On the Motion of a Compressible Gravity Water Wave with Vorticity

Abstract

We prove a priori estimates for the compressible Euler equations modeling the motion of a liquid with moving physical vacuum boundary with unbounded initial domain. The liquid is under influence of gravity but without surface tension. Our fluid is not assumed to be irrotational. But the physical sign condition needs to be assumed on the free boundary. We generalize the method used in Lindblad and Luo (Commun Pure Appl Math, 2008) to prove the energy estimates in an unbounded domain up to arbitrary order. In addition to that, the a priori energy estimates are in fact uniform in the sound speed $\kappa$. As a consequence, we obtain the convergence of solutions of compressible Euler equations with a free boundary to solutions of the incompressible equations, generalizing the result of Lindblad and Luo (2008) to when you have an unbounded domain. On the other hand, we prove that there are initial data satisfying the compatibility condition in some weighted Sobolev spaces, and this will propagate within a short time interval, which is essential for proving long time existence for slightly compressible irrotational water waves.

Appendix

List of Notations

• $D_{t}$: the material derivative

• $\partial _i$: partial derivative with respect to Eulerian coordinate $x_i$

• $\mathcal {D}_t\in \mathbb {R}^n$: the domain occupied by fluid particles at time t in Eulerian coordinate

• $\Omega \in \mathbb {R}^n$: the domain occupied by fluid particles in Lagrangian coordinate

• $\partial _a = \frac{\partial }{\partial y_a}$: partial derivative with respect to Lagrangian coordinate $y_a$

• $\nabla _a$: covariant derivative with respect to $y_a$

• $\Pi S$: projected tensor S on the boundary

• $\overline{\nabla },\overline{\partial }$: projected derivative on the boundary

• N: the outward unit normal of the boundary

• $\theta =\overline{\nabla }N$: the second fundamental form of the boundary

• $\sigma = tr(\theta )$: the mean curvature

• $\kappa =\kappa (x)$: the sound speed

• $L^{p}_w(\Omega )$: The weighted $L^p$ space

• $W^{s,p}_w(\Omega )$: The weighted Sobolev space

Mixed norms

• $\langle \langle \cdot \rangle \rangle _r = \sum _{k+s=r}||\nabla ^sD_t^k\cdot ||_{L^{2}(\partial \Omega )}$

• $||\cdot ||_{r,0} = \sum _{s+k=r,k<r}||\nabla ^s D_t^k\cdot ||_{L^{2}(\Omega )}$

• $||\cdot ||_{r} = ||\cdot ||_{r,0}+||\sqrt{e'(h)}D_t^r\cdot ||_{L^{2}(\Omega )}$

• $||\cdot ||_{r,1,0} = \sum _{k+s=r, k<r-1}||\nabla ^s D_t^k \cdot ||_{L^{2}(\Omega )} + ||\sqrt{e'(h)}\nabla D_t^{r-1}\cdot ||_{L^{2}(\Omega )}$,

• $||\cdot ||_{r,1} = ||\cdot ||_{r,1,0} + ||e'(h)D_t^r\cdot ||_{L^{2}(\Omega )}$.

Weighted norms

• $||u||_{L^p_w(\Omega )} = (\int _{\Omega }|u(x)|^p w(x)\,dx)^{1/p}$

• $||u||_{W^{s,p}_w(\Omega )} = \sum _{|\alpha |\le s}||\nabla ^{\alpha }u||_{L^p_w(\Omega )}$

The Geometry of the Boundary, Extension of Normal to the Interior and the Geodesic Normal Coordinate

The definition of our energy (1.12) relies on extending the normal to the interior, which is done by foliating the domain close to the boundary into the surface that do not self-intersect. We also want to control the time evolution of the boundary, which can be measured by the time derivative of the normal in the Lagrangian coordinate. We conclude the above statements by the following two lemmas, whose proof can be found in [5].

