_v(t)} x)|}{|\tilde{x} -x|^{d-1}} \,{\mathrm {d}}S(x). \end{aligned}$$
(127)
Using \(|\nabla h^+_{e(t)}|\leqq Cr_c^{-2}\) and \(|\nabla ^2 h^+_{e(t)}|\leqq Cr_c^{-4}e(t)^{-1}\) from Proposition 27 as well, we get
$$\begin{aligned}&\int _{F\cap B_{2^{k+1}}(\tilde{x})} C (2^k)^{2-d} |\nabla (W(P_{h^+_{e(t)}} x) \cdot (\nabla P_{I_v(t)})^T(x) \nabla h^+_{e(t)}(P_{I_v(t)}x))| \,{\mathrm {d}}S(x)\nonumber \\&\quad \leqq C 2^k \big (e(t)^{-1} r_c^{-5} \Vert v\Vert _{W^{1,\infty }}+ r_c^{-4}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))} \big ) \end{aligned}$$
(128)
and
$$\begin{aligned}&C \Vert \nabla ^2 h^+_{e(t)}\Vert _{L^\infty } (2^k)^{d+1}\nonumber \\&\qquad \times \big \Vert \nabla \big (\theta _k(x-\tilde{x}) \otimes \big (W(P_{h^+_{e(t)}} x)\cdot (\nabla P_{I_v(t)})^T(x) \nabla h^+_{e(t)}(P_{I_v(t)} x)\big )\big )\big \Vert _{L^\infty }\nonumber \\&\quad \leqq C r_c^{-4} e(t)^{-1} 2^k r_c^{-3} \Vert v\Vert _{W^{1,\infty }}\nonumber \\&\qquad + C r_c^{-4} e(t)^{-1} (2^k)^2 \big (e(t)^{-1} r_c^{-5} \Vert v\Vert _{W^{1,\infty }}+ r_c^{-4}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))} \big ). \end{aligned}$$
(129)
Using (126), (127), (128) and (129) to estimate the term in (125), we get
$$\begin{aligned} III&\leqq C\frac{\Vert v\Vert _{W^{1,\infty }}}{r_c^3}\sum _{k=-\infty }^{\lfloor \log e^2(t) \rfloor -1} \chi _{\{{\text {dist}}(\tilde{x},F)< 2^{k}\}}\frac{{\text {dist}}(\tilde{x}, F)}{2^k} \nonumber \\&\qquad \qquad \qquad \qquad \qquad \quad \qquad \times \int _{F\cap B_{2^{k+1}}(\tilde{x}){\setminus } B_{2^{k-1}}(\tilde{x})}\frac{|\nabla h^+_{e(t)}(P_{I_v(t)} x)|}{|\tilde{x} -x|^{d-1}} \,{\mathrm {d}}S(x)\nonumber \\&\quad +Cr_c^{-9}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}e(t). \end{aligned}$$
(130)
In turn, combining this with (123) and (124) and also gathering (120), (121) and (117), as well as the corresponding bounds for \(\nabla w^-\) and \(\nabla (\theta *\nabla \cdot w^-)\), we then finally deduce (99).
Step 4:\(L^2L^\infty \)-estimate for\(\nabla w\). By making use of the precise formula (106) for \(\nabla w^+\) and the definition of the vector field W in (97), we immediately get
$$\begin{aligned}&\int _{I_v(t)}\sup _{y\in [-r_c,r_c]}|(\nabla w^+)^T(x+y\mathrm {n}_v(x,t))\cdot \mathrm {n}_v(x,t)|^2\,{\mathrm {d}}S(x)\nonumber \\&\quad \leqq Cr_c^{-2}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}\int _{I_v(t)} |h^+_{e(t)}|^2+|\nabla h^+_{e(t)}|^2\,{\mathrm {d}}S. \end{aligned}$$
(131)
To estimate the contribution from \(|\nabla (\theta *(\nabla \cdot w^+))|\) we use the same dyadic decomposition as in (119). We start with the terms in the range \(k=\lfloor \log e^2(t)\rfloor ,\ldots ,0\).
Let \(x\in I_v(t)\) and \(y\in (-r_c,r_c)\) be fixed. We abbreviate \(\bar{x}:=x+y\mathrm {n}_v(x,t)\). Denote by \(F_x\) the tangent plane of the interface \(I_v(t)\) at the point x. Let \(\Phi _{F_x}:F_x\times \mathbb {R}\rightarrow {\mathbb {R}^d}\) be the diffeomorphism given by \(\Phi _{F_x}(\hat{x},\hat{y}):=\hat{x}+\hat{y}\mathrm {n}_{F_x}(\hat{x})\). We start estimating using the change of variables \(\Phi _{F_x}\), the bound \(|\nabla \theta _k(x)|\leqq C\chi _{2^{k-1}\leqq |x|\leqq 2^{k+1}}|x|^{-d}\), as well as the fact that \(\hat{x} + y\mathrm {n}_{F_x}(\hat{x})=\hat{x} + y\mathrm {n}_v(x,t)\) is exactly the point on the ray originating from \(\hat{x}\in F_x\) in normal direction which is closest to \(\bar{x}\):
$$\begin{aligned}&|\big (\nabla (\theta _k *(\nabla \cdot w^+))\big )^T(x+y\mathrm {n}_v(x,t))|\\&\quad \leqq \int _{(B_{2^{k+1}}(\bar{x}){\setminus } B_{2^{k-1}}(\bar{x}))\cap (I_v(t)+B_{r_c/2})} |\nabla \theta _k(\bar{x}{-}\tilde{x})||(\nabla \cdot w^+)(\tilde{x})| \,\mathrm{d}\tilde{x}\\&\quad \leqq C\int _{F_x\cap (B_{2^{k+1}}(x){\setminus } B_{2^{k-1}}(x))} \sup _{\hat{y}\in [-r_c,r_c]} \frac{|(\nabla \cdot w^+)(\hat{x} {+} \hat{y}\mathrm {n}_{F_x}(\hat{x}))|}{|x-\hat{x}|^{d-1}}\,{\mathrm {d}}S(\hat{x}). \end{aligned}$$
Note that the right hand side is independent of y. Hence, we may estimate, with Minkowski’s inequality,
$$\begin{aligned}&\bigg (\int _{I_v(t)}\sup _{y\in [-r_c,r_c]}\bigg |\sum _{k=\lfloor \log e^2(t)\rfloor -1}^0 \nabla (\theta _k *(\nabla \cdot w^+))(x+y\mathrm {n}_v(x,t))\bigg |^2\,{\mathrm {d}}S(x)\bigg )^\frac{1}{2}\\&\quad \leqq C|\log e(t)| \bigg (\int _{I_v(t)}\bigg |\int _{F_x} \sup _{\hat{y}\in [-r_c,r_c]} \frac{|(\nabla \cdot w^+)(\hat{x} {+} \hat{y}\mathrm {n}_{F_x}(\hat{x}))|}{|x-\hat{x}|^{d-1}}\,{\mathrm {d}}S(\hat{x})\bigg |^2\,{\mathrm {d}}S(x)\bigg )^\frac{1}{2}. \end{aligned}$$
The inner integral is to be understood in the Cauchy principal value sense. To proceed we use the \(L^2\)-theory for singular operators of convolution type, the precise formula (108) for \(\nabla \cdot w^+\) as well as (19) and (28) which entails
$$\begin{aligned}&\bigg (\int _{I_v(t)}\bigg |\int _{F_x} \sup _{\hat{y}\in [-r_c,r_c]} \frac{|(\nabla \cdot w^+)(\hat{x} {+} \hat{y}\mathrm {n}_{F_x}(\hat{x}))|}{|x-\hat{x}|^{d-1}}\,{\mathrm {d}}S(\hat{x})\bigg |^2\,{\mathrm {d}}S(x)\bigg )^\frac{1}{2}\\&\quad \leqq C\bigg (\int _{I_v(t)}\sup _{y\in [-r_c,r_c]}|(\nabla \cdot w^+)(x{+}y\mathrm {n}_v(x,t))|^2 {\mathrm {d}}S(x)\bigg )^\frac{1}{2}\\&\quad \leqq Cr_c^{-1}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}^\frac{1}{2} \bigg (\int _{I_v(t)} |h^+_{e(t)}|^2 + |\nabla h^+_{e(t)}|^2 \,{\mathrm {d}}S\bigg )^\frac{1}{2}. \end{aligned}$$
An application of (78a) and the assumption \(E[\chi _u,u,V|\chi _v,v](t)\leqq e^2(t)\) finally yields
$$\begin{aligned}&\bigg (\int _{I_v(t)}\sup _{y\in [-r_c,r_c]}\bigg |\sum _{k=\lfloor \log e^2(t)\rfloor -1}^0 \nabla (\theta _k *(\nabla \cdot w^+))(x+y\mathrm {n}_v(x,t))\bigg |^2\,{\mathrm {d}}S(x)\bigg )^{1/2}\nonumber \\&\quad \leqq Cr_c^{-5}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}|\log e(t)|e(t). \end{aligned}$$
(132)
We move on with the contributions in the range \(k=1,\ldots ,\infty \). Note that by (121) we may directly infer from (78a) and the assumption \(E[\chi _u,u,V|\chi _v,v](t)\leqq e^2(t)\)
$$\begin{aligned}&\int _{I_v(t)}\sup _{y\in [-r_c,r_c]}\Big |\sum _{k=1}^\infty \big (\nabla (\theta _k *(\nabla \cdot w^+))\big )^T(x+y\mathrm {n}_v(x,t)) \cdot \mathrm {n}_v(x,t)\Big |^2\,{\mathrm {d}}S(x)\nonumber \\&\quad \leqq Cr_c^{-8}\Vert v\Vert ^2_{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}\mathcal {H}^{d-1}(I_v(t))^2 e^2(t). \end{aligned}$$
(133)
Moreover, the contributions estimated in (123) and (124) result in a bound of the form (recall that \(e(t)<r_c\))
$$\begin{aligned} Cr_c^{-4}\Vert v\Vert ^2_{W^{3,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}e^2(t) + Cr_c^{-8}\Vert v\Vert _{W^{1,\infty }}^2 e^2(t). \end{aligned}$$
(134)
Note that when summing the respective bounds from (128) and (129) over the relevant range \(k=-\infty ,\ldots ,\lfloor \log e^2(t) \rfloor - 1\), we actually gain a factor e(t), that is, the contributions estimated in (128) and (129) then directly yield a bound of the form
$$\begin{aligned} Cr_c^{-18}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}^2e^2(t). \end{aligned}$$
(135)
Finally, the contribution from (127) may be estimated as follows. Let \(x\in I_v(t)\), \(y\in [-r_c,r_c]\) and denote by \(F_{\bar{x}}\) the tangent plane to the manifold \(\{{\text {dist}}^{\pm }(x,I_v(t))=h^+_{e(t)}(P_{I_v(t)}x)\}\) at the nearest point to \(\bar{x} = x+y\mathrm {n}_v(x,t)\). In light of (127), we start estimating for \(k \leqq \lfloor \log e^2(t) \rfloor - 1\) by using Jensen’s inequality, the bound \(|\nabla h^+_{e(t)}|\leqq Cr_c^{-2}\) from Proposition 27, as well as the fact that \(|\bar{x} - \tilde{x}|\geqq |x-\tilde{x}|\) for all \(\tilde{x}\in I_v(t)\) (since \(x=P_{I_v(t)}\bar{x}\) is the closest point to \(\bar{x}\) on the interface \(I_v(t)\))
$$\begin{aligned}&\bigg |\int _{F_{\bar{x}}\cap B_{2^{k+1}}(\bar{x}){\setminus } B_{2^{k-1}}(\bar{x})} \frac{|\nabla h^+_{e(t)}(P_{I_v(t)} \tilde{x})|}{|\bar{x} - \tilde{x}|^{d-1}} \,{\mathrm {d}}S(\tilde{x})\bigg |^2\\&\quad \leqq \int _{F_{\bar{x}}\cap B_{2^{k+1}}(\bar{x}){\setminus } B_{2^{k-1}}(\bar{x})} \frac{|\nabla h^+_{e(t)}(P_{I_v(t)} \tilde{x})|^2}{|\bar{x} - \tilde{x}|^{d-1}} \,{\mathrm {d}}S(\tilde{x})\\&\quad \leqq Cr_c^{-2(d-1)}\int _{I_v(t)\cap B_{Cr_c^{-2}2^{k+1}}(x)} \frac{|\nabla h^+_{e(t)}(\tilde{x})|^2}{|x - \tilde{x}|^{d-1}} \,{\mathrm {d}}S(\tilde{x}). \end{aligned}$$
Since this bound does not depend anymore on \(y\in [-r_c,r_c]\), we may estimate the contributions from (127) using Minkowski’s inequality as well as once more the \(L^2\)-theory for singular operators of convolution type to reduce everything to the \(H^1\)-bound (78a) for the local interface error heights. All in all, the contributions from (127) are therefore bounded by
$$\begin{aligned} Cr_c^{-14}\Vert v\Vert _{W^{1,\infty }}^2e^2(t). \end{aligned}$$
(136)
The asserted bound (100) then finally follows from collecting the estimates (131), (132), (133), (134), (135) and (136) together with the analogous bounds for \(\nabla w^-\) and \(\nabla (\theta *\nabla \cdot w^-)\).
