Traveling Wave Solutions to the One-Phase Muskat Problem: Existence and Stability

Abstract

We study the Muskat problem for one fluid in an arbitrary dimension, bounded below by a flat bed and above by a free boundary given as a graph. In addition to a fixed uniform gravitational field, the fluid is acted upon by a generic force field in the bulk and an external pressure on the free boundary, both of which are posited to be in traveling wave form. We prove that, for sufficiently small force and pressure data in Sobolev spaces, there exists a locally unique traveling wave solution in Sobolev-type spaces. The free boundary of the traveling wave solutions is either periodic or asymptotically flat at spatial infinity. Moreover, we prove that small periodic traveling wave solutions induced by external pressure only are asymptotically stable. These results provide the first class of nontrivial stable solutions for the problem.

Appendix A. Analytic Tools

This appendix collects a number of analysis tools used throughout the paper.

1.1 A.1 Specialized Scales of Anisotropic Sobolev Spaces

In this subsection we recall the properties of a scale of anisotropic Sobolev spaces introduced in [9].

Definition A.1

Let $0 \leqq s \in \mathbb {R}$.

• (1) We define the anisotropic Sobolev-type space

  $$\begin{aligned} \mathcal {H}^s(\mathbb {R}^d) = \{ f \in \mathscr {S}'(\mathbb {R}^d) \;\vert \;f = \bar{f}, \hat{f} \in L^1_{loc}(\mathbb {R}^d), \text { and } \left\| f\right\| _{\mathcal {H}^s} < \infty \}, \nonumber \\ \end{aligned}$$

  (A.1)

  where the square-norm is defined by

  $$\begin{aligned} \left\| f\right\| _{\mathcal {H}^s}^2 = \int _{B(0,1)} \frac{\xi _1^2 + \left| \xi \right| ^4 }{\left| \xi \right| ^2} \left| \hat{f}(\xi )\right| ^2 \textrm{d}\xi + \int _{B(0,1)^c} \langle \xi \rangle ^{2s} \left| \hat{f}(\xi )\right| ^2 \textrm{d}\xi .\nonumber \\ \end{aligned}$$

  (A.2)

  We endow the space $\mathcal {H}^s(\mathbb {R}^d)$ with the obvious associated inner-product. We write $\mathcal {H}^s(\Sigma ) = \mathcal {H}^s(\mathbb {R}^{n-1})$ with the usual identification of $\Sigma \simeq \mathbb {R}^{n-1}$.

• (2) We define

  $$\begin{aligned} \mathcal {H}^s(\mathbb {T}^d) = \mathring{H}^s(\mathbb {T}^d) = \left\{ f \in H^s(\mathbb {T}^d) \;\vert \;\int _{\mathbb {T}^d} f = 0 \right\} \end{aligned}$$

  (A.3)

  with the usual norm.

• (3) We write $\mathcal {H}^s(\Sigma ) = \mathcal {H}^s(\Gamma )$ via the natural identification of $\Sigma = \Gamma \times \{0\}$ with $\Gamma \in \{\mathbb {R}^{n-1},\mathbb {T}^{n-1}\}$.

The following result summarizes the fundamental properties of this space:

Theorem A.2

Let $0 \leqq s \in \mathbb {R}$. Then the following hold.

• (1) $\mathcal {H}^s(\mathbb {R}^d)$ is a Hilbert space, and the set of real-valued Schwartz functions is dense in $\mathcal {H}^s(\mathbb {R}^d)$.

• (2) $H^s(\mathbb {R}^d) \hookrightarrow \mathcal {H}^s(\mathbb {R}^d)$, and this inclusion is strict for $d\geqq 2$. If $d =1$, then $H^s(\mathbb {R}) = \mathcal {H}^s(\mathbb {R})$.

• (3) If $t \in \mathbb {R}$ and $s < t$, then we have the continuous inclusion $\mathcal {H}^t(\mathbb {R}^d) \hookrightarrow \mathcal {H}^s(\mathbb {R}^d)$.