Lemma A.1

let $l_0$ be the injective radius (1.10), and let $d(y)=dist_g(y,\partial \Omega )$ be the geodesic distance in the metric g from y to $\partial \Omega$. Then the co-normal $n=\nabla d$ to the set $S_a=\partial \{y\in \Omega :d(y)=a\}$ satisfies, when $d(y)\le \frac{l_0}{2}$ that

$$\begin{aligned}&|\nabla n| \lesssim |\theta |_{L^{\infty }(\partial \Omega )}, \end{aligned}$$

(A.1)

$$\begin{aligned}&|D_t n| \lesssim |D_t g|_{L^{\infty }(\Omega )}, \end{aligned}$$

(A.2)

where we have used the convention that $A\lesssim B$ means $A\le CB$ for universal constant C.

Lemma A.2

let $l_0$ be the injective radius (1.10),and let $d_0$ be a fixed number such that $\frac{l_0}{16}\le d_0\le \frac{l_0}{2}$. Let $\eta$ be a smooth cut-off function satisfying $0\le \eta (d)\le 1$, $\eta (d)=1$ when $d\le \frac{d_0}{4}$ and $\eta (d)=0$ when $d>\frac{d_0}{2}$. Then the psudo-Riemannian metric $\gamma$ given by

$$\begin{aligned} \gamma _{ab}=g_{ab}-{\tilde{n}}_a{\tilde{n}}_b, \end{aligned}$$

where ${\tilde{n}}_c=\eta (\frac{d}{d_0})\nabla _cd$ satisfies

$$\begin{aligned} |\nabla \gamma |_{L^{\infty }(\Omega )}\lesssim \left( |\theta |_{L^{\infty }(\partial \Omega )}+\frac{1}{l_0}\right)&\end{aligned}$$

(A.3)

$$\begin{aligned} |D_t\gamma (t,y)|\lesssim |D_t g|_{L^{\infty }(\Omega )}.&\end{aligned}$$

(A.4)

Remark

The above two lemmas yield that the quantities $|D_t n|$ and $|D_t\gamma (t,y)|$ involved in the Q-inner product is controlled by the a priori assumptions (1.24)–(1.29),since $D_tg$ behaves like $\nabla v$ by (2.8). Hence, the time derivative on the coefficients of the Q-inner product generates only lower-order terms. In addition, by (1.24) , $|\nabla n|$ and $|\nabla \gamma |$ are controlled by K, which is essential when proving the Christodoulou-Lindblad type elliptic estimates.

The next lemma introduces the partition of unity $\{\chi _i\}$ in a domain with sufficient regular boundary.

Lemma A.3

Suppose that $\Omega \in \mathbb {R}^n$ is a domain whose boundary satisfying the condition $|\theta |+\frac{1}{l_0}\le K$. Then there are functions $\chi _i\in C_c^{\infty }(\mathbb {R}^n), i=1,2,\cdots$, such that

$$\begin{aligned} 0\le \chi _i\le 1,\quad \sum \chi _i =1,\quad \sum |\partial ^{\alpha }\chi _i|\le C_{\alpha }K^{|\alpha |},\quad diam(supp(\chi _i))\le K^{-1}, \end{aligned}$$

(A.5)

and for each $x\in \mathbb {R}^n$, there are at most $16^n$ i’s such that $\chi _i(x)\ne 0$. Furthermore, either $supp(\chi _i)\cup \partial \Omega$ is empty or is part of a graph contained in $\partial \Omega$, for which (possibly after a rotation) is given by

$$\begin{aligned} x_n = f_i(x'),\quad |\partial f_i|\le c_1,\quad N(x_i)=e_n,\quad \text {for}\,\, |x'-x_i'|\le l_0. \end{aligned}$$

(A.6)

Proof

See [5]. $\square$

Sobolev Lemmas

Let us now state some Sobolev lemmas in a domain with boundary, whose proofs are standard and can be found in [5, 11, 25].