Step 5: Estimate on the time derivative\(\partial _t w\). To estimate \(\partial _t w^+\), we first deduce using (107), \(|\partial _t \eta |\leqq C r_c^{-1} \Vert v\Vert _{L^\infty }\), \(|\frac{\mathrm {d}}{{\mathrm {d}}t}\mathrm {n}_v(P_{I_v(t)}x)|\leqq \frac{C}{r_c^2}\Vert v\Vert _{W^{1,\infty }}\) (which follows from (34)), (21) and finally (72) that
$$\begin{aligned}&\partial _t w^+(x,t) \\&\quad =\chi _{0\leqq {\text {dist}}^{\pm }(x,I_v)\leqq h^+_{e(t)}(P_{I_v(t)} x)} W (x) \partial _t {\text {dist}}^{\pm }(x,I_v(t))\\&\qquad +\eta \, \chi _{{\text {dist}}^{\pm }(x,I_v)> h^+_{e(t)}(P_{I_v(t)} x)} W (P_{h^+_{e(t)}} x) \\&\qquad \qquad \times \big (\partial _t h^+_{e(t)}(P_{I_v(t)}x)+\partial _t P_{I_v(t)}x \cdot \nabla h^+_{e(t)}(P_{I_v(t)}x)\big )\\&\qquad +\tilde{g}^+ \end{aligned}$$
for some vector field \(\tilde{g}^+\) subject to \(\Vert \tilde{g}^+(\cdot ,t)\Vert _{L^2}\leqq Cr_c^{-2}(1+\Vert v\Vert _{W^{1,\infty }}) (\Vert v\Vert _{W^{1,\infty }} +\Vert \partial _t \nabla v\Vert _{L^\infty ({\mathbb {R}^d}{\setminus } I_v(t))} +\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}) (\int _{I_v(t)} |h^+_{e(t)}(\cdot ,t)|^2 \,{\mathrm {d}}S)^{1/2}\). Using (106), (21) as well as (72) we may compute
$$\begin{aligned}&(v(x)\cdot \nabla )w^+(x,t)\\&\qquad + \chi _{0\leqq {\text {dist}}^{\pm }(x,I_v(t))\leqq h^+_{e(t)}(P_{I_v(t)} x)} W (x) \partial _t {\text {dist}}^{\pm }(x,I_v(t))\\&\qquad +\eta \, \chi _{{\text {dist}}^{\pm }(x,I_v(t))> h^+_{e(t)}(P_{I_v(t)} x)} W (P_{h^+_{e(t)}} x) \partial _t P_{I_v(t)}x \cdot \nabla h^+_{e(t)}(P_{I_v(t)}x)\\&\quad =\eta \, \chi _{{\text {dist}}^{\pm }(x,I_v(t))>h^+_{e(t)}(P_{I_v(t)} x)} W(P_{h^+_{e(t)}} x) (\mathrm {Id}{-}\mathrm {n}_v\otimes \mathrm {n}_v)v(P_{I_v(t)}x) \cdot \nabla h^+_{e(t)}(P_{I_v(t)}x)\\&\qquad +\chi _{0\leqq {\text {dist}}^{\pm }(x,I_v(t))\leqq h^+_{e(t)}(P_{I_v(t)} x)} W(x) \big ((v(x)-v(P_{I_v(t)}x)\big )\cdot \mathrm {n}_v(P_{I_v(t)})\\&\qquad + \eta \, \chi _{{\text {dist}}^{\pm }(x,I_v(t))>h^+_{e(t)}(P_{I_v(t)} x)} W(P_{h^+_{e(t)}} x) \\&\qquad \qquad \cdot \big (\nabla P_{I_v(t)}(x)v(x)-v(P_{I_v(t)}x)\big )\cdot \nabla h^+_{e(t)}(P_{I_v(t)}x)\\&\qquad + \tilde{g}_1^+, \end{aligned}$$
for some \(\Vert \tilde{g}_1^+\Vert _{L^2}\leqq Cr_c^{-2}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}(\int _{I_v(t)} |h^+_{e(t)}(\cdot ,t)|^2 + |\nabla h^+_{e(t)}(\cdot ,t)|^2 \,{\mathrm {d}}S)^{\frac{1}{2}}\). This computation in turn implies
$$\begin{aligned}&\partial _t w^+(x,t)\nonumber \\&\quad =-(v(x)\cdot \nabla )w^+(x,t)\nonumber \\&\qquad +\eta \, \chi _{{\text {dist}}^{\pm }(x,I_v(t))> h^+_{e(t)}(P_{I_v(t)} x)} W (P_{h^+_{e(t)}} x) \nonumber \\&\qquad \qquad \times \big (\partial _t h^+_{e(t)}(P_{I_v(t)}x)+(\mathrm {Id}{-}\mathrm {n}_v\otimes \mathrm {n}_v)v(P_{I_v(t)}x) \cdot \nabla h^+_{e(t)}(P_{I_v(t)}x)\big )\nonumber \\&\qquad +g^+ \end{aligned}$$
(137)
for some \(g^+\) with
$$\begin{aligned}&\Vert g^+\Vert _{L^2}\\&\quad \leqq C r_c^{-2} (1{+}\Vert v\Vert _{W^{1,\infty }}) (\Vert \partial _t \nabla v\Vert _{L^\infty ({\mathbb {R}^d}{\setminus } I_v(t))}{+}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))})\\&\qquad \times \bigg (\int _{I_v(t)} |h^+_{e(t)}|^2 {+} |\nabla h^+_{e(t)}|^2 \,{\mathrm {d}}S\bigg )^{\frac{1}{2}}. \end{aligned}$$
We now aim to make use of (78d) to further estimate the second term in the right hand side of (137). To establish the corresponding \(L^2\)- resp. \(L^\frac{4}{3}\)-contributions, we first need to perform an integration by parts in order to use (78d). The resulting curvature term as well as all other terms which do not appear in the third term of (137) can be directly bounded by a term whose associated \(L^2\)-norm is controlled by \(Cr_c^{-1}\Vert v\Vert _{W^{1,\infty }} \Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}(\int _{I_v(t)} |h^+_{e(t)}(\cdot ,t)|^2 {+} |\nabla h^+_{e(t)}(\cdot ,t)|^2 \,{\mathrm {d}}S)^{\frac{1}{2}}\). Hence, using (78d) in (137) implies
$$\begin{aligned} \partial _t w^+(x,t)&=-(v\cdot \nabla )w^+(x,t) +\bar{g}^+ +\hat{g}^+ \end{aligned}$$
(138)
with the corresponding \(L^2\)-bound
$$\begin{aligned}&\Vert \bar{g}^+\Vert _{L^2({\mathbb {R}^d})}\nonumber \\&\quad \leqq C \frac{1{+}\Vert v\Vert _{W^{1,\infty }}}{r_c^2} (\Vert \partial _t \nabla v\Vert _{L^\infty ({\mathbb {R}^d}{\setminus } I_v(t))}{+}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t)})\nonumber \\&\qquad ~~~~ \times \bigg (\int _{I_v(t)} |h^+_{e(t)}|^2 {+} |\nabla h^+_{e(t)}|^2 \,{\mathrm {d}}S\bigg )^\frac{1}{2}\nonumber \\&\qquad +C\frac{\Vert v\Vert _{W^{1,\infty }}(1+\Vert v\Vert _{W^{1,\infty }})}{e(t)r_c} \int _{{\mathbb {R}^d}} 1-\xi \cdot \frac{\nabla \chi _u}{|\nabla \chi _u|} \,\mathrm{d}|\nabla \chi _u|\nonumber \\&\qquad +Cr_c^{-2}\Vert v\Vert _{W^{1,\infty }}^2(1+e'(t)) \big (\Vert h^\pm (\cdot ,t)\Vert _{L^2(I_v(t))} +\Vert \nabla h^\pm _{e(t)}(\cdot ,t)\Vert _{L^2(I_v(t))}\big )\nonumber \\&\qquad +C\frac{\Vert v\Vert _{W^{1,\infty }}(1{+}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))})}{r_c} \bigg (\int _{{\mathbb {R}^d}}|\chi _u{-}\chi _v| \min \Big \{\frac{{\text {dist}}(x,I_v(t))}{r_c},1\Big \} \,{\mathrm {d}}x\bigg )^\frac{1}{2}\nonumber \\&\qquad +C\Vert v\Vert _{W^{1,\infty }}\bigg (\int _{I_v(t)}|u-v|^2\,{\mathrm {d}}S\bigg )^\frac{1}{2} \end{aligned}$$
(139)
and \(L^\frac{4}{3}\)-estimate
$$\begin{aligned}&\Vert \hat{g}^+\Vert _{L^{\frac{4}{3}}({\mathbb {R}^d})}\nonumber \\&\quad \leqq C\frac{\Vert v\Vert _{W^{1,\infty }}}{e(t)r_c^2}\bigg (\int _{I_v(t)} |\bar{h}^\pm |^4 \,{\mathrm {d}}S\bigg )^\frac{1}{4}\nonumber \\&\qquad ~~~~\times \bigg (\int _{I_v(t)} \sup _{y\in [-r_c,r_c]} |u{-}v|^2(x{+}y\mathrm {n}_v(x,t),t) \,{\mathrm {d}}S(x)\bigg )^\frac{1}{2}\nonumber \\&\qquad +C\frac{\Vert v\Vert _{W^{1,\infty }}(1+\Vert v\Vert _{W^{1,\infty }})}{e(t)} \int _{{\mathbb {R}^d}} 1-\xi \cdot \frac{\nabla \chi _u}{|\nabla \chi _u|} \,\mathrm{d}|\nabla \chi _u|. \end{aligned}$$
(140)
In both bounds, we add and subtract the compensation function w and therefore obtain together with (98) and (42)
$$\begin{aligned} \int _{I_v(t)}|u-v|^2\,{\mathrm {d}}S&\leqq \int _{I_v(t)} \sup _{y\in [-r_c,r_c]} |u{-}v|^2(x{+}y\mathrm {n}_v(x,t),t) \,{\mathrm {d}}S(x)\nonumber \\&\leqq \int _{I_v(t)} \sup _{y\in [-r_c,r_c]} |u-v-w|^2(x+y\mathrm {n}_v(x,t),t) \,{\mathrm {d}}S(x)\nonumber \\&\quad +\int _{I_v(t)} \sup _{y\in [-r_c,r_c]} |w(x+y\mathrm {n}_v(x,t),t)|^2\,{\mathrm {d}}S(x)\nonumber \\&\leqq C r_c^{-4}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}^2 \int _{I_v(t)} |h^\pm _{e(t)}|^2 + |\nabla h^\pm _{e(t)}|^2 \,{\mathrm {d}}S\nonumber \\&\quad +C(\Vert u{-}v{-}w\Vert _{L^2}\Vert \nabla (u{-}v{-}w)\Vert _{L^2}+\Vert u{-}v{-}w\Vert _{L^2}^2). \end{aligned}$$
(141)
Analogous estimates may be derived for \(w^-\). We therefore proceed with the terms related to \(\theta *\nabla \cdot w^\pm \). First of all, note that the singular integral operator \((\theta *\nabla \cdot )\) satisfies (see Theorem 38)
$$\begin{aligned} \Vert \theta *\nabla \cdot \hat{g}\Vert _{L^{\frac{4}{3}}({\mathbb {R}^d})} \leqq C \Vert \hat{g}\Vert _{L^{\frac{4}{3}}({\mathbb {R}^d})}, \quad \Vert \theta *\nabla \cdot \bar{g}\Vert _{L^{2}({\mathbb {R}^d})} \leqq C \Vert \bar{g}\Vert _{L^{2}({\mathbb {R}^d})}. \end{aligned}$$
(142)
Furthermore, to estimate \(\Vert \theta *\nabla \cdot ((v\cdot \nabla )w^+) - (v\cdot \nabla ) (\theta *\nabla \cdot w^+)\Vert _{L^{2}({\mathbb {R}^d})}\) we first replace v with its normal velocity \(V_{\mathrm {n}}(x):=(v(x)\cdot \mathrm {n}_v(P_{I_v(t)}x))\mathrm {n}_v(P_{I_v(t)}x)\). We want to exploit the fact that the vector field \(V_{\mathrm {n}}\) has bounded derivatives up to second order, see (43) and (44). Moreover, the kernel \(\nabla ^2 \theta (x-\tilde{x}) \otimes (\tilde{x}-x)\) gives rise to a singular integral operator of convolution type, as does \(\nabla \theta \). To see this, we need to check whether its average over \({\mathbb {S}^{d-1}}\) vanishes. We write \(x\otimes \nabla ^2\theta (x) = \nabla F(x)-\delta _{ij}e_i\otimes \nabla \theta \otimes e_j\), where \(F(x) = x\otimes \nabla \theta (x)\). Now, since \(\nabla \theta \) is homogeneous of degree \(-d\), F itself is homogeneous of degree \(-(d-1)\). Hence, we compute \(\int _{B_1{\setminus } B_r}\nabla F\,{\mathrm {d}}x=\int _{{\mathbb {S}^{d-1}}}\mathrm {n}\otimes F\,{\mathrm {d}}S- \int _{r{\mathbb {S}^{d-1}}}\mathrm {n}\otimes F\,{\mathrm {d}}S= 0\) for every \(0<r<1\). Passing to the limit \(r\rightarrow 1\) shows that \(\nabla F\), and therefore also \(\nabla ^2\theta (x)\otimes x\), have vanishing average on \({\mathbb {S}^{d-1}}\). We may now compute (where the integrals are well defined in the Cauchy principal value sense due to the above considerations) for almost every \(x\in {\mathbb {R}^d}\)