• (4) For $R \in \mathbb {R}_+$ and $f \in \mathcal {H}^s(\mathbb {R}^d)$ define the low-frequency localization $f_{l,R} = (\hat{f} {\chi }_{B(0,R)})^{\vee }$ and the high-frequency localization $f_{h,R} = (\hat{f} {\chi }_{B(0,R)^c})^\vee$. Then $f_{l,R} \in \bigcap _{t \geqq 0} \mathcal {H}^t(\mathbb {R}^d) \cap \bigcap _{k\in \mathbb {N}} C^k_0(\mathbb {R}^d)$, and $f_{h,R} \in \mathcal {H}^s(\mathbb {R}^d) \cap H^s(\mathbb {R}^d)$. Moreover, we have the estimates

  $$\begin{aligned} \left\| f_{l,R}\right\| _{\mathcal {H}^s} \leqq \left\| f\right\| _{\mathcal {H}^s} \text { and } \left\| f_{h,R}\right\| _{\mathcal {H}^s} \leqq \left\| f\right\| _{\mathcal {H}^s} \end{aligned}$$

  (A.4)

  as well as

  $$\begin{aligned} \left\| f_{l,R}\right\| _{C^k_0} \lesssim _k \left\| f\right\| _{\mathcal {H}^s}, \left\| f_{l,R}\right\| _{\mathcal {H}^t} \lesssim \left\| f\right\| _{\mathcal {H}^s}, \text { and } \left\| f_{h,R}\right\| _{H^s} \lesssim \left\| f\right\| _{\mathcal {H}^s}. \nonumber \\ \end{aligned}$$

  (A.5)

• (5) For each $k \in \mathbb {N}$ we have the continuous inclusion $\mathcal {H}^s(\mathbb {R}^d) \hookrightarrow C^k_0(\mathbb {R}^d) + H^s(\mathbb {R}^d)$.

• (6) If $s > d/2$ then $\hat{f}\in L^1(\mathbb {R}^d;\mathbb {C})$, and

  $$\begin{aligned} \left\| \hat{f}\right\| _{L^1} \lesssim \left\| f\right\| _{\mathcal {H}^s}. \end{aligned}$$

  (A.6)

• (7) If $k \in \mathbb {N}$ and $s >k+ d/2$, then we have the continuous inclusion $\mathcal {H}^s(\mathbb {R}^d) \hookrightarrow C^k_0(\mathbb {R}^d)$.

• (8) If $s \geqq 1$, then $\left\| \nabla f\right\| _{H^{s-1}} \lesssim \left\| f\right\| _{\mathcal {H}^s}$ for each $f \in \mathcal {H}^s(\mathbb {R}^d)$. In particular, we have that $\nabla : \mathcal {H}^s(\mathbb {R}^d) \rightarrow H^{s-1}(\mathbb {R}^d;\mathbb {R}^d)$ is a bounded linear map, and this map is injective.

• (9) If $s \geqq 1$, then $\left[ \partial _1 f\right] _{\dot{H}^{-1}} \lesssim \left\| f\right\| _{\mathcal {H}^s}$ for $f \in \mathcal {H}^s(\mathbb {R}^d)$. In particular, we have that $\partial _1: \mathcal {H}^s(\mathbb {R}^d) \rightarrow H^{s-1}(\mathbb {R}^d)\cap \dot{H}^{-1}(\mathbb {R}^d)$ is a bounded linear map, and this map is injective.

Proof

These are proved in Proposition 5.2 and Theorems 5.5 and 5.6 of [9]. $\square$

Next we recall another space introduced in [9] that will be useful in working with the Poisson extension of functions in $\mathcal {H}^s(\Sigma )$.

Definition A.3

Let $0 \leqq s \in \mathbb {R}$ and $n \geqq 2$.