Lemma A.4

(Interior Sobolev inequalities) Suppose $\frac{1}{l_0}\le K$ and $\alpha$ is a (0, r) tensor, then

$$\begin{aligned} ||\alpha ||_{L^{\frac{2n}{n-2s}}(\Omega )}\lesssim _{K} \sum _{l=0}^s||\nabla ^l\alpha ||_{L^{2}(\Omega )},\quad 2s<n,&\end{aligned}$$

(A.7)

$$\begin{aligned} ||\alpha ||_{L^{\infty }(\Omega )}\lesssim _{K} \sum _{l=0}^{s}||\nabla ^l\alpha ||_{L^{2}(\Omega )},\quad 2s>n.&\end{aligned}$$

(A.8)

These inequalities remains valid in weighted spaces $L^p_w(\Omega )$ if the weight satisfies $|\partial ^r w|\le C_r w/(1+|x|)^r$.

Proof

See [5]. $\square$

Similarly, on the boundary $\partial \Omega$, we have

Lemma A.5

(Boundary Sobolev inequalities)

$$\begin{aligned} ||\alpha ||_{L^{\frac{2(n-1)}{n-1-2s}}(\Omega )}\lesssim _{K} \sum _{l=0}^s||\nabla ^l\alpha ||_{L^{2}(\partial \Omega )},\quad 2s<n-1,&\end{aligned}$$

(A.9)

$$\begin{aligned} ||\alpha ||_{L^{\infty }(\Omega )}\lesssim _{K} \delta ||\nabla ^s\alpha ||_{L^{2}(\partial \Omega )}+\delta ^{-1}\sum _{l=0}^{s-1}||\nabla ^l\alpha ||_{L^{2}(\partial \Omega )},\quad 2s>n-1,&\end{aligned}$$

(A.10)

for any $\delta >0$. These inequalities remain valid in weighted spaces $L^p_w(\Omega )$ as well. In addition, for the boundary we can also interpret the norm be given by the inner product $\langle \alpha , \alpha \rangle =\gamma ^{IJ}\alpha _I\alpha _J$, and the covariant derivative is then given by $\overline{\nabla }$.

Interpolation on Spatial Derivatives

We shall first record spatial interpolation inequalities. Most of the results are are standard in $\mathbb {R}^n$, but we must control how it depends on the geometry of our evolving domain. The coefficients involved in our inequalities depend on K, whose reciprocal is the lower bound for the injective radius $l_0$.

Theorem A.6

(Interior interpolation) Let u be a (0, s) tensor, and suppose $\frac{1}{l_0}\le K$, we have

$$\begin{aligned} \sum _{j=0}^{l}||\nabla ^ju||_{L^{\frac{2r}{k}}(\Omega )}\lesssim ||u||_{L^{\frac{2(r-l)}{k-l}}(\Omega )}^{1-\frac{l}{r}}\left( \sum _{i=0}^{r}||\nabla ^iu||_{L^{2}(\Omega )}K^{r-i}\right) ^{\frac{l}{r}}. \end{aligned}$$

(A.11)

In particular, if $k=l$,

$$\begin{aligned} \sum _{j=0}^{k}||\nabla ^ju||_{L^{\frac{2r}{k}}(\Omega )}\lesssim ||u||_{L^{\infty }(\Omega )}^{1-\frac{k}{r}}\left( \sum _{i=0}^{r}||\nabla ^iu||_{L^{2}(\Omega )}K^{r-i}\right) ^{\frac{k}{r}}. \end{aligned}$$

(A.12)

These inequalities remains valid when $L^p(\Omega )$ is replaced by $L^p_w(\Omega )$ if $w\ge 0$ satisfies $|\partial ^r w|\le C_r w / (1+|x|)^r$.