$$\begin{aligned}&\int _{\mathbb {R}^d}\nabla \theta (x-\tilde{x}) \cdot (V_{\mathrm {n}}(\tilde{x},t)\cdot \nabla _{\tilde{x}}) w^+(\tilde{x},t)- (V_{\mathrm {n}}(x,t)\cdot \nabla _x) \nabla \theta (x-\tilde{x}) \cdot w^+(\tilde{x},t) \,\mathrm{d}\tilde{x}\\&\quad =\int _{\mathbb {R}^d}\nabla \theta (x-\tilde{x}) ((V_{\mathrm {n}}(\tilde{x},t)-V_{\mathrm {n}}(x,t)) \cdot \nabla _{\tilde{x}}) w^+(\tilde{x},t) \,\mathrm{d}\tilde{x}\\&\quad =\int _{\mathbb {R}^d}\nabla ^2 \theta (x-\tilde{x}) : (V_{\mathrm {n}}(\tilde{x},t)-V_{\mathrm {n}}(x,t)-(\tilde{x}-x)\cdot \nabla V_{\mathrm {n}}(\tilde{x},t)) \otimes w^+(\tilde{x},t) \,\mathrm{d}\tilde{x}\\&\qquad -\int _{\mathbb {R}^d}\nabla \theta (x-\tilde{x}) \cdot (\nabla \cdot V_{\mathrm {n}})(\tilde{x},t) w^+(\tilde{x},t) \,\mathrm{d}\tilde{x}\\&\qquad +\int _{\mathbb {R}^d}\nabla ^2 \theta (x-\tilde{x}) : ((\tilde{x}-x) \cdot \nabla ) V_{\mathrm {n}}(\tilde{x},t) \otimes \,w^+(\tilde{x},t) \,\mathrm{d}\tilde{x}. \end{aligned}$$
Note that we have \(|V_{\mathrm {n}}(\tilde{x},t)-V_{\mathrm {n}}(x,t)-(\tilde{x}-x)\cdot \nabla V_{\mathrm {n}}(x,t)|\leqq \Vert \nabla ^2 V_{\mathrm {n}}\Vert _{L^\infty } |\tilde{x}-x|^2\) and \(|V_{\mathrm {n}}(\tilde{x},t)-V_{\mathrm {n}}(x,t)-(\tilde{x}-x)\cdot \nabla V_{\mathrm {n}}(x,t)|\leqq ||\nabla V_{\mathrm n}||_{L^\infty } |\tilde{x}-x|\). We then estimate using Young’s inequality for convolutions and \(|\nabla ^2 \theta (x)|\leqq |x|^{-d-1}\)
$$\begin{aligned}&\int _{{\mathbb {R}^d}{\setminus } B_{3R}(0)}\bigg |\int _{B_R(0)} \nabla ^2 \theta (x-\tilde{x}) \nonumber \\&~~~~~~~~~~~~~~~~~~~~~~~~\qquad :(V_{\mathrm {n}}(\tilde{x})-V_{\mathrm {n}}(x)-(\tilde{x}-x)\cdot \nabla V_{\mathrm {n}}(\tilde{x})) \otimes w^+(\tilde{x}) \,\mathrm{d}\tilde{x}\bigg |^2{\mathrm {d}}x\nonumber \\&\quad \leqq C\Vert \nabla V_{\mathrm {n}}\Vert _{L^\infty }^2 \int _{{\mathbb {R}^d}{\setminus } B_{3R}(0)}\bigg |\int _{B_R(0)} \frac{1}{|x-\tilde{x}|^{d}} |w^+(\tilde{x})| \,\mathrm{d}\tilde{x}\bigg |^2{\mathrm {d}}x\nonumber \\&\quad \leqq C\Vert \nabla V_{\mathrm {n}}\Vert _{L^\infty }^2 ||\,|\cdot |^{-d}\,||_{L^2({\mathbb {R}^d}{\setminus } B_R)}^2 \bigg |\int _{B_R(0)}|w^+| \,{\mathrm {d}}x\bigg |^2\nonumber \\&\quad \leqq C R^{-d} R^d \int _{B_R(0)}|w^+|^2 \,{\mathrm {d}}x. \end{aligned}$$
(143)
As a consequence, we obtain from (143), Young’s inequality for convolutions, (114) and (44) that
$$\begin{aligned}&\int _{{\mathbb {R}^d}}\bigg |\int _{{\mathbb {R}^d}} \nabla ^2 \theta (x-\tilde{x}) : (V_{\mathrm {n}}(\tilde{x})-V_{\mathrm {n}}(x)-(\tilde{x}-x)\cdot \nabla V_{\mathrm {n}}(\tilde{x})) \otimes w^+(\tilde{x}) \,\mathrm{d}\tilde{x}\bigg |^2{\mathrm {d}}x\nonumber \\&\quad \leqq C\Vert \nabla ^2 V_{\mathrm {n}}\Vert _{L^\infty }^2 \int _{B_{3R}(0)}\bigg |\int _{{\mathbb {R}^d}}\frac{|w^+(\tilde{x})|}{|x-\tilde{x}|^{d-1}}\,\mathrm{d}\tilde{x}\bigg |^2\,{\mathrm {d}}x+C\Vert \nabla V_{\mathrm {n}}\Vert _{L^\infty }^2 \int _{B_R(0)}|w^+|^2\,{\mathrm {d}}x\nonumber \\&\quad \leqq Cr_c^{-4}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}^2(1{+}R^2)\int _{I_v(t)} |h^+_{e(t)}|^2 \,{\mathrm {d}}S. \end{aligned}$$
(144)
Applying Theorem 38 to the singular integral operators \(\nabla \theta \) resp. \(\nabla ^2\theta \otimes x\) as well as making use of (43), (114) and (144) we then obtain the estimate
$$\begin{aligned}&\int _{\mathbb {R}^d}|\theta *\nabla \cdot ((V_{\mathrm {n}}\cdot \nabla )w^+) -(V_{\mathrm {n}}\cdot \nabla )(\theta *\nabla \cdot w^+)|^2 \,{\mathrm {d}}x\nonumber \\&\quad \leqq Cr_c^{-4}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}^2(1{+}R^2)\int _{I_v(t)} |h^+_{e(t)}|^2 \,{\mathrm {d}}S\nonumber \\&\qquad +C\Vert \nabla V_{\mathrm {n}}\Vert _{L^\infty }^2 \int _{{\mathbb {R}^d}}|w^+|^2\,{\mathrm {d}}x\nonumber \\&\quad \leqq Cr_c^{-4}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}^2(1{+}R^2)\int _{I_v(t)} |h^+_{e(t)}|^2 \,{\mathrm {d}}S. \end{aligned}$$
(145)
It remains to estimate \(\Vert \theta *\nabla \cdot ((V_{\mathrm {tan}}\cdot \nabla )w^+) - (V_{\mathrm {tan}}\cdot \nabla ) (\theta *\nabla \cdot w^+)\Vert _{L^{2}({\mathbb {R}^d})}\) with \(V_{\mathrm {tan}}(x)=(\mathrm {Id}-\mathrm {n}_v(P_{I_v(t)}x)\otimes \mathrm {n}_v(P_{I_v(t)}x))v(x)\) denoting the tangential velocity of v. To this end, note that we may rewrite
$$\begin{aligned}&\int _{\mathbb {R}^d}\nabla \theta (x-\tilde{x}) \cdot (V_{\mathrm {tan}}(\tilde{x},t)\cdot \nabla _{\tilde{x}}) w^+(\tilde{x},t) - (\nabla \cdot w^+(\tilde{x},t))\\&\qquad ~~~~~~~~ (V_{\mathrm {tan}}(x,t)\cdot \nabla _x)\theta (x-\tilde{x}) \,\mathrm{d}\tilde{x}\\&\quad =\int _{\mathbb {R}^d}\nabla \theta (x{-}\tilde{x}) \\&\qquad ~~~~~~~~~~ \big (\nabla w^+(\tilde{x}) {-} \chi _{0\leqq {\text {dist}}^{\pm }(\tilde{x},I_v(t))\leqq h^+_{e(t)}(P_{I_v(t)} \tilde{x})}W(\tilde{x}) \otimes \mathrm {n}_v(P_{I_v(t)}\tilde{x})\big )V_{\mathrm {tan}}(\tilde{x},t)\,\mathrm{d}\tilde{x}\\&\qquad -\int _{{\mathbb {R}^d}}(\nabla \cdot w^+(\tilde{x},t))(V_{\mathrm {tan}}(x,t)\cdot \nabla _x) \theta (x-\tilde{x}) \,\mathrm{d}\tilde{x}. \end{aligned}$$
Using Theorem 38, (111) as well as (112) we then obtain
$$\begin{aligned}&\Vert \theta *\nabla \cdot ((V_{\mathrm {tan}}\cdot \nabla )w^+) - (V_{\mathrm {tan}}\cdot \nabla ) (\theta *\nabla \cdot w^+)\Vert _{L^{2}({\mathbb {R}^d})}^2\nonumber \\&\leqq Cr_c^{-4}\Vert v\Vert _{L^\infty }^2\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}^2 \int _{I_v(t)} |h^+_{e(t)}|^2 + |\nabla h^+_{e(t)}|^2 \,{\mathrm {d}}S. \end{aligned}$$
(146)
Putting the estimates (139), (140), (141), (142), (145) and (146) together, we get
$$\begin{aligned} \partial _t w(x,t)+(v\cdot \nabla )w(x,t) = g + \hat{g} \end{aligned}$$
with the asserted bounds. This concludes the proof. \(\quad \square \)
Estimate for the Additional Surface Tension Terms
Having established all the relevant properties of the compensating vector field w in Proposition 28, we can now estimate the additional terms in the relative entropy inequality from Proposition 10. To this end, we start with the additional surface tension terms given by
$$\begin{aligned} A_{surTen}&= -\sigma \int _0^T\int _{\mathbb {R}^d\times \mathbb {S}^{d-1}}(s{-}\xi ) \cdot \big ((s{-}\xi )\cdot \nabla \big ) w \,\mathrm{d}V_t(x,s)\,{\mathrm {d}}t\nonumber \\&\quad +\sigma \int _0^T\int _{\mathbb {R}^d}(1-\theta _t)\, \xi \cdot (\xi \cdot \nabla )w\,\mathrm{d}|V_t|_{\mathbb {S}^{d-1}}(x)\,{\mathrm {d}}t\nonumber \\&\quad +\sigma \int _0^T\int _{\mathbb {R}^d}(\chi _u-\chi _v)(w\cdot \nabla )(\nabla \cdot \xi )\,{\mathrm {d}}x\,{\mathrm {d}}t\nonumber \\&\quad +\sigma \int _0^T\int _{\mathbb {R}^d}(\chi _u-\chi _v) \nabla w :\nabla \xi ^T \,{\mathrm {d}}x\,{\mathrm {d}}t\nonumber \\&\quad -\sigma \int _0^T\int _{\mathbb {R}^d}\xi \cdot \big ((\mathrm {n}_u-\xi ) \cdot \nabla \big ) w \,\mathrm{d}|\nabla \chi _u|\,{\mathrm {d}}t\nonumber \\&=: I + II + III + IV + V. \end{aligned}$$
(147)
A precise estimate for these terms is the content of the following result:
Lemma 29
Let the assumptions and notation of Proposition 28 be in place. In particular, we assume that there exists a \(C^1\)-function \(e:[0,{T_{strong}})\rightarrow [0,r_c)\) such that the relative entropy is bounded by \(E[\chi _u,u,V,|\chi _v,v](t)\leqq e^2(t)\). Then the additional surface tension terms \(A_{surTen}\) are bounded by a Gronwall-type term