• (1) When $\Gamma = \mathbb {R}^{n-1}$ we define the space

  $$\begin{aligned} \mathbb {P}^s(\Omega )= & {} H^s(\Omega ) + \mathcal {H}^s(\Sigma ) = \{f \in L^1_{\text {loc}}(\Omega ) \;\vert \;\text {there exist } g \in H^s(\Omega ) \text { and } h \in \mathcal {H}^s(\Sigma ) \nonumber \\{} & {} \text { such that } f(x) = g(x) + h(x') \text { for almost every }x \in \Omega \}. \end{aligned}$$

  (A.7)

  We endow $\mathbb {P}^s(\Omega )$ with the norm

  $$\begin{aligned} \left\| f\right\| _{\mathbb {P}^s} = \inf \{ \left\| g\right\| _{H^s} + \left\| h\right\| _{\mathcal {H}^s} \;\vert \;f = g +h \text { for } g \in H^s(\Omega ), h\in \mathcal {H}^s(\mathbb {R}^{n-1}) \}. \nonumber \\ \end{aligned}$$

  (A.8)

• (2) When $\Gamma = \mathbb {T}^{n-1}$ we define the space $\mathbb {P}^s(\Omega ) = H^s(\Omega )$ and endow it with the usual $H^s(\Omega )$ norm.

The key properties of this space are recorded in the following.

Theorem A.4

Let $0 \leqq s \in \mathbb {R}$ and $n \geqq 2$. Then the following hold.

• (1) If $t \in \mathbb {R}$ and $s < t$, then we have the continuous inclusion $\mathbb {P}^t(\Omega ) \subset \mathbb {P}^s(\Omega )$.

• (2) For each $f \in \mathcal {H}^s(\Sigma )$ we have that $\left\| f\right\| _{\mathbb {P}^s} \leqq \left\| f\right\| _{\mathcal {H}^s}$, and hence we have the continuous inclusion $\mathcal {H}^s(\Sigma ) \subset \mathbb {P}^s(\Omega )$.

• (3) If $k \in \mathbb {N}$ and $s >k+ n/2$, then $\left\| f\right\| _{C^k_b} \lesssim \left\| f\right\| _{\mathbb {P}^s}$ for all $f \in \mathbb {P}^s(\Omega )$. Moreover, we have the continuous inclusion

  $$\begin{aligned} \mathbb {P}^s(\Omega ) \subseteq \{f \in C^k_b(\Omega ) \;\vert \;\lim _{\left| x'\right| \rightarrow \infty } \partial ^\alpha f(x) = 0 \text { for } \left| \alpha \right| \leqq k\} \subset C^k_b(\Omega ). \nonumber \\ \end{aligned}$$

  (A.9)

• (4) If $s \geqq 1$, then $\left\| \nabla f\right\| _{H^{s-1}} \lesssim \left\| f\right\| _{\mathbb {P}^s}$ for each $f \in \mathbb {P}^s(\Omega )$. In particular, we have that $\nabla : \mathbb {P}^s(\Omega ) \rightarrow H^{s-1}(\Omega ;\mathbb {R}^n)$ is a bounded linear map.

• (5) If $s > 1/2$, then the trace map ${{\,\textrm{Tr}\,}}: H^s(\Omega ) \rightarrow H^{s-1/2}(\Sigma )$ extends to a bounded linear map ${{\,\textrm{Tr}\,}}: \mathbb {P}^s(\Omega ) \rightarrow \mathcal {H}^{s-1/2}(\Sigma )$. More precisely, the following hold.

  • (a) If $f \in C^0(\bar{\Omega }) \cap \mathbb {P}^s(\Omega )$, then ${{\,\textrm{Tr}\,}}f = f \vert _{\Sigma }$.

  • (b) If $\varphi \in C_c^1(\mathbb {R}^{n-1} \times (-b,0])$, then

    $$\begin{aligned} \int _{\Sigma } {{\,\textrm{Tr}\,}}f \varphi = \int _{\Omega } \partial _n f \varphi + f \partial _n \varphi \text { for all } f \in \mathbb {P}^s(\Omega ). \end{aligned}$$

    (A.10)

  • (c) We have the bound $\left\| {{\,\textrm{Tr}\,}}f\right\| _{\mathcal {H}^{s-1/2}} \lesssim \left\| f\right\| _{\mathbb {P}^s}$ for all $f\in \mathbb {P}^s(\Omega )$.