Proof

It suffices to prove (A.11) with $s=0$, i.e., when u is a function, since u can be replaced by its magnitude |u|. Furthermore, since (A.11) is equivalent to

$$\begin{aligned} \sum _{j\le l}||\nabla ^j u||_{L^s(\Omega )}\le C(K)||u||_{L^q(\Omega )}^{1-a}\left( \sum _{i\le r}||\nabla ^i u||_{L^p(\Omega )}\right) ^a, \end{aligned}$$

(A.13)

where $a=l/r$ and $\frac{r}{s}=\frac{l}{p}+\frac{r-l}{q}$. We can further reduce (A.13) to the case when $r=2$ and $s=1$, because the general cases follow from the logarithmic convexity.

Using the partition of unity $\{\chi _i\}$ defined in Lemma A.3, we write $u=\sum u_i$, where $u_i=\chi _iu$. In a neighbourhood of $supp(\chi _i)$, we can then write $\Omega$ as a graph after a rotation:

$$\begin{aligned} x_n = f(x'),\quad |\partial f|\le C. \end{aligned}$$

We now define the reflection

$$\begin{aligned} {\tilde{u}}_i(x)= {\left\{ \begin{array}{ll} u_i(x),\quad \text {when}\,x\in \Omega \\ u_i({\tilde{x}}),\quad \text {when}\, x\in \Omega ^c \end{array}\right. } \end{aligned}$$

Here, ${\tilde{x}}=(x',x_n-2(x_n-f(x'))$. Then by the interpolation in $\mathbb {R}^n$, we have

$$\begin{aligned} ||\nabla {\tilde{u}}_i||_{L^s(\mathbb {R}^n)}^2\le ||{\tilde{u}}_i||_{L^q(\mathbb {R}^n)}||\nabla ^2{\tilde{u}}_i||_{L^p(\mathbb {R}^n)}. \end{aligned}$$

But since for every $1\le p'\le \infty$ and $|{\partial {\tilde{x}}^i}/{\partial x^j}|\le C$,

$$\begin{aligned} ||\nabla ^{\alpha }{\tilde{u}}_i||_{L^{p'}(\mathbb {R}^n)}\le C(||\nabla ^{\alpha }u_i||_{L^{p'}(\Omega )}+||\nabla ^{\alpha }{\tilde{u}}_i||_{L^{p'}(\Omega ^c)}) \le C||\nabla ^{\alpha }u_i||_{L^{p'}(\Omega )}, \end{aligned}$$

for $|\alpha |\le 2$. Furthermore, we have

$$\begin{aligned}&||\nabla u_i||_{L^{p'}(\Omega )} \le C||(\nabla \chi _i)u||_{L^{p'}(\Omega )}+C||\chi _i \nabla u||_{L^{p'}(\Omega )},\\&\quad ||\nabla ^2 u_i||_{L^{p'}(\Omega )} \le C||(\nabla ^2\chi _i)u||_{L^{p'}(\Omega )}+C||(\nabla \chi _i)\nabla u||_{L^{p'}(\Omega )}+C||\chi _i \nabla ^2 u||_{L^{p'}(\Omega )} \end{aligned}$$

and this gives (A.13) via Lemma A.3 for $l=1$ and $r=2$. The general case follows by letting $M_k=\sum _{i\le k}||\nabla ^i u||_{L^{s(k)}}$, and so far we have proven $M_1\lesssim M_0M_2$, and hence we get $M_k^2\lesssim M_{k-1}M_{k+1}$ follows from this special case. But the logarithmic convexity then gives $M_k\lesssim M_0^{(r-l)/r}M_r^{l/r}$. Finally, the weighted case follow from the non-weighted case since $|\partial ^r w|\lesssim |w|/(1+|x|)^r$. $\square$

Interpolation on $\partial \Omega$

Theorem A.7

(Boundary interpolation) Let u be a (0, s) tensor, then

$$\begin{aligned} ||\overline{\nabla }^l u||_{L^{\frac{2r}{k}}(\partial \Omega )}\lesssim ||u||_{L^{\frac{2(r-l)}{k-l}}(\partial \Omega )}^{1-\frac{l}{r}}||\overline{\nabla }^ru||_{L^{2}(\partial \Omega )}^{\frac{l}{r}}. \end{aligned}$$

(A.14)