$$\begin{aligned} A_{surTen}&\leqq \frac{C}{r_c^{10}}(1+\Vert v\Vert _{L^\infty _tW^{2,\infty }_x({\mathbb {R}^d}{\setminus } I_v(t))}^2 +\Vert v\Vert _{L^\infty _tW^{3,\infty }_x({\mathbb {R}^d}{\setminus } I_v(t))})\nonumber \\&\quad \quad \quad \quad ~~\times \int _0^T(1+|\log e(t)|)E[\chi _u,u,V|\chi _v,v](t)\,\mathrm{d}t\nonumber \\&\quad +\frac{C}{r_c^{10}}(1+\Vert v\Vert _{L^\infty _tW^{2,\infty }_x({\mathbb {R}^d}{\setminus } I_v(t))}^2 +\Vert v\Vert _{L^\infty _tW^{3,\infty }_x({\mathbb {R}^d}{\setminus } I_v(t))})\nonumber \\&\quad \quad \quad \quad ~~\times \int _0^T (1+|\log e(t)|)e(t)E[\chi _u,u,V|\chi _v,v]^\frac{1}{2}(t)\,\mathrm{d}t. \end{aligned}$$
(148)
Proof
We estimate term by term in (147). A straightforward estimate for the first two terms using also the coercivity property (39) yields
$$\begin{aligned} I + II&\leqq C\int _0^T\Vert \nabla w(t)\Vert _{L^\infty _{x}}\int _{\mathbb {R}^d\times \mathbb {S}^{d-1}}|s-\xi |^2\,\mathrm{d}V_t(x,s)\,{\mathrm {d}}t\nonumber \\&\quad +C\int _0^T\Vert \nabla w(t)\Vert _{L^\infty _{x}}\int _{\mathbb {R}^d}(1-\theta _t)\,\mathrm{d}|V_t|_{\mathbb {S}^{d-1}}(x)\,{\mathrm {d}}t\nonumber \\&\leqq C\int _0^T\Vert \nabla w(t)\Vert _{L^\infty _{x}}E[\chi _u,u,V|\chi _v,v](t)\,\mathrm{d}t. \end{aligned}$$
(149)
Making use of (19), a change of variables \(\Phi _t\), Hölder’s and Young’s inequality, (98), (41), (78a) as well as the coercivity property (36) the term III may be bounded by
$$\begin{aligned} III&\leqq \frac{C}{r_c^2}\int _0^T\int _{I_v(t)}\sup _{y\in [-r_c,r_c]}|w(x{+}y\mathrm {n}_v(x,t))| \int _{-r_c}^{r_c}|\chi _u{-}\chi _v|(x{+}y\mathrm {n}_v(x,t))\,{\mathrm {d}}y\,{\mathrm {d}}S\,{\mathrm {d}}t\nonumber \\&\leqq \frac{C}{r_c^2}\int _0^T\int _{I_v(t)}\sup _{y\in [-r_c,r_c]}|w(x{+}y\mathrm {n}_v(x,t))|^2\,{\mathrm {d}}S\,{\mathrm {d}}t\nonumber \\&\quad +\frac{C}{r_c^2}\int _0^T\int _{I_v(t)}\bigg |\int _{-r_c}^{r_c}|\chi _u{-}\chi _v|(x{+}y\mathrm {n}_v(x,t))\,{\mathrm {d}}y\bigg |^2\,{\mathrm {d}}S\,{\mathrm {d}}t\nonumber \\&\leqq \frac{C}{r_c^6}\Vert v\Vert _{L^\infty _tW^{2,\infty }_x({\mathbb {R}^d}{\setminus } I_v(t))}^2 \int _0^T\int _{I_v(t)} |h^\pm _{e(t)}|^2 + |\nabla h^\pm _{e(t)}|^2 \,{\mathrm {d}}S\,{\mathrm {d}}t\nonumber \\&\quad +\frac{C}{r_c^2}\int _0^T\int _{\mathbb {R}^d}|\chi _u{-}\chi _v| \min \Big \{\frac{{\text {dist}}(x,I_v(t))}{r_c},1\Big \} \,{\mathrm {d}}x\,{\mathrm {d}}t\nonumber \\&\leqq \frac{C}{r_c^{10}}\Vert v\Vert _{L^\infty _tW^{2,\infty }_x({\mathbb {R}^d}{\setminus } I_v(t))}^2 \int _0^T\int _{\mathbb {R}^d}1-\xi \cdot \frac{\nabla \chi _u}{|\nabla \chi _u|} \,\mathrm{d}|\nabla \chi _u|\,{\mathrm {d}}t\nonumber \\&\quad + \frac{C}{r_c^{10}}(1+\Vert v\Vert _{L^\infty _tW^{2,\infty }_x({\mathbb {R}^d}{\setminus } I_v(t))}^2) \int _0^T\int _{\mathbb {R}^d}|\chi _u{-}\chi _v| \min \Big \{\frac{{\text {dist}}(x,I_v(t))}{r_c},1\Big \} \,{\mathrm {d}}x\,{\mathrm {d}}t\nonumber \\&\leqq \frac{C}{r_c^{10}}(1+\Vert v\Vert _{L^\infty _tW^{2,\infty }_x({\mathbb {R}^d}{\setminus } I_v(t))}^2) \int _0^TE[\chi _u,u,V|\chi _v,v](t)\,\mathrm{d}t. \end{aligned}$$
(150)
For the term IV, we first add zero, then perform an integration by parts which is followed by an application of Hölder’s inequality to obtain
$$\begin{aligned} IV&\leqq C\int _0^T\bigg (\int _{\mathbb {R}^d}|\chi _u-\chi _{v,h^+_{e(t)},h^-_{e(t)}}|\,{\mathrm {d}}x\bigg )^\frac{1}{2} \bigg (\int _{\mathbb {R}^d}|(\nabla w)^T:\nabla \xi |^2\,{\mathrm {d}}x\bigg )^\frac{1}{2}\,{\mathrm {d}}t\nonumber \\&\quad +C\int _0^T\bigg |\int _{{\mathbb {R}^d}}(\chi _v-\chi _{v,h^+_{e(t)},h^-_{e(t)}})(w\cdot \nabla ) (\nabla \cdot \xi )\,{\mathrm {d}}x\bigg |\,{\mathrm {d}}t\nonumber \\&\quad +C\int _0^T\bigg |\int _{{\mathbb {R}^d}}((w\cdot \nabla )\xi )\cdot \mathrm {d}\nabla (\chi _v-\chi _{v,h^+_{e(t)},h^-_{e(t)}})\bigg |\,{\mathrm {d}}t\nonumber \\&=: (IV)_a + (IV)_b + (IV)_c. \end{aligned}$$
(151)
By definition of \(\xi \) (see (32)) recall that
$$\begin{aligned} \nabla \xi&= \frac{\zeta '\big (\frac{{\text {dist}}^{\pm }(x,I_v(t))}{r_c}\big )}{r_c} \mathrm {n}_v(P_{I_v(t)}x)\otimes \mathrm {n}_v(P_{I_v(t)}x) \\&\quad + \zeta \Big (\frac{{\text {dist}}^{\pm }(x,I_v(t))}{r_c}\Big ) \nabla ^2{\text {dist}}^{\pm }(x,I_v(t)). \end{aligned}$$
Recalling also (96), (97) and (113) as well as making use of (78c), (19), (28), (78a) and finally the coercivity property (36) the term \((IV)_a\) from (151) is estimated by
$$\begin{aligned} (IV)_a&\leqq \frac{C}{r_c}\int _0^T E[\chi _u,u,V|\chi _v,v](t) + e(t)E[\chi _u,u,V|\chi _v,v]^\frac{1}{2}(t)\,{\mathrm {d}}t\nonumber \\&\quad +\frac{C}{r_c^4}\Vert v\Vert _{L^\infty _tW^{2,\infty }_x({\mathbb {R}^d}{\setminus } I_v(t))}^2\int _0^T \int _{I_v(t)}|h^\pm _{e(t)}|^2+|\nabla h^\pm _{e(t)}|^2\,{\mathrm {d}}S\,{\mathrm {d}}t\nonumber \\&\leqq \frac{C}{r_c^8}(1{+}\Vert v\Vert _{L^\infty _tW^{2,\infty }_x({\mathbb {R}^d}{\setminus } I_v(t))}^2) \nonumber \\&\qquad ~~~~~~ \times \int _0^T E[\chi _u,u,V|\chi _v,v](t) {+} e(t)E[\chi _u,u,V|\chi _v,v]^\frac{1}{2}(t)\,{\mathrm {d}}t. \end{aligned}$$
(152)
Recalling from (78b) the definition of \(\chi _{v,h^+_{e(t)},h^-_{e(t)}}\), we may estimate the term \((IV)_b\) from (151) by a change of variables \(\Phi _t\), (19), Hölder’s and Young’s inequality, (98) as well as (78a)
$$\begin{aligned} (IV)_b&\leqq \frac{C}{r_c^2}\int _0^T\int _{I_v(t)}|h^\pm _{e(t)}|^2\,{\mathrm {d}}S\,{\mathrm {d}}t\nonumber \\&\quad +\frac{C}{r_c^2}\int _0^T\int _{I_v(t)}\sup _{y\in [-r_c,r_c]}|w(x{+}y\mathrm {n}_v(x,t))|^2\,{\mathrm {d}}S\,{\mathrm {d}}t\nonumber \\&\leqq \frac{C}{r_c^{10}}\Vert v\Vert _{L^\infty _tW^{2,\infty }_x({\mathbb {R}^d}{\setminus } I_v(t))}^2\int _0^T E[\chi _u,u,V|\chi _v,v](t)\,{\mathrm {d}}t. \end{aligned}$$
(153)
To estimate the term \((IV)_c\) from (151), we again make use of the definition of \(\chi _{v,h^+_{e(t)},h^-_{e(t)}}\), (19), Hölder’s and Young’s inequality, (98) as well as (78a) which yields the following bound:
$$\begin{aligned} (IV)_c&\leqq \frac{C}{r_c}\int _0^T\int _{I_v(t)}|\nabla h^\pm _{e(t)}| \sup _{y\in [-r_c,r_c]}|w(x{+}y\mathrm {n}_v(x,t))|\,{\mathrm {d}}S\,{\mathrm {d}}t\nonumber \\&\leqq \frac{C}{r_c^9}\Vert v\Vert _{L^\infty _tW^{2,\infty }_x({\mathbb {R}^d}{\setminus } I_v(t))} \int _0^T E[\chi _u,u,V|\chi _v,v](t)\,{\mathrm {d}}t. \end{aligned}$$
(154)
Hence, taking together the bounds from (152), (153) and (154), we obtain
$$\begin{aligned} IV&\leqq \frac{C}{r_c^{10}}(1{+}\Vert v\Vert _{L^\infty _tW^{2,\infty }_x({\mathbb {R}^d}{\setminus } I_v(t))}^2) \int _0^T E[\chi _u,u,V|\chi _v,v](t)\,{\mathrm {d}}t\nonumber \\&\quad +\frac{C}{r_c^{10}}(1{+}\Vert v\Vert _{L^\infty _tW^{2,\infty }_x({\mathbb {R}^d}{\setminus } I_v(t))}^2) \int _0^Te(t)E^\frac{1}{2}[\chi _u,u,V|\chi _v,v](t)\,{\mathrm {d}}t. \end{aligned}$$
(155)
In order to estimate the term V, we argue as follows: in a first step, we split \({\mathbb {R}^d}\) into the region \(I_v(t)+B_{r_c}\) near to and the region \({\mathbb {R}^d}{\setminus }(I_v(t)+B_{r_c})\) away from the interface of the strong solution. Recall then that the indicator function \(\chi _u(\cdot ,t)\) of the varifold solution is of bounded variation in \(I_v(t)+B_{r_c}\). In particular, \(E^+:=\{x\in {\mathbb {R}^d}:\chi _u>0\}\cap (I_v(t)+B_{r_c})\) is a set of finite perimeter in \(I_v(t)+B_{r_c}\). Applying Theorem 39 in local coordinates, the sections
$$\begin{aligned} E^+_x = \{y\in (-r_c,r_c):\chi _u(x+y\mathrm {n}_v(x,t))>0\} \end{aligned}$$
are guaranteed to be one-dimensional Caccioppoli sets in \((-r_c,r_c)\), and such that all of the four properties listed in Theorem 39 hold true for \(\mathcal {H}^{d-1}\)-almost every \(x\in I_v(t)\). Recall from [11, Proposition 3.52] that one-dimensional Caccioppoli sets are in fact finite unions of disjoint intervals. We then distinguish for \(\mathcal {H}^{d-1}\)-almost every \(x\in I_v(t)\) between the cases that \(\mathcal {H}^0(\partial ^*E_x^+)\leqq 2\) or \(\mathcal {H}^0(\partial ^*E_x^+)> 2\). In other words, we distinguish between those sections which consist of at most one interval and those which consist of at least two intervals. It also turns out to be useful to further keep track of whether \(\mathrm {n}_v\cdot \mathrm {n}_u\leqq \frac{1}{2}\) or \(\mathrm {n}_v\cdot \mathrm {n}_u\geqq \frac{1}{2}\) holds.