Proof

In the case $\Gamma = \mathbb {R}^{n-1}$ these are proved in Theorems 5.7, 5.9, and 5.11 of [9]. In the case $\Gamma = \mathbb {T}^{n-1}$ they follow from standard properties of Sobolev spaces. $\square$

Next we record a crucial fact about the $\mathbb {P}^s$ spaces: they give rise to standard $H^s$ multipliers.

Theorem A.5

Let $n \geqq 2$ and $s > n/2$. Then for $0 \leqq r \leqq s$

$$\begin{aligned} \left\| fg\right\| _{H^r} \lesssim \left\| f\right\| _{\mathbb {P}^s} \left\| g\right\| _{H^r} \text { for all }f\in \mathbb {P}^s(\Omega ) \text { and }g \in H^r(\Omega ). \end{aligned}$$

(A.11)

In particular, for $1 \leqq k \in \mathbb {N}$ the mapping

$$\begin{aligned} H^r(\Omega ) \times \prod _{j=1}^k \mathbb {P}^s(\Omega ) \ni (g,f_1,\dotsc ,f_k) \mapsto g \prod _{j=1}^k f_j \in H^r(\Omega ) \end{aligned}$$

(A.12)

is a bounded $(k+1)-$linear map.

Proof

When $\Gamma = \mathbb {T}^{n-1}$ this follows from the standard properties of Sobolev spaces. Assume then that $\Gamma = \mathbb {R}^{n-1}$. The bound (A.11) is proved for $r=s$ in Theorem 5.14 in [9]. When $r=0$, the bound (A.11) follows immediately from the third item of Theorem A.4. The general case $0< r < s$ then follows from these endpoint bounds and interpolation (see, for instance, [2, 15]). $\square$

1.2 A.2 Poisson Extension

We now study the Poisson extension operator, first on standard Sobolev spaces.

Proposition A.6

Let $-1/2 \leqq s \in \mathbb {R}$. For $\eta \in H^s(\Sigma )$ define $\mathfrak {P}\eta : \Omega \rightarrow \mathbb {R}$ via

$$\begin{aligned} \mathfrak {P}\eta (x) = \frac{1}{(2\pi )^{n-1}} \int _{\hat{\Gamma }} e^{i x'\cdot \xi } e^{\left| \xi \right| x_n} \hat{\eta }(\xi ) \textrm{d}\xi . \end{aligned}$$

(A.13)

Then $\mathfrak {P}\eta \in H^{s+1/2}(\Omega )$ and $\left\| \mathfrak {P}\eta \right\| _{H^{s+1/2}} \lesssim \left\| \eta \right\| _{H^{s}}$. In particular, $\mathfrak {P}: H^{s}(\Sigma ) \rightarrow H^{s+1/2}(\Omega )$ defines a bounded linear map.

Proof

We’ll only present the proof in the case $\Gamma = \mathbb {R}^{n-1}$, and the case $\Gamma = \mathbb {T}^{n-1}$ is similar and simpler. First note that

$$\begin{aligned} \int _{-b}^0 \left| e^{\left| \xi \right| x_n}\right| ^2 \textrm{d}x_n = \frac{1}{2 \left| \xi \right| }(1 - e^{-2 \left| \xi \right| b}) \asymp {\left\{ \begin{array}{ll} b &{} \text {for } \left| \xi \right| \asymp 0 \\ \frac{1}{2 \left| \xi \right| } &{}\text {for } \left| \xi \right| \asymp \infty . \end{array}\right. } \end{aligned}$$

(A.14)