In particular, if $k=l$,

$$\begin{aligned} ||\overline{\nabla }^k u||_{L^{\frac{2r}{k}}(\partial \Omega )}\lesssim ||u||_{L^{\infty }(\partial \Omega )}^{1-\frac{k}{r}}||\overline{\nabla }^ru||_{L^{2}(\partial \Omega )}^{\frac{k}{r}}. \end{aligned}$$

(A.15)

Furthermore, if $w\ge 0$ satisfies $|\partial ^r w|\le C_rw/(1+|x|)^r$, then

$$\begin{aligned} ||\overline{\nabla }^l u||_{L^{\frac{2r}{k}_w}(\partial \Omega )}\lesssim ||u||_{L^{\frac{2(r-l)}{k-l}_w}(\partial \Omega )}^{1-\frac{l}{r}}\left( \sum _{i\le r}||\overline{\nabla }^i u||_{L^2_w(\partial \Omega )}^{\frac{l}{r}}\right) . \end{aligned}$$

(A.16)

Proof

The proof for (A.14) can be found in [5], and 5.25 follows from the same proof and the lower order terms on the RHS is generated when the derivatives fall on the weight function w. $\square$

Elliptic Estimates in Weighted Sobolev Spaces

This section is devoted to set up the elliptic estimates in weighted Sobolev spaces $H^{s}_{w}(\Omega )$ (Definition 7.1) with weight $w(x) = (1+|x|^2)^\mu$, $\mu \ge 2$, where $\Omega \subset \mathbb {R}^n, n=2,3$ be a smooth domain, diffeomorphic to the half space $\{x\in \mathbb {R}^n:x_n\le 0\}$. Consider the Dirichlet boundary value problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta u = f,\quad \text {in}\,\, \Omega \\ u=0,\quad \text {in}\,\, \partial \Omega \end{array}\right. } \end{aligned}$$

(A.17)

then the following $L^2$ elliptic estimate holds.

Theorem A.8

(Boccia-Salvato-Transirico [2]) Fix $s\ge 2$ and $p\in (0,\infty )$, then

$$\begin{aligned} ||u||_{W^{s,p}_w(\Omega )} \le C(||f||_{W^{s-2, p}_w(\Omega )}+||u||_{L^p_w(\Omega )}), \end{aligned}$$

(A.18)

holds for all $u\in W^{s,p}_w(\Omega )$ that solves (A.17).

Now we show that the $||u||_{L^p_w(\Omega )}$ on the RHS of (A.18) can in fact be dropped. It is worth to mention here that we have no problem to drop this term if $\Omega$ were bounded, since $\lambda =0$ is not an eigenvalue of $\Delta$ in this case (e.g, chapter 6.2 in Evans [11]). However, it is in general impossible to drop the term $||u||_{L^2}$ in elliptic estimates when $\Omega$ is unbounded, unless u is sufficiently smooth and decays fast enough at infinity.

Theorem A.9

(Rellich-Kondrachov embedding for weighted spaces) The spaces $H^1_{0,w}(\Omega )$ (the space consists of $u\in H^{1}_w(\Omega )$ with $u|_{\partial \Omega }=0$) are compactly embedding in the spaces $L^q(\Omega )$ for any $q<2n/(n-2)$.

Proof

We follow the proof given by Gilbarg-Trudinger [13] with some modifications. We initially assume $q=1$. Let $\mathcal {A}$ be a bounded subset in $H^{1}_{0,w}(\Omega )$. Without loss of generality we assume that $\mathcal {A}\in C_c^1(\Omega )$ and that $||u||_{H^{1}_w(\Omega )}\le 1$. For fixed $\delta >0$, let $\mathcal {A}_\delta :=\{u_{\delta }:u\in \mathcal {A}\}$, where $u_\delta$ is the mollification of u, i.e., $u_{\delta } = \eta _{\delta }*u$, where $\eta (x)$ is a smooth bump function supported in the unit ball satisfying $\int \eta (x)\,dx=1$, and $\eta _{\delta } = \delta ^{-n}\eta (\delta ^{-1}x)$.