We then obtain by Young’s and Hölder’s inequality, as well as the fact that due to Definition 13 the vector field \(\xi \) is supported in \(I_v(t)+B_{r_c}\), that
$$\begin{aligned} V&\leqq \int _0^T\bigg (\int _{\begin{array}{c} \{x{+}y\mathrm {n}_v(x,t)\in \partial ^*E^+:x\in I_v(t),\,|y|<r_c,\\ \mathcal {H}^0(\partial ^*E_x^+)\leqq 2,\, \mathrm {n}_v(x)\cdot \mathrm {n}_u(x{+}y\mathrm {n}_v(x,t))\geqq \frac{1}{2}\} \end{array}} |(\nabla w)^T \xi |^2 \,\mathrm{d}\mathcal {H}^{d-1}\bigg )^{1/2}\nonumber \\&\qquad \times \bigg (\int _{\mathbb {R}^d}|\mathrm {n}_u-\xi |^2 \,\mathrm{d}|\nabla \chi _u|\bigg )^{1/2}\,{\mathrm {d}}t\nonumber \\&\quad +C\int _0^T\Vert \nabla w(t)\Vert _{L^\infty _x}\bigg ( \int _{\begin{array}{c} \{x{+}y\mathrm {n}_v(x,t)\in \partial ^*E^+:x\in I_v(t),\,|y|<r_c,\quad \\ \mathcal {H}^0(\partial ^*E_x^+)> 2,\, \mathrm {n}_v(x)\cdot \mathrm {n}_u(x{+}y\mathrm {n}_v(x,t))\geqq \frac{1}{2}\} \end{array}} 1\,\mathrm{d}\mathcal {H}^{d-1}\bigg )\,{\mathrm {d}}t\nonumber \\&\quad +C\int _0^T\Vert \nabla w(t)\Vert _{L^\infty _x}\bigg ( \int _{\begin{array}{c} \{x{+}y\mathrm {n}_v(x,t)\in \partial ^*E^+:x\in I_v(t),\,|y|<r_c,\\ \mathrm {n}_v(x)\cdot \mathrm {n}_u(x{+}y\mathrm {n}_v(x,t))\leqq \frac{1}{2}\} \end{array}} 1\,\mathrm{d}\mathcal {H}^{d-1}\bigg )\,{\mathrm {d}}t\nonumber \\&\quad +C\int _0^T\Vert \nabla w(t)\Vert _{L^\infty _x}\bigg (\int _{{\mathbb {R}^d}{\setminus } (I_v(t)+ B_{r_c})} 1\,\mathrm{d}|\nabla \chi _u|\bigg )\,{\mathrm {d}}t\nonumber \\&\leqq C\int _0^T\Vert \nabla w(t)\Vert _{L^\infty _x}E[\chi _u,u,V|\chi _v,v](t)\,{\mathrm {d}}t\nonumber \\&\qquad +C\int _0^T\bigg (\int _{\begin{array}{c} \{x{+}y\mathrm {n}_v(x,t)\in \partial ^*E^+:x\in I_v(t),\,|y|<r_c,\\ \mathcal {H}^0(\partial ^*E_x^+)\leqq 2,\, \mathrm {n}_v(x)\cdot \mathrm {n}_u(x{+}y\mathrm {n}_v(x,t))\geqq \frac{1}{2}\} \end{array}} |(\nabla w)^T \xi |^2 \,\mathrm{d}\mathcal {H}^{d-1}\bigg )^\frac{1}{2}\nonumber \\&\qquad ~~~~~~~~~~~~~~~~~\times \bigg (\int _{\mathbb {R}^d}|\mathrm {n}_u-\xi |^2 \,\mathrm{d}|\nabla \chi _u|\bigg )^{1/2}\,{\mathrm {d}}t\nonumber \\&\qquad +C\int _0^T\Vert \nabla w(t)\Vert _{L^\infty _x}\bigg ( \int _{\begin{array}{c} \{x{+}y\mathrm {n}_v(x,t)\in \partial ^*E^+:x\in I_v(t),\,|y|<r_c,~~~\\ \mathcal {H}^0(\partial ^*E_x^+)> 2,\, \mathrm {n}_v(x)\cdot \mathrm {n}_u(x{+}y\mathrm {n}_v(x,t))\geqq \frac{1}{2}\} \end{array}} 1\,\mathrm{d}\mathcal {H}^{d-1}\bigg )\,{\mathrm {d}}t\nonumber \\&\quad =: C\int _0^T\Vert \nabla w(t)\Vert _{L^\infty _x}E[\chi _u,u,V|\chi _v,v](t)\,{\mathrm {d}}t+ V_a + V_b. \end{aligned}$$
(156)
To estimate \(V_a\) from (156), we use the co-area formula for rectifiable sets (see [11, (2.72)]), (100), Hölder’s inequality and the coercivity property (38), which together yield (we abbreviate in the first line \(F(x,y,t):=(\nabla w)^T(x{+}y\mathrm {n}_v(x,t))\mathrm {n}_v(x,t)\))
$$\begin{aligned} V_a&\leqq C\int _0^T\bigg (\int _{\{x\in I_v(t):\mathcal {H}^0(\partial ^*E_x^+)\leqq 2\}} \int _{\{y\in \partial ^*E^+_x:\mathrm {n}_v(x)\cdot \mathrm {n}_u(x{+}y\mathrm {n}_v(x,t))\geqq \frac{1}{2}\}}\nonumber \\&\qquad ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~|F(x,y,t)|^2 \,\mathrm{d}\mathcal {H}^0(y)\,{\mathrm {d}}S(x)\bigg )^\frac{1}{2}\nonumber \\&\qquad \qquad ~~~~~~~\times \bigg (\int _{\mathbb {R}^d}|\mathrm {n}_u-\xi |^2 \,\mathrm{d}|\nabla \chi _u|\bigg )^{1/2}\,{\mathrm {d}}t\nonumber \\&\leqq C\int _0^T\bigg (\int _{I_v(t)}\sup _{y\in [-r_c,r_c]} |(\nabla w)^T(x{+}y\mathrm {n}_v(x,t))\cdot \mathrm {n}_v(x,t)|^2\,{\mathrm {d}}S(x)\bigg )^\frac{1}{2}\nonumber \\&\qquad \qquad ~~~~~~~~\times \bigg (\int _{\mathbb {R}^d}|\mathrm {n}_u-\xi |^2 \,\mathrm{d}|\nabla \chi _u|\bigg )^{1/2}\,{\mathrm {d}}t\nonumber \\&\leqq \frac{C}{r_c^{9}}\Vert v\Vert _{L^\infty _tW^{3,\infty }_x({\mathbb {R}^d}{\setminus } I_v(t))} \int _0^T (1+|\log e(t)|)e(t)E[\chi _u,u,V|\chi _v,v]^\frac{1}{2}(t)\,{\mathrm {d}}t. \end{aligned}$$
(157)
It remains to bound the term \(V_b\) from (156). To this end, we make use of the fact that it follows from property iv) in Theorem 39 that every second point \(y\in \partial ^*E^+_x\cap (-r_c,r_c)\) has to have the property that \(\mathrm {n}_v(x)\cdot \mathrm {n}_u(x{+}y\mathrm {n}_v(x,t))<0\), that is, \(1\leqq 1-\mathrm {n}_v(x)\cdot \mathrm {n}_u(x{+}y\mathrm {n}_v(x,t))\). We may therefore estimate, with the help of the co-area formula for rectifiable sets (see [11, (2.72)]) and the bound (99),
$$\begin{aligned} V_b&\leqq C\int _0^T\Vert \nabla w(t)\Vert _{L^\infty _x} \int _{\{x\in I_v(t):\mathcal {H}^0(\partial ^*E_x^+)> 2\}} \int _{\{y\in \partial ^*E^+_x:\mathrm {n}_v(x)\cdot \mathrm {n}_u(x{+}y\mathrm {n}_v(x,t))\geqq \frac{1}{2}\}} \nonumber \\&\qquad ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1\,\mathrm{d}\mathcal {H}^0(y)\,{\mathrm {d}}S(x)\,{\mathrm {d}}t\nonumber \\&\leqq C \int _0^T \Vert \nabla w(t)\Vert _{L^\infty _x} \int _{I_v(t)}\int _{\partial ^*E^+_x}1-\mathrm {n}_v(x,t)\cdot \mathrm {n}_u(x{+}y\mathrm {n}_v(x,t)) \,\mathrm{d}\mathcal {H}^0(y)\,{\mathrm {d}}S(x)\,{\mathrm {d}}t\nonumber \\&\leqq \frac{C}{r_c^9}|\log e(t)| \Vert v\Vert _{L^\infty _tW^{3,\infty }_x({\mathbb {R}^d}{\setminus } I_v(t))} \int _0^TE[\chi _u,u,V|\chi _v,v](t)\,{\mathrm {d}}t. \end{aligned}$$
(158)
All in all, we obtain from the assumption \(E[\chi _u,u,V|\chi _v,v](t)\leqq e^2(t)\) as well as (156), (157), (158) and (99)
$$\begin{aligned} V&\leqq \frac{C}{r_c^{9}}\Vert v\Vert _{L^\infty _tW^{3,\infty }_x({\mathbb {R}^d}{\setminus } I_v(t))} \int _0^T (1+|\log e(t)|)e(t)E[\chi _u,u,V|\chi _v,v]^\frac{1}{2}(t)\,{\mathrm {d}}t. \end{aligned}$$
(159)
Hence, we deduce from the bounds (149), (150), (155), (159) as well as (99) the asserted estimate for the additional surface tension terms. \(\quad \square \)
Estimate for the Viscosity Terms
In contrast to the case of equal shear viscosities \(\mu _+=\mu _-\), we have to deal with the problematic viscous stress term given by \((\mu (\chi _v)-\mu (\chi _u))(\nabla v+\nabla v^T)\). We now show that the choice of w indeed compensates for (most of) this term in the sense that the viscosity terms from Proposition 10
$$\begin{aligned} R_{visc}+A_{visc}&= -\int _0^T \int _{\mathbb {R}^d} 2\big (\mu (\chi _u)-\mu (\chi _v)\big ) {D^{{\text {sym}}}}v:{D^{{\text {sym}}}}(u-v) \,{\mathrm {d}}x\,{\mathrm {d}}t\nonumber \\&\quad +\int _0^T \int _{\mathbb {R}^d} 2\big (\mu (\chi _u)-\mu (\chi _v)\big ) {D^{{\text {sym}}}}v:{D^{{\text {sym}}}}w \,{\mathrm {d}}x\,{\mathrm {d}}t\nonumber \\&\quad -\int _0^T \int _{\mathbb {R}^d} 2\mu (\chi _u) {D^{{\text {sym}}}}w:{D^{{\text {sym}}}}(u-v-w) \,{\mathrm {d}}x\,{\mathrm {d}}t\end{aligned}$$
(160)
may be bounded by a Gronwall-type term.
Lemma 30
Let the assumptions and notation of Proposition 28 be in place. In particular, we assume that there exists a \(C^1\)-function \(e:[0,{T_{strong}})\rightarrow [0,r_c)\) such that the relative entropy is bounded by \(E[\chi _u,u,V,|\chi _v,v](t)\leqq e^2(t)\).
Then, for any \(\delta >0\) there exists a constant \(C>0\) such that the viscosity terms \(R_{visc}+A_{visc}\) may be estimated by
$$\begin{aligned} R_{visc}+A_{visc}&\leqq \frac{C}{r_c^{8}}\Vert v\Vert _{L^\infty _tW^{2,\infty }_x({\mathbb {R}^d}{\setminus } I_v(t))}^2 \int _0^TE[\chi _u,u,V|\chi _v,v](t)\,{\mathrm {d}}t\nonumber \\&\quad +\frac{C}{r_c}\Vert v\Vert _{L^\infty _tW^{1,\infty }_x}^2\int _0^Te(t)E[\chi _u,u,V|\chi _v,v]^\frac{1}{2}(t)\,{\mathrm {d}}t\nonumber \\&\quad +\delta \int _0^T\int _{{\mathbb {R}^d}}|{D^{{\text {sym}}}}(u-v-w)|^2\,{\mathrm {d}}x\,{\mathrm {d}}t. \end{aligned}$$
(161)
Proof
We argue pointwise for the time variable and start by adding zero:
$$\begin{aligned}&R_{visc}+A_{visc}\nonumber \\&\quad =-2\int _{\mathbb {R}^d}(\mu (\chi _u)-\mu (\chi _v)){D^{{\text {sym}}}}v:{D^{{\text {sym}}}}(u{-}v{-}w) \,{\mathrm {d}}x\nonumber \\&\qquad -2\int _{\mathbb {R}^d}\mu (\chi _u){D^{{\text {sym}}}}w:{D^{{\text {sym}}}}(u-v-w) \,{\mathrm {d}}x\nonumber \\&\quad =-2\int _{\mathbb {R}^d}\big (\mu (\chi _u)-\mu (\chi _v)-(\mu ^--\mu ^+)\chi _{0\leqq {\text {dist}}^{\pm }(x,I_v(t))\leqq h^+_{e(t)}(P_{I_v(t)}x)}\nonumber \\&\qquad ~~~~~~~~~~~~~~~~~ -(\mu ^+ -\mu ^-)\chi _{-h^-_{e(t)}(P_{I_v(t)}x)\leqq {\text {dist}}^{\pm }(x,I_v(t)) \leqq 0}\big ){D^{{\text {sym}}}}v:{D^{{\text {sym}}}}(u{-}v{-}w) \,{\mathrm {d}}x\nonumber \\&\qquad -2\int _{\mathbb {R}^d}\chi _{{\text {dist}}^{\pm }(x,I_v(t))\notin [-h^-_{e(t)} (P_{I_v(t)}x),h^+_{e(t)}(P_{I_v(t)}x)]} \mu (\chi _u) {D^{{\text {sym}}}}w:{D^{{\text {sym}}}}(u{-}v{-}w) \,{\mathrm {d}}x\nonumber \\&\qquad -2\int _{\mathbb {R}^d}\chi _{0\leqq {\text {dist}}^{\pm }(x,I_v(t))\leqq h^+_{e(t)}(P_{I_v(t)}x)} (\mu (\chi _u)-\mu ^-) {D^{{\text {sym}}}}w:{D^{{\text {sym}}}}(u{-}v{-}w) \,{\mathrm {d}}x\nonumber \\&\qquad -2\int _{\mathbb {R}^d}\chi _{-h^-_{e(t)}(P_{I_v(t)}x)\leqq {\text {dist}}^{\pm }(x,I_v(t))\leqq 0} (\mu (\chi _u)-\mu ^+) {D^{{\text {sym}}}}w:{D^{{\text {sym}}}}(u{-}v{-}w) \,{\mathrm {d}}x\nonumber \\&\qquad -2\int _{\mathbb {R}^d}\chi _{0\leqq {\text {dist}}^{\pm }(x,I_v(t))\leqq h^+_{e(t)}(P_{I_v(t)}x)} ((\mu ^-{-}\mu ^+){D^{{\text {sym}}}}v+\mu ^- {D^{{\text {sym}}}}w):\nabla (u{-}v{-}w) \,{\mathrm {d}}x\nonumber \\&\qquad -2\int _{\mathbb {R}^d}\chi _{-h^-_{e(t)}(P_{I_v(t)}x)\leqq {\text {dist}}^{\pm }(x,I_v(t))\leqq 0} ((\mu ^+{-}\mu ^-){D^{{\text {sym}}}}v+\mu ^+ {D^{{\text {sym}}}}w):\nabla (u{-}v{-}w) \,{\mathrm {d}}x\nonumber \\&=: I + II + III + IV + V + VI. \end{aligned}$$
(162)
We start by estimating the first four terms. Note that \(\mu (\chi _u)-\mu ^- = (\mu _+-\mu _-)\chi _u\). Recalling the definition of \(\chi _{v,h^+_{e(t)},h^-_{e(t)}}\) from (78b) we see that