Suppose initially that $m \in \mathbb {N}$ and that $\eta \in H^{m-1/2}(\Sigma )$. Using (A.14), we may readily bound

$$\begin{aligned} \left\| \mathfrak {P}\eta \right\| _{H^m(\Omega )}^2{} & {} \lesssim \int _{\mathbb {R}^{n-1}} \langle \xi \rangle ^{2m} \left| \hat{\eta }(\xi )\right| ^2 \int _{-b}^0\left| e^{\left| \xi \right| x_n}\right| ^2 \textrm{d}x_n \textrm{d}\xi \nonumber \\{} & {} \lesssim \int _{\mathbb {R}^{n-1}} \langle \xi \rangle ^{2m-1} \left| \hat{\eta }(\xi )\right| ^2 \textrm{d}\xi = \left\| \eta \right\| _{H^{m-1/2}}^2. \end{aligned}$$

(A.15)

Thus, $\mathfrak {P}: H^{m-1/2}(\Sigma ) \rightarrow H^m(\Omega )$ defines a bounded linear operator for every $m \in \mathbb {N}$. Standard interpolation theory (see, for instance, [2, 15]) then shows that $\mathfrak {P}: H^{t-1/2}(\Sigma ) \rightarrow H^t(\Omega )$ defines a bounded linear operator for every $0 \leqq t \in \mathbb {R}$, which is the desired result upon setting $t-1/2 = s$. $\square$

Next we consider the Poisson extension on the anisotropic spaces $\mathcal {H}^s(\Sigma )$, which requires the use of the $\mathbb {P}$ spaces from Definition A.3.

Theorem A.7

Let $0 \leqq s \in \mathbb {R}$ and $\Gamma = \mathbb {R}^{n-1}$. For $\eta \in \mathcal {H}^s(\Sigma )$ define $\mathfrak {P}\eta : \Omega \rightarrow \mathbb {R}$ via

$$\begin{aligned} \mathfrak {P}\eta (x) = \frac{1}{(2\pi )^{n-1}} \int _{\mathbb {R}^{n-1}} e^{i x'\cdot \xi } e^{\left| \xi \right| x_n} \hat{\eta }(\xi ) \textrm{d}\xi . \end{aligned}$$

(A.16)

Then the following hold:

• (1) $\mathfrak {P}\eta - \eta _l \in H^{s+1/2}(\Omega )$ and $\left\| \mathfrak {P}\eta - \eta _l\right\| _{H^{s+1/2}} \lesssim \left\| \eta \right\| _{\mathcal {H}^s}$, where $\eta _l = \eta _{l,1} \in \mathcal {H}^{s+1/2}(\Sigma ) \cap \bigcap _{k \in \mathbb {N}} C^k_0(\Sigma )$ in the notation of Theorem A.2.

• (2) $\mathfrak {P}\eta \in \mathbb {P}^{s+1/2}(\Omega )$ and $\left\| \mathfrak {P}\eta \right\| _{\mathbb {P}^{s+1/2}} \lesssim \left\| \eta \right\| _{\mathcal {H}^{s}}$.

• (3) The induced map $\mathfrak {P}: \mathcal {H}^{s}(\Sigma ) \rightarrow \mathbb {P}^{s+1/2}(\Omega )$ is bounded and linear.

Proof

We split $\eta$ into its high and low frequency parts: $\eta = \eta _h + \eta _l$, where $\hat{\eta }_h = {\chi }_{B(0,1)^c} \hat{\eta }$ and $\hat{\eta }_{l} = {\chi }_{B(0,1)} \hat{\eta }$. Then we know from Theorem A.2 that $\eta _l,\eta _h \in \mathcal {H}^s(\Sigma )$ with $\left\| \eta _{j}\right\| _{\mathcal {H}^s} \leqq \left\| \eta \right\| _{\mathcal {H}^s}$ for $j \in \{l,h\}$. We also know that $\eta _h \in H^s(\Sigma )$ with $\left\| \eta _h\right\| _{H^s} \lesssim \left\| \eta \right\| _{\mathcal {H}^s}$. Consequently, Proposition A.6 shows that $\mathfrak {P}\eta _h \in H^{s+1/2}(\Omega )$ with

$$\begin{aligned} \left\| \mathfrak {P}\eta _h\right\| _{H^{s+1/2}} \lesssim \left\| \eta _h\right\| _{H^s} \lesssim \left\| \eta \right\| _{\mathcal {H}^s}. \end{aligned}$$