For each $u\in \mathcal {A}$, we have

$$\begin{aligned} ||u_{\delta }(x)||_{L^{\infty }(\Omega )}\le \delta ^{-n}||\eta ||_{L^{\infty }(\Omega )}||u||_{H^{1}_w(\Omega )},&\\ ||\nabla u_{\delta }(x)||_{L^{\infty }(\Omega )}\le \delta ^{-n-1}||\nabla \eta ||_{L^{\infty }(\Omega )}||u||_{H^{1}_w(\Omega )},&\end{aligned}$$

and so $\mathcal {A}_\delta$ is a bounded, equicontinuous subset of $C_c(\Omega )$ and hence precompact in $C_c(\Omega )$, and consequently also precompact in $L^1(\Omega )$. Nevertheless, we have

$$\begin{aligned}&|u(x) - u_{\delta }(x)|\le \int _{|z|\le 1}\eta (z)|u(x)-u(x-\delta z)\,dz\\&\quad \le \int _{|z|\le 1}\eta (z)\int _0^{\delta |z|}|\nabla _r u\left( x-r\frac{z}{|z|}\right) \,dr\,dz, \end{aligned}$$

and hence

$$\begin{aligned}&\int _{\Omega }|u(x) - u_{\delta }(x)|\,dx\le \delta \int _{\Omega }|\nabla u|\,dx\\&\quad \le \delta \left( \int _{\mathbb {R}^n}\frac{1}{(1+|x|^2)^\mu }\,dx\right) ^{1/2}||u||_{H^1_w(\Omega )}. \end{aligned}$$

But since $\int _{\mathbb {R}^n}\frac{1}{(1+|x|^2)^\mu }\,dx<\infty$ when $n\le 3$ and so $u_\delta$ is uniformly close to u in $L^1(\Omega )$. It then follows that $\mathcal {A}$ is precompact in $L^1(\Omega )$. Now, for any $q<2n/(n-2)$, we have

$$\begin{aligned} ||u||_{L^q(\Omega )} \lesssim ||u||_{L^1}^a||u||_{L^{2n/(n-2)}}^{1-a} \end{aligned}$$

for some $0<a<1$ via interpolation. In addition, we have

$$\begin{aligned} ||u||_{L^{2n/(n-2)}}\lesssim ||u||_{H^1_w}, \end{aligned}$$

by Sobolev lemma and the fact that $w(x)\ge 1$. This concludes that a bounded set in $H^1_{0,w}(\Omega )$ must be precompact in $L^q(\Omega )$. $\square$

Remark

The classical Rellich-Kondrachov embedding theorem yields that $H^1(\Omega )$ is compactly embedding in the spaces $L^q(\Omega )$ when $\Omega$ is bounded.

Theorem A.10

(Improved elliptic estimates) Let $u\in H^s_w(\Omega )\cap H^1_{0,w}(\Omega )$ be a function that solves (A.17), and if $f\in H^{s-2}_w(\Omega )$ then

$$\begin{aligned} ||u||_{H^{s}_w(\Omega )} \le C||f||_{H^{s-2}_w(\Omega )}. \end{aligned}$$

(A.19)

Proof

It suffices to prove (A.19) when $s=2$. If (A.19) is not true, then there exists a sequence $\{u_m\}\subset H^2_w(\Omega )\cap H^1_{0,w}(\Omega )$ satisfying

$$\begin{aligned} ||u_m||_{L^2_w(\Omega )}=1,\quad ||u_m||_{L^2(\Omega )}\le 1,\quad ||\Delta u_m||_{L^2_w(\Omega )}\rightarrow 0. \end{aligned}$$