$$\begin{aligned} \chi _{0\leqq {\text {dist}}^{\pm }(x,I_v(t))\leqq h^+_{e(t)}(P_{I_v(t)}x)}\chi _u = \chi _{0\leqq {\text {dist}}^{\pm }(x,I_v(t))\leqq h^+_{e(t)}(P_{I_v(t)}x)}(\chi _u-\chi _{v,h^+_{e(t)},h^-_{e(t)}}). \end{aligned}$$
Hence, we may rewrite
$$\begin{aligned} III&= -2\int _{\mathbb {R}^d}\chi _{0\leqq {\text {dist}}^{\pm }(x,I_v(t))\leqq h^+_{e(t)}(P_{I_v(t)}x)} (\mu _+-\mu _-) (\chi _u-\chi _{v,h^+_{e(t)},h^-_{e(t)}})\\&\qquad ~~~~~~~~~~~~ \times (W\otimes \mathrm {n}_v(P_{I_v(t)}x)):{D^{{\text {sym}}}}(u{-}v{-}w) \,{\mathrm {d}}x\\&\qquad -2\int _{\mathbb {R}^d}\chi _{0\leqq {\text {dist}}^{\pm }(x,I_v(t))\leqq h^+_{e(t)}(P_{I_v(t)}x)} (\mu _+-\mu _-)\\&\qquad ~~~~~~~~~~~~ \times (\nabla w-W\otimes \mathrm {n}_v(P_{I_v(t)}x)):{D^{{\text {sym}}}}(u{-}v{-}w) \,{\mathrm {d}}x. \end{aligned}$$
Carrying out an analogous computation for IV, using again the definition of the smoothed approximation \(\chi _{v,h^+_{e(t)},h^-_{e(t)}}\) for \(\chi _u\) from (78b) and using (96) as well as (97), we then get the bound
$$\begin{aligned}&I + II + III + IV\\&\quad \leqq C \Vert v\Vert _{W^{1,\infty }} \bigg (\int _{\mathbb {R}^d}|\chi _u-\chi _{v,h^+_{e(t)},h^-_{e(t)}}| \,{\mathrm {d}}x\bigg )^{1/2} \bigg (\int _{\mathbb {R}^d}|{D^{{\text {sym}}}}(u{-}v{-}w)|^2 \,{\mathrm {d}}x\bigg )^{1/2}\\&\qquad +\frac{C}{r_c^2}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))} \bigg (\int _{I_v(t)} |h^\pm _{e(t)}|^2 {+} |\nabla h^\pm _{e(t)}|^2 \,{\mathrm {d}}S\bigg )^{1/2}\\&\qquad \qquad \qquad \times \bigg (\int _{\mathbb {R}^d}|{D^{{\text {sym}}}}(u{-}v{-}w)|^2 \,{\mathrm {d}}x\bigg )^{1/2}. \end{aligned}$$
Plugging in the estimates (78a) and (78c), we obtain, by Young’s inequality,
$$\begin{aligned} I + II + III + IV&\leqq \frac{C\delta ^{-1}}{r_c^8 }\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))} ^2 E[\chi _u,u,V|\chi _v,v](t)\nonumber \\&\quad + \frac{C\delta ^{-1}}{r_c}\Vert v\Vert _{W^{1,\infty }}^2e(t)E[\chi _u,u,V|\chi _v,v]^\frac{1}{2}(t)\nonumber \\&\quad +C\delta ^{-1}\Vert v\Vert _{W^{1,\infty }}^2E[\chi _u,u,V|\chi _v,v](t)\nonumber \\&\quad +\delta \Vert {D^{{\text {sym}}}}(u-v-w)\Vert _{L^2} \end{aligned}$$
(163)
for every \(\delta \in (0,1)\). To estimate the last two terms V and VI in (162), we may rewrite, making use of the definition (97) of the vector field W and abbreviating \(\mathrm {n}_v=\mathrm {n}_v(P_{I_v(t)}x)\), \({\text {dist}}^{\pm }={\text {dist}}^{\pm }(x,I_v(t))\) as well as \(h_{e(t)}^+=h_{e(t)}^+(P_{I_v(t)}x)\),
$$\begin{aligned}&-\int _{\mathbb {R}^d}\chi _{0\leqq {\text {dist}}^{\pm }\leqq h^+_{e(t)}} ((\mu ^-{-}\mu ^+){D^{{\text {sym}}}}v+\mu ^- {D^{{\text {sym}}}}w):\nabla (u{-}v{-}w) \,{\mathrm {d}}x\\&\quad = -\int _{\mathbb {R}^d}\chi _{0\leqq {\text {dist}}^{\pm }\leqq h^+_{e(t)}} ((\mu ^-{-}\mu ^+)({\text {Id}}-\mathrm {n}_v\otimes \mathrm {n}_v) ({D^{{\text {sym}}}}v\cdot \mathrm {n}_v)\otimes \mathrm {n}_v+\mu ^- {D^{{\text {sym}}}}w)\\&\quad \quad ~~~~~~~~~~~~ :\nabla (u{-}v{-}w) \,{\mathrm {d}}x\\&\qquad -\int _{\mathbb {R}^d}\chi _{0\leqq {\text {dist}}^{\pm }\leqq h^+_{e(t)}} (\mu ^-{-}\mu ^+){D^{{\text {sym}}}}v\, ({\text {Id}}-\mathrm {n}_v\otimes \mathrm {n}_v):\nabla (u{-}v{-}w) \,{\mathrm {d}}x\\&\qquad -\int _{\mathbb {R}^d}\chi _{0\leqq {\text {dist}}^{\pm }\leqq h^+_{e(t)}} (\mu ^-{-}\mu ^+)(\mathrm {n}_v\cdot {D^{{\text {sym}}}}v\cdot \mathrm {n}_v) (\mathrm {n}_v\otimes \mathrm {n}_v): \nabla (u{-}v{-}w) \,{\mathrm {d}}x\\&\quad = -\int _{\mathbb {R}^d}\chi _{0\leqq {\text {dist}}^{\pm }\leqq h^+_{e(t)}} ((\mu ^-{-}\mu ^+)({\text {Id}}-\mathrm {n}_v\otimes \mathrm {n}_v)({D^{{\text {sym}}}}v\cdot \mathrm {n}_v)\otimes \mathrm {n}_v+\mu ^- {D^{{\text {sym}}}}w)\\&\qquad ~~~~~~~~~~~~:\nabla (u{-}v{-}w) \,{\mathrm {d}}x\\&\qquad -\int _{\mathbb {R}^d}\chi _{0\leqq {\text {dist}}^{\pm }\leqq h^+_{e(t)}} (\mu ^-{-}\mu ^+){D^{{\text {sym}}}}v\, ({\text {Id}}-\mathrm {n}_v\otimes \mathrm {n}_v):\nabla (u{-}v{-}w) \,{\mathrm {d}}x\\&\qquad +\int _{\mathbb {R}^d}\chi _{0\leqq {\text {dist}}^{\pm }\leqq h^+_{e(t)}} (\mu ^-{-}\mu ^+)(\mathrm {n}_v\cdot {D^{{\text {sym}}}}v\cdot \mathrm {n}_v) ({\text {Id}}-\mathrm {n}_v\otimes \mathrm {n}_v):\nabla (u{-}v{-}w) \,{\mathrm {d}}x,\\&\quad = \frac{1}{2}\int _{\mathbb {R}^d}\chi _{0\leqq {\text {dist}}^{\pm }\leqq h^+_{e(t)}} ((W\otimes \mathrm {n}_v-\nabla w)+(W\otimes \mathrm {n}_v-\nabla w)^T):\nabla (u{-}v{-}w) \,{\mathrm {d}}x\\&\qquad +(\mu ^--\mu ^+)\int _{\mathbb {R}^d}\chi _{0\leqq {\text {dist}}^{\pm }\leqq h^+_{e(t)}} ((\mathrm {Id}{-}\mathrm {n}_v\otimes \mathrm {n}_v)({D^{{\text {sym}}}}v\cdot \mathrm {n}_v)\otimes \mathrm {n}_v):\nabla (u{-}v{-}w) \,{\mathrm {d}}x\\&\qquad -\int _{\mathbb {R}^d}\chi _{0\leqq {\text {dist}}^{\pm }\leqq h^+_{e(t)}} (\mu ^-{-}\mu ^+){D^{{\text {sym}}}}v\, ({\text {Id}}-\mathrm {n}_v\otimes \mathrm {n}_v):\nabla (u{-}v{-}w) \,{\mathrm {d}}x\\&\qquad +\int _{\mathbb {R}^d}\chi _{0\leqq {\text {dist}}^{\pm }\leqq h^+_{e(t)}} (\mu ^-{-}\mu ^+)(\mathrm {n}_v\cdot {D^{{\text {sym}}}}v\cdot \mathrm {n}_v) ({\text {Id}}-\mathrm {n}_v\otimes \mathrm {n}_v):\nabla (u{-}v{-}w) \,{\mathrm {d}}x, \end{aligned}$$
where in the penultimate step we have used the fact that \(\nabla \cdot (u-v-w)=0\), and in the last step we added zero. This yields, after an integration by parts,
$$\begin{aligned}&-\int _{\mathbb {R}^d}\chi _{0\leqq {\text {dist}}^{\pm }\leqq h_{e(t)}^+} ((\mu ^-{-}\mu ^+){D^{{\text {sym}}}}v+\mu ^- {D^{{\text {sym}}}}w):\nabla (u{-}v{-}w) \,{\mathrm {d}}x\\&\quad = \frac{1}{2}\int _{\mathbb {R}^d}\chi _{0\leqq {\text {dist}}^{\pm }\leqq h_{e(t)}^+} ((W\otimes \mathrm {n}_v-\nabla w)+(W\otimes \mathrm {n}_v-\nabla w)^T):\nabla (u{-}v{-}w) \,{\mathrm {d}}x\\&\qquad -(\mu ^-{-}\mu ^+)\int _{\mathbb {R}^d}\chi _{0\leqq {\text {dist}}^{\pm }\leqq h_{e(t)}^+} \nabla \cdot (\mathrm {n}_v\otimes (\mathrm {Id}{-}\mathrm {n}_v\otimes \mathrm {n}_v)({D^{{\text {sym}}}}v\cdot \mathrm {n}_v)) \cdot (u{-}v{-}w) \,{\mathrm {d}}x\\&\qquad +(\mu ^-{-}\mu ^+)\int _{\mathbb {R}^d}(\mathrm {n}_v\cdot (u{-}v{-}w)) (\mathrm {Id}{-}\mathrm {n}_v\otimes \mathrm {n}_v)({D^{{\text {sym}}}}v\cdot \mathrm {n}_v) \cdot \,\mathrm{d}\nabla \chi _{0\leqq {\text {dist}}^{\pm }\leqq h_{e(t)}^+}\\&\qquad +(\mu ^-{-}\mu ^+)\int _{\mathbb {R}^d}\chi _{0\leqq {\text {dist}}^{\pm }\leqq h_{e(t)}^+} \nabla \cdot \big (({D^{{\text {sym}}}}v{-}(\mathrm {n}_v\cdot {D^{{\text {sym}}}}v\cdot \mathrm {n}_v) {\text {Id}}) ({\text {Id}}{-}\mathrm {n}_v\otimes \mathrm {n}_v)\big )\\&\qquad \qquad ~~~~~~~~~~~~~~~~~~~~~~~~\cdot (u{-}v{-}w) \,{\mathrm {d}}x\\&\qquad +(\mu ^-{-}\mu ^+)\int _{\mathbb {R}^d}(u{-}v{-}w)\\&\qquad \qquad ~~~~~~~~~~~~~~~~~~~~~~~~\cdot ({D^{{\text {sym}}}}v{-}(\mathrm {n}_v\cdot {D^{{\text {sym}}}}v\cdot \mathrm {n}_v) {\text {Id}}) ({\text {Id}}{-}\mathrm {n}_v\otimes \mathrm {n}_v) \,\mathrm{d}\nabla \chi _{0\leqq {\text {dist}}^{\pm }\leqq h_{e(t)}^+}. \end{aligned}$$
As a consequence of (96), (78a), (19) and the global Lipschitz estimate \(|\nabla h_e^\pm (\cdot ,t)|\leqq Cr_c^{-2}\) from Proposition 27, we obtain
$$\begin{aligned}&\bigg |\int _{\mathbb {R}^d}\chi _{0\leqq {\text {dist}}^{\pm }(x,I_v(t))\leqq h_{e(t)}^+(P_{I_v(t)}x)} ((\mu ^--\mu ^+){D^{{\text {sym}}}}v+\mu ^- {D^{{\text {sym}}}}w)\\&\qquad :\nabla (u-v-w) \,{\mathrm {d}}x\bigg |\\&\quad \leqq \frac{C}{r_c^{7/2}} \Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))} E\big [\chi _u,u,V\big |\chi _v,v\big ]^{1/2} \Vert \nabla (u-v-v)\Vert _{L^2}\\&\qquad +\frac{C}{r_c}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))} \int _{\mathbb {R}^d}\chi _{0\leqq {\text {dist}}^{\pm }(x,I_v(t))\leqq h_{e(t)}^+(P_{I_v(t)}x)} |u-v-w| \,{\mathrm {d}}x\\&\qquad +\frac{C}{r_c^2} \Vert v\Vert _{W^{1,\infty }} \int _{I_v(t)} \sup _{y\in (-r_c,r_c)} |u-v-w|(x+y\mathrm {n}_v(x,t)) |\nabla h_{e(t)}^+(x)| \,{\mathrm {d}}S(x). \end{aligned}$$
By a change of variables \(\Phi _t\), (18), (42), (78a) and an application of Young’s and Korn’s inequality, the latter two terms may be further estimated by
$$\begin{aligned}&\frac{C}{r_c^2}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))} \bigg (\int _{I_v(t)}\sup _{y\in (-r_c,r_c)} |u-v-w|^2(x+y\mathrm {n}_v(x,t))\,{\mathrm {d}}S\bigg )^\frac{1}{2}\\&\qquad \times \bigg (\int _{I_v(t)}|h_{e(t)}^+|^2+|\nabla h_{e(t)}^+|^2\,{\mathrm {d}}S\bigg )^\frac{1}{2}\\&\quad \leqq \frac{C}{r_c^{3}}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))} E[\chi _u,u,V|\chi _v,v]^\frac{1}{2}(t)\Vert u-v-w\Vert _{L^2}\\&\qquad +\frac{C}{r_c^{2}}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}E[\chi _u,u,V|\chi _v,v]^\frac{1}{2}(t) \Vert \nabla (u-v-w)\Vert _{L^2}\\&\quad \leqq \frac{C\delta ^{-1}}{r_c^{4}}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}^2 E[\chi _u,u,V|\chi _v,v](t) +\delta \Vert {D^{{\text {sym}}}}(u-v-w)\Vert _{L^2} \end{aligned}$$
for every \(\delta \in (0,1]\). In total, we obtain the bound
$$\begin{aligned} V&\leqq \frac{C\delta ^{-1}}{r_c^{4}}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}^2 E[\chi _u,u,V|\chi _v,v](t) +\delta \Vert {D^{{\text {sym}}}}(u-v-w)\Vert _{L^2}, \end{aligned}$$
(164)
where \(\delta \in (0,1)\) is again arbitrary. Analogously, one can derive a bound of the same form for the last term VI in (162). Together with the bounds from (163) as well as (164) this concludes the proof. \(\quad \square \)
Estimate for Terms with the Time Derivative of the Compensation Function
We proceed with the estimate for the terms from the relative entropy inequality of Proposition 10:
$$\begin{aligned} A_{dt}:=&-\int _0^T \int _{\mathbb {R}^d} \rho (\chi _u) (u-v-w)\cdot \partial _t w \,{\mathrm {d}}x\,{\mathrm {d}}t\nonumber \\&-\int _0^T\int _{\mathbb {R}^d}\rho (\chi _u) (u-v-w) \cdot (v\cdot \nabla ) w\,{\mathrm {d}}x\,{\mathrm {d}}t, \end{aligned}$$
(165)
which are related to the time derivative of the compensation function w.