(A.17)

Now consider the function $\mathfrak {P}\eta _l - \eta _l: \Omega \rightarrow \mathbb {R}$, which satisfies

$$\begin{aligned} \mathfrak {P}\eta _l(x) - \eta _l(x') = \frac{1}{(2\pi )^{n-1}} \int _{B(0,1)} e^{i x'\cdot \xi } (e^{ \left| \xi \right| b}-1) \hat{\eta }(\xi ) \textrm{d}\xi . \end{aligned}$$

(A.18)

We calculate

$$\begin{aligned} \int _{-b}^0 \left| e^{\left| \xi \right| x_n}-1\right| ^2 \textrm{d}x_n{} & {} = \frac{1}{2 \left| \xi \right| } (-3+ 2 \left| \xi \right| b +4 e^{- \left| \xi \right| b} - e^{-2 \left| \xi \right| b})\nonumber \\{} & {} \asymp {\left\{ \begin{array}{ll} \frac{ \left| \xi \right| ^2 b^3}{3} &{} \text {for } \left| \xi \right| \asymp 0 \\ b &{}\text {for } \left| \xi \right| \asymp \infty . \end{array}\right. } \end{aligned}$$

(A.19)

For any $m \in \mathbb {N}$ this allows us to bound

$$\begin{aligned}{} & {} \left\| \mathfrak {P}\eta _l - \eta _l\right\| _{H^m}^2 \lesssim \int _{B(0,1)} \langle \xi \rangle ^{2m} \left| \hat{\eta }(\xi )\right| ^2 \int _{-b}^0\left| e^{ \left| \xi \right| x_n}-1\right| ^2 \textrm{d}x_n \textrm{d}\xi \nonumber \\{} & {} \quad \lesssim \int _{B(0,1)} \langle \xi \rangle ^{2m-1} \left| \xi \right| ^{2}\left| \hat{\eta }(\xi )\right| ^2 \textrm{d}\xi \nonumber \\{} & {} \quad \lesssim \int _{B(0,1)} \left| \xi \right| ^{2}\left| \hat{\eta }(\xi )\right| ^2 \textrm{d}\xi \lesssim \left\| \eta \right\| _{\mathcal {H}^s}^2. \end{aligned}$$

(A.20)

Thus, $\mathfrak {P}\eta _l - \eta _l \in \bigcap _{m \in \mathbb {N}} H^m(\Omega )$, but in particular we can choose a fixed $s+1/2 \leqq m \in \mathbb {N}$ to see that $\mathfrak {P}\eta _l - \eta _l \in H^{s+1/2}(\Omega )$ with

$$\begin{aligned} \left\| \mathfrak {P}\eta _l - \eta _l\right\| _{H^{s+1/2}} \lesssim \left\| \mathfrak {P}\eta _l - \eta _l\right\| _{H^{m}} \lesssim \left\| \eta \right\| _{\mathcal {H}^s}. \end{aligned}$$

(A.21)

Finally, note that $\eta _l \in \bigcap _{t > 0} \mathcal {H}^t(\Sigma )$ and that

$$\begin{aligned} \left\| \eta _l\right\| _{\mathcal {H}^t} = \left\| \eta _l\right\| _{\mathcal {H}^s} \text { for all } 0 \leqq t \in \mathbb {R}. \end{aligned}$$

(A.22)

In particular, $\eta _l \in \mathcal {H}^{s+1/2}(\Sigma )$ with $\left\| \eta _l\right\| _{\mathcal {H}^{s+1/2}} \leqq \left\| \eta \right\| _{\mathcal {H}^s}$. We may thus combine (A.17), (A.21), and (A.22) to see that $\mathfrak {P}\eta = [\mathfrak {