By virtue of the apriori estimate (A.18), Theorem A.9, and the weakly compactness of bounded subsets in $H^2_w(\Omega )$, there exists a subsequence, relabelled as $\{u_m\}$, converging weakly to a function $u\in H^2_w(\Omega )\cap H^1_{0,w}(\Omega )$ satisfying $||u||_{L^2_w(\Omega )}= 1$. However, for any $\phi \in L_w^2(\Omega )$, we must have

$$\begin{aligned} \int _{\Omega } \phi (\Delta u)w = 0. \end{aligned}$$

Hence, $\Delta u =0$ and so $u=0$ by the uniqueness assertion (e.g. G-T [13], Theorem 8.9 or maximum principle since u decays to 0 at $\infty$). But this implies $||u||_{L^2_w}=0$, a contradiction. $\square$

Gagliardo-Nirenberg Interpolation Inequality

Theorem A.11

Let u be a (0, r) tensor defined on $\partial \Omega \in \mathbb {R}^2$ and suppose $\frac{1}{l_0}\le K$, we have

$$\begin{aligned} ||u||_{L^4{(\partial \Omega })}^2 \lesssim _K ||u||_{L^{2}(\partial \Omega )}||u||_{H^1(\partial \Omega )}, \end{aligned}$$

(A.20)

where $H^1(\partial \Omega )$ is defined via tangential derivative $\overline{\nabla }$. Furthermore, (A.20) remains valid in the case of weighted Sobolev spaces.

Proof

It suffices for us to work in the local coordinate charts $\{U_i\}$ of $\partial \Omega$. We consider the corresponding partition of unity $\{\chi _i\}$, where each $\chi _i$ is supported in $U_i$ and vanishing on the boundary of $U_i$. As proved in Lemma A.3, $\chi _i$ can be chosen to satisfy

$$\begin{aligned} \sum _i |\overline{\nabla }\chi _i| \le C(K). \end{aligned}$$

Now by the result of Constantin and Seregin [6], we have

$$\begin{aligned} ||u_i||_{L^4(U_i)}^2 \lesssim ||u_i||_{L^2(U_i)}||\overline{\nabla }u_i||_{L^2(U_i)}, \end{aligned}$$

where $u_i=\chi _i u$. But since

$$\begin{aligned} ||\overline{\nabla }u_i||_{L^2(U_i)} = ||\overline{\nabla }(\chi _i u)||_{L^2(U_i)} \le |\overline{\nabla }\chi _i|_{L^{\infty }}||u||_{L^2(U_i)}+||\chi _i\overline{\nabla }u||_{L^2(U_i)} . \end{aligned}$$

Hence, (A.20) follows by summing up (A.11). This proof remains valid with $L^p$ being replaced by $L^p_w$, where w is defined in Section A.5. $\square$

The Trace Theorem

Theorem A.12

(The trace theorem) Let $\alpha$ be a (0, r) tensor, and assume that $|\theta |_{L^{\infty }(\partial \Omega )}+\frac{1}{l_0}\le K$, then

$$\begin{aligned} ||\alpha ||_{L^{2}(\partial \Omega )}\lesssim _{K} \sum _{j\le 1}||\nabla ^j\alpha ||_{L^{2}(\Omega )}. \end{aligned}$$

(A.21)

Furthermore,

$$\begin{aligned} ||\alpha ||_{L^2_w(\partial \Omega )}\lesssim _{K} \sum _{j\le 1}||\nabla ^j\alpha ||_{L^2_w(\Omega )}. \end{aligned}$$

(A.22)

Here, w is defined in Section A.5.

Proof

It suffices to show (A.22) only, since the proof for (A.21) is almost identical. Let N be the extension of the normal in the interior of $\Omega$ given by the geodesic normal coordinate (i.e., Lemma A.1). Then

$$\begin{aligned} \int _{\partial \Omega }|\alpha |^2w\,d\mu _{\gamma } = \int _{\Omega }\nabla _k(N^k|\alpha |^2w)\,d\mu _{g}. \end{aligned}$$

(A.23)

But since $|\nabla N|\le K$ and $|\nabla w|\le Cw$, (A.22) follows. $\square$