Lemma 31
Let the assumptions and notation of Proposition 28 be in place. In particular, we assume that there exists a \(C^1\)-function \(e:[0,{T_{strong}})\rightarrow [0,r_c)\) such that the relative entropy is bounded by \(E[\chi _u,u,V,|\chi _v,v](t)\leqq e^2(t)\).
Then, for any \(\delta >0\) there exists a constant \(C>0\) such that \(A_{dt}\) may be estimated by
$$\begin{aligned} A_{dt}&\leqq \frac{C}{r_c^{22}} \Vert v\Vert _{L^\infty _tW^{1,\infty }_x}^2(1{+}\Vert v\Vert _{L^\infty _tW^{2,\infty }_x({\mathbb {R}^d}{\setminus } I_v(t))})^2\nonumber \\&\qquad ~~~~~\times \int _0^T(1{+}|\log e(t)|)E[\chi _u,u,V|\chi _v,v](t)\,{\mathrm {d}}t\nonumber \\&\quad + \frac{C}{r_c^{11}} \Vert v\Vert _{L^\infty _tW^{1,\infty }_x}(1{+}\Vert v\Vert _{L^\infty _tW^{2,\infty }_x({\mathbb {R}^d}{\setminus } I_v(t))})\nonumber \\&\qquad ~~~~~\times \int _0^T(1{+}|\log e(t)|)E[\chi _u,u,V|\chi _v,v](t)\,{\mathrm {d}}t\nonumber \\&\quad +\frac{C}{r_c^{8}}(1{+}\Vert v\Vert _{L^\infty _tW^{1,\infty }_x}) (\Vert \partial _t \nabla v\Vert _{L^\infty _{x,t}}{+}(R^2{+}1)\Vert v\Vert _{L^\infty _t W^{2,\infty }_x({\mathbb {R}^d}{\setminus } I_v(t))})\nonumber \\&\quad ~~~~~~~~~~\times \int _0^TE[\chi _u,u,V|\chi _v,v](t)\,{\mathrm {d}}t\nonumber \\&\quad +\frac{C}{r_c^2}\Vert v\Vert _{L^\infty _t W^{1,\infty }_x}^2\int _0^T(1+e'(t)) E[\chi _u,u,V|\chi _v,v](t)\,{\mathrm {d}}t\nonumber \\&\quad +\delta \int _0^T\int _{{\mathbb {R}^d}}|{D^{{\text {sym}}}}(u-v-w)|^2\,{\mathrm {d}}x\,{\mathrm {d}}t. \end{aligned}$$
(166)
Proof
To estimate the terms involving the time derivative of w we make use of the decomposition of \(\partial _t w + (v\cdot \nabla )w\) from (101):
$$\begin{aligned}&\bigg |-\int _0^T\int _{\mathbb {R}^d}\rho (\chi _u) (u{-}v{-}w) \cdot \partial _t w \,{\mathrm {d}}x\,{\mathrm {d}}t-\int _0^T\int _{\mathbb {R}^d}\rho (\chi _u) (u{-}v{-}w) \cdot (v\cdot \nabla ) w \,{\mathrm {d}}x\,{\mathrm {d}}t\bigg |\\&\quad \leqq \int _0^T\Vert g\Vert _{L^2}\Vert u-v-w\Vert _{L^2}\,{\mathrm {d}}t+ \int _0^T\Vert \hat{g}\Vert _{L^\frac{4}{3}}\Vert u-v-w\Vert _{L^4}\,{\mathrm {d}}t. \end{aligned}$$
Employing the bounds (59a), (59b) and the assumption \(E[\chi _u,u,V|\chi _v,v](t)\leqq e(t)^2\) together with the Orlicz-Sobolev embedding (228) from Proposition 41 or (231) from Lemma 42 depending on the dimension, we obtain
$$\begin{aligned} \bigg (\int _{I_v(t)} |\bar{h}^\pm |^4\,\mathrm{d}S\bigg )^\frac{1}{4} \leqq \frac{C}{r_c^6}e(t)\Big (1+\log \frac{1}{e(t)}\Big )^\frac{1}{4}. \end{aligned}$$
(167)
Making use of (78a), the bound for the vector field \(\hat{g}\) from (102), the Gagliardo-Nirenberg-Sobolev embedding \(\Vert u{-}v{-}w\Vert _{L^4}\leqq C\Vert \nabla (u{-}v{-}w)\Vert ^{1-\alpha }_{L^2} \Vert u{-}v{-}w\Vert ^{\alpha }_{L^2}\), with \(\alpha =\frac{1}{2}\) for \(d=2\) and \(\alpha =\frac{1}{4}\) for \(d=3\), as well as the assumption \(E[\chi _u,u,V|\chi _v,v](t)\leqq e(t)^2\), we obtain
$$\begin{aligned}&\Vert \hat{g}\Vert _{L^\frac{4}{3}}\Vert u-v-w\Vert _{L^4}\nonumber \\&\quad \leqq C\frac{\Vert v\Vert _{W^{1,\infty }}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}}{r_c^{11}}\Big (1+\log \frac{1}{e(t)}\Big )^\frac{1}{4}\nonumber \\&\qquad \times (\Vert \nabla (u{-}v{-}w)\Vert _{L^2}+ \Vert u{-}v{-}w\Vert _{L^2}) E[\chi _u,u,V|\chi _v,v]^\frac{1}{2}(t)\nonumber \\&\qquad + C\frac{\Vert v\Vert _{W^{1,\infty }}}{r_c^8}\Big (1{+}\log \frac{1}{e(t)}\Big )^\frac{1}{4} (\Vert \nabla (u{-}v{-}w)\Vert _{L^2}{+} \Vert u{-}v{-}w\Vert _{L^2})\Vert u{-}v{-}w\Vert _{L^2}\nonumber \\&\qquad + C\frac{\Vert v\Vert _{W^{1,\infty }}}{r_c^8}\Big (1+\log \frac{1}{e(t)}\Big )^\frac{1}{4} \Vert \nabla (u{-}v{-}w)\Vert ^{\frac{3}{2}-\alpha }_{L^2} \Vert u{-}v{-}w\Vert ^{\frac{1}{2}+\alpha }_{L^2}\nonumber \\&\qquad +C\Vert v\Vert _{W^{1,\infty }}(1{+}\Vert v\Vert _{W^{1,\infty }}) E[\chi _u,u,V|\chi _v,v]^\frac{1}{2}(t) \nonumber \\&\qquad ~~~~\times (\Vert \nabla (u{-}v{-}w)\Vert _{L^2}+\Vert u{-}v{-}w\Vert _{L^2}). \end{aligned}$$
(168)
Now, by an application of Young’s and Korn’s inequality for all the terms on the right hand side of (168) which include an \(L^2\)-norm of the gradient of \(u-v-w\) (in the case \(d=3\) we use \(a^\frac{5}{4}b^\frac{3}{4}=(a(8\delta /5)^\frac{1}{2})^\frac{5}{4} (b(8\delta /5)^{-\frac{5}{6}})^\frac{3}{4} \leqq \delta a^2 + \frac{3}{8}\big (\frac{8}{5}\big )^{-\frac{5}{3}}\delta ^{-\frac{5}{3}}b^2\), which follows from Young’s inequality with exponents \(p=\frac{8}{5}\) and \(q=\frac{8}{3}\)), we obtain
$$\begin{aligned}&\Vert \hat{g}\Vert _{L^\frac{4}{3}}\Vert u-v-w\Vert _{L^4}\nonumber \\&\quad \leqq \frac{C}{\delta ^\frac{5}{3} r_c^{22}} \Vert v\Vert _{W^{1,\infty }}^2(1{+}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))})^2 (1{+}|\log e(t)|)E[\chi _u,u,V|\chi _v,v](t)\nonumber \\&\qquad + \frac{C}{r_c^{11}} \Vert v\Vert _{W^{1,\infty }}(1{+}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))}) (1{+}|\log e(t)|)E[\chi _u,u,V|\chi _v,v](t)\nonumber \\&\qquad +\delta \Vert {D^{{\text {sym}}}}(u{-}v{-}w)\Vert ^2_{L^2}, \end{aligned}$$
(169)
where \(\delta \in (0,1)\) is arbitrary. This gives the desired bound for the \(L^\frac{4}{3}\)-contribution of \(\partial _t w + (v\cdot \nabla )w\). Concerning the \(L^2\)-contribution, we estimate using (59a), (78a), the bound for \(\Vert g\Vert _{L^2}\) from (103) as well as the assumption \(E[\chi _u,u,V|\chi _v,v](t)\leqq e(t)^2\) to get
$$\begin{aligned}&\Vert g\Vert _{L^2}\Vert u-v-w\Vert _{L^2} \nonumber \\&\quad \leqq C \frac{1{+}\Vert v\Vert _{W^{1,\infty }}}{r_c^{8}} (\Vert \partial _t \nabla v\Vert _{L^\infty ({\mathbb {R}^d}{\setminus } I_v(t))}{+}(R^2{+}1)\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))})\nonumber \\&\qquad ~~~~\times E[\chi _u,u,V|\chi _v,v]^\frac{1}{2}(t)\Vert u{-}v{-}w\Vert _{L^2}\nonumber \\&\qquad +C\Vert v\Vert _{W^{1,\infty }}(1+\Vert v\Vert _{W^{1,\infty }}) E[\chi _u,u,V|\chi _v,v]^\frac{1}{2}(t)\Vert u{-}v{-}w\Vert _{L^2}\nonumber \\&\qquad +\frac{C}{r_c^2}(1+e'(t))\Vert v\Vert _{W^{1,\infty }}^2 E[\chi _u,u,V|\chi _v,v]^\frac{1}{2}(t)\Vert u{-}v{-}w\Vert _{L^2}\nonumber \\&\qquad +C\frac{\Vert v\Vert _{W^{1,\infty }}(1{+}\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))})}{r_c} E[\chi _u,u,V|\chi _v,v]^\frac{1}{2}(t)\Vert u{-}v{-}w\Vert _{L^2}\nonumber \\&\qquad +C\Vert v\Vert _{W^{1,\infty }}(\Vert \nabla (u{-}v{-}w)\Vert _{L^2}+\Vert u{-}v{-}w\Vert _{L^2})\Vert u{-}v{-}w\Vert _{L^2}. \end{aligned}$$
(170)
Hence, by another application of Young’s and Korn’s inequality, we may bound
$$\begin{aligned}&\Vert g\Vert _{L^2}\Vert u-v-w\Vert _{L^2} \nonumber \\&\quad \leqq \frac{C}{r_c^{8}}(1{+}\Vert v\Vert _{W^{1,\infty }}) (\Vert \partial _t \nabla v\Vert _{L^\infty ({\mathbb {R}^d}{\setminus } I_v(t))}{+}(R^2{+}1)\Vert v\Vert _{W^{2,\infty }({\mathbb {R}^d}{\setminus } I_v(t))})\nonumber \\&\qquad ~~~~ \times E[\chi _u,u,V|\chi _v,v](t)\nonumber \\&\qquad +\frac{C}{r_c^2}\Vert v\Vert _{W^{1,\infty }}^2(1+e'(t)) E[\chi _u,u,V|\chi _v,v](t)\nonumber \\&\qquad +C\delta ^{-1}\Vert v\Vert _{W^{1,\infty }}^2E[\chi _u,u,V|\chi _v,v](t)\nonumber \\&\qquad +\delta \Vert {D^{{\text {sym}}}}(u{-}v{-}w)\Vert ^2_{L^2}, \end{aligned}$$
(171)
where \(\delta \in (0,1]\) is again arbitrary. All in all, (169) and (171) therefore imply the desired bound. \(\quad \square \)
Estimate for the Additional Advection Terms
We move on with the additional advection terms from the relative entropy inequality of Proposition 10:
$$\begin{aligned} A_{adv} =&-\int _0^T\int _{\mathbb {R}^d}\rho (\chi _u) (u-v-w)\cdot (w\cdot \nabla )(v+w) \,{\mathrm {d}}x\,{\mathrm {d}}t\\\nonumber&-\int _0^T\int _{\mathbb {R}^d}\rho (\chi _u) (u-v-w)\cdot \big ((u-v-w)\cdot \nabla \big )w \,{\mathrm {d}}x\,{\mathrm {d}}t. \end{aligned}$$
(172)
A precise estimate is the content of
Lemma 32
Let the assumptions and notation of Proposition 28 be in place. In particular, we assume that there exists a \(C^1\)-function \(e:[0,{T_{strong}})\rightarrow [0,r_c)\) such that the relative entropy is bounded by \(E[\chi _u,u,V,|\chi _v,v](t)\leqq e^2(t)\). Then the additional advection terms \(A_{adv}\) may be bounded by a Gronwall-type term
$$\begin{aligned} A_{adv} \leqq \frac{C}{r_c^{14}}(1{+}R)\Vert v\Vert _{L^\infty _tW^{3,\infty }_x({\mathbb {R}^d}{\setminus } I_v(t))}^2\int _0^T(1{+}|\log e(t)|) E[\chi _u,u,V|\chi _v,v](t)\,{\mathrm {d}}t. \end{aligned}$$
(173)
Proof
A straightforward estimate yields
$$\begin{aligned} A_{adv}&\leqq C(\Vert v\Vert _{L^\infty _tW^{1,\infty }_x}{+}\Vert \nabla w\Vert _{L^\infty _{x,t}}) \Vert u{-}v{-}w\Vert _{L^2_{x,t}}\bigg (\int _0^T\int _{\mathbb {R}^d}|w|^2\,{\mathrm {d}}x\,{\mathrm {d}}t\bigg )^\frac{1}{2}\\&\quad +C\Vert \nabla w\Vert _{L^\infty _{x,t}}\Vert u{-}v{-}w\Vert _{L^2_{x,t}}^2. \end{aligned}$$
Making use of (95), (99) as well as (78a) immediately shows that the desired bound holds true. \(\quad \square \)
Estimate for the Additional Weighted Volume Term
It finally remains to state the estimate for the additional weighted volume term from the relative entropy inequality of Proposition 10:
$$\begin{aligned} A_{weightVol} := \int _0^T\int _{\mathbb {R}^d}(\chi _u{-}\chi _v)(w\cdot \nabla ) \beta \Big (\frac{{\text {dist}}^{\pm }(\cdot ,I_v)}{r_c}\Big )\,{\mathrm {d}}x\,{\mathrm {d}}t. \end{aligned}$$
(174)
Lemma 33
Let the assumptions and notation of Proposition 28 be in place. In particular, we assume that there exists a \(C^1\)-function \(e:[0,{T_{strong}})\rightarrow [0,r_c)\) such that the relative entropy is bounded by \(E[\chi _u,u,V,|\chi _v,v](t)\leqq e^2(t)\). Then the additional weighted volume term \(A_{weightVol}\) may be bounded by a Gronwall term
$$\begin{aligned} A_{weightVol}\leqq \frac{C}{r_c^{10}}(1+\Vert v\Vert _{L^\infty _tW^{2,\infty }_x({\mathbb {R}^d}{\setminus } I_v(t))}^2) \int _0^TE[\chi _u,u,V|\chi _v,v](t)\,\mathrm{d}t. \end{aligned}$$
(175)
Proof
We may use the exact same argument as in the derivation of the estimate for the term III from the additional surface tension terms \(A_{surTen}\), see (150). \(\quad \square \)
The Weak–Strong Uniqueness Principle with Different Viscosities
Before we proceed with the proof of Theorem 1, let us summarize the estimates from the previous sections in the form of a post-processed relative entropy inequality. The proof is a direct consequence of the relative entropy inequality from Proposition 10 and the bounds (46), (54), (55), (56), (148), (161), (166), (173) and (175).
Proposition 34
(Post-processed relative entropy inequality) Let \(d\leqq 3\). Let \((\chi _u,u,V)\) be a varifold solution to the free boundary problem for the incompressible Navier–Stokes equation for two fluids (1a)–(1c) in the sense of Definition 2 on some time interval \([0,{T_{vari}})\). Let \((\chi _v,v)\) be a strong solution to (1a)–(1c) in the sense of Definition 6 on some time interval \([0,{T_{strong}})\) with \({T_{strong}}\leqq {T_{vari}}\).
Let \(\xi \) be the extension of the inner unit normal vector field \(\mathrm {n}_v\) of the interface \(I_v(t)\) from Definition 13. Let w be the vector field contructed in Proposition 28. Let \(\beta \) be the truncation of the identity from Proposition 10, and let \(\theta \) be the density \(\theta _t=\frac{\mathrm {d}|\nabla \chi _u(\cdot ,t)|}{\mathrm {d}|V_t|_{\mathbb {S}^{d-1}}}\). Let \(e:[0,{T_{strong}})\rightarrow (0,r_c]\) be a \(C^1\)-function and assume that the relative entropy
$$\begin{aligned} E\big [\chi _u,u,V\big |\chi _v,v\big ](T)&:= \sigma \int _{\mathbb {R}^d}1-\xi (\cdot ,T)\cdot \frac{\nabla \chi _u(\cdot ,T)}{|\nabla \chi _u(\cdot ,T)|} \,\mathrm{d}|\nabla \chi _u(\cdot ,T)|\nonumber \\&\qquad + \int _{{\mathbb {R}^d}} \frac{1}{2} \rho \big (\chi _u(\cdot ,T)\big ) \big |u-v-w\big |^2(\cdot ,T) \,{\mathrm {d}}x\nonumber \\&\qquad +\int _{\mathbb {R}^d}\big |\chi _u(\cdot ,T)-\chi _v(\cdot ,T)\big |\,\Big |\beta \Big (\frac{{\text {dist}}^{\pm }(\cdot ,I_v(T))}{r_c}\Big )\Big |\,{\mathrm {d}}x\nonumber \\&\qquad +\sigma \int _{\mathbb {R}^d}1-\theta _T\,\mathrm{d}|V_T|_{\mathbb {S}^{d-1}} \end{aligned}$$
is bounded by \(E[\chi _u,u,V|\chi _v,v](t)\leqq e(t)^2\).
Then the relative entropy is subject to the estimate
$$\begin{aligned}&E[\chi _u,u,V|\chi _v,v](T) +c\int _0^T \int _{\mathbb {R}^d}|\nabla (u-v-w)|^2 {\mathrm {d}}x~{\mathrm {d}}t\nonumber \\&\quad \leqq E[\chi _u,u,V|\chi _v,v](0)\nonumber \\&\qquad + C\int _0^T (1+|\log e(t)|)\,E[\chi _u,u,V|\chi _v,v](t)\,{\mathrm {d}}t\nonumber \\&\qquad + C\int _0^T (1+|\log e(t)|)\,e(t)\sqrt{E[\chi _u,u,V|\chi _v,v](t)}\,{\mathrm {d}}t\nonumber \\&\qquad + C\int _0^T \Big (\frac{\mathrm {d}}{{\mathrm {d}}t}e(t)\Big )E[\chi _u,u,V|\chi _v,v](t)\,{\mathrm {d}}t\end{aligned}$$
(176)
for almost every \(T\in [0,{T_{strong}})\). Here, \(C>0\) is a constant which is structurally of the form \(C=\widetilde{C}r_c^{-22}\) with a constant \(\tilde{C}=\tilde{C}(r_c,\Vert v\Vert _{L^\infty _tW^{3,\infty }_x},\Vert \partial _t v\Vert _{L^\infty _tW^{1,\infty }_x})\), depending on the various norms of the velocity field of the strong solution, the regularity parameter \(r_c\) of the interface of the strong solution, and the physical parameters \(\rho ^\pm \), \(\mu ^\pm \), and \(\sigma \).
We have everything in place to to prove the main result of this work.
Proof of Theorem 1
The proof of Theorem 1 is based on the post-processed relative entropy inequality of Proposition 34. It amounts to nothing but a more technical version of the upper bound
$$\begin{aligned} E(t) \leqq e^{e^{-C t} \log E(0)}, \end{aligned}$$
valid for all solutions of the differential inequality \(\frac{\hbox {d}}{\hbox {d}t} E(t)\leqq C E(t) |\log E(t)|\). However, it is made more technical by the more complex right-hand side (34) in the relative entropy inequality (which involves the anticipated upper bound \(e(t)^2\)) and the smallness assumption on the relative entropy \(E[\chi _u,u,V|\chi _v,v](t)\) needed for the validity of the relative entropy inequality.
We start the proof with the precise choice of the function e(t) as well as the necessary smallness assumptions on the initial relative entropy. We then want to exploit the post-processed form of the relative entropy inequality from Proposition 34 to compare \(E[\chi _u,u,V|\chi _v,v](t)\) with e(t).
Let \(C>0\) be the constant from Proposition 34 and choose \(\delta >0\) such that \(\delta <\frac{1}{6(C+1)}\). Let \(\varepsilon >0\) (to be chosen in a moment, but finally we will let \(\varepsilon \rightarrow 0\)) and consider the strictly increasing function
$$\begin{aligned} e(t) := e^{\frac{1}{2}e^{-\frac{t}{\delta }}\log (E[\chi _u,u,V|\chi _v,v](0)+\varepsilon )}. \end{aligned}$$
(177)
Note that \(e^2(0) = E[\chi _u,u,V|\chi _v,v](0)+\varepsilon \) which strictly dominates the relative entropy at the initial time. To ensure the smallness of this function, let us choose \(c>0\) small enough such that whenever we have \(E[\chi _u,u,V|\chi _v,v](0)<c\) and \(\varepsilon <c\), it holds that
$$\begin{aligned} e(t)<\frac{1}{3C}\wedge r_c \end{aligned}$$
(178)
for all \(t\in [0,{T_{strong}})\). This is indeed possible since the condition in (178) is equivalent to \(\frac{1}{2} \log (E[\chi _u,u,V|\chi _v,v](0)+\varepsilon )<e^{\frac{{T_{strong}}}{\delta }}\log (\frac{1}{3C}\wedge r_c)\). For technical reasons to be seen later, we will also require \(c>0\) be small enough such that
$$\begin{aligned} e^{-\frac{{T_{strong}}}{\delta }}\frac{1}{6\delta } \big |\log (E[\chi _u,u,V|\chi _v,v](0)+\varepsilon )\big |>C \end{aligned}$$
(179)
whenever \(E[\chi _u,u,V|\chi _v,v](0)<c\) and \(\varepsilon <c\). We proceed with some further computations. We start with
$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t} e(t) =\frac{1}{2\delta }|\log (E[\chi _u,u,V|\chi _v,v](0)+\varepsilon )|e(t)e^{-\frac{t}{\delta }} = \frac{1}{\delta }|\log e(t)|e(t). \end{aligned}$$
(180)
This, in particular, entails
$$\begin{aligned} e^2(T) -e^2(\tau )&= \int _\tau ^T \frac{\mathrm {d}}{\mathrm {d}t} e^2(t)\,{\mathrm {d}}t\nonumber \\&=\frac{1}{\delta }|\log (E[\chi _u,u,V|\chi _v,v](0)+\varepsilon )| \int _\tau ^T e^2(t)e^{-\frac{t}{\delta }}\,{\mathrm {d}}t. \end{aligned}$$
(181)
After these preliminary considerations, let us consider the relative entropy inequality from Proposition 10. Arguing similarly to the derivation of the relative entropy inequality in Proposition 10 but using the energy dissipation inequality in its weaker form \(E[\chi _u,u,V|\chi _v,v](T)\leqq E[\chi _u,u,V|\chi _v,v](\tau )\) for a. e. \(\tau \in [0,T]\), we may deduce (upon modifying the solution on a subset of \([0,{T_{strong}})\) of vanishing measure)
$$\begin{aligned} \limsup _{T\downarrow \tau }E[\chi _u,u,V|\chi _v,v](T)\leqq E[\chi _u,u,V|\chi _v,v](\tau ) \end{aligned}$$
(182)
for all \(\tau \in [0,{T_{strong}})\). Now, consider the set \(\mathcal {T}\subset [0,{T_{strong}})\) which contains all \(\tau \in [0,{T_{strong}})\) such that \(\limsup _{T\downarrow \tau }E[\chi _u,u,V|\chi _v,v](T)>e^2(\tau )\). Arguing by contradiction, we assume \(\mathcal {T}\ne \emptyset \) and define
$$\begin{aligned} T^* := \inf \mathcal {T}. \end{aligned}$$
Since \(E[\chi _u,u,V|\chi _v,v](0)<e^2(0)\) and \(e^2\) is strictly increasing, we deduce by the same argument which established (182) that \(T^*>0\). Hence, we can apply Proposition 34 at least for times \(T<T^*\) (with \(\tau = 0\)). However, by the same argument as before the relative entropy inequality from Proposition 10 shows that \(E[\chi _u,u,V|\chi _v,v](T^*)\leqq E[\chi _u,u,V|\chi _v,v](T)+C(T^*-T)\) for all \(T<T^*\), whereas \(E[\chi _u,u,V|\chi _v,v](T)\) may be bounded by means of the post-processed relative entropy inequality. Hence, we obtain using also (177) and (180)
$$\begin{aligned} E[\chi _u,u,V|\chi _v,v](T^*)&\leqq E[\chi _u,u,V|\chi _v,v](0)\nonumber \\&\quad + C\int _0^{T^*} e^2(t)\,{\mathrm {d}}t\nonumber \\&\quad + C\frac{1}{2\delta }\big |\log (E[\chi _u,u,V|\chi _v,v](0)+\varepsilon )\big |\int _0^{T^*} e^3(t)e^{-\frac{t}{\delta }}\,{\mathrm {d}}t\nonumber \\&\quad + C\frac{1}{2}\big |\log (E[\chi _u,u,V|\chi _v,v](0)+\varepsilon )\big |\int _0^{T^*} e^2(t)e^{-\frac{t}{\delta }}\,{\mathrm {d}}t. \end{aligned}$$
(183)
We compare this to the equation (181) for \(e^2(t)\) (with \(\tau =0\) and \(T=T^*\)). Recall that \(e^2(0)\) strictly dominates the relative entropy at the initial time. Because of (179), the second term on the right hand side of (183) is dominated by one third of the right hand side of (181). Because of (178) and the choice \(\delta <\frac{1}{6(C+1)}\) the same is true for the other two terms on the right hand side of (183). In particular, we obtain, using (182) as well, that
$$\begin{aligned} \limsup _{T\downarrow T^*}E[\chi _u,u,V|\chi _v,v](T) - e^2(T^*) \leqq E[\chi _u,u,V|\chi _v,v](T^*) - e^2(T^*) < 0, \end{aligned}$$
which contradicts the definition of \(T^*\). This concludes the proof since the asserted stability estimate as well as the weak–strong uniqueness principle is now a consequence of letting \(\varepsilon \rightarrow 0\). \(\quad \square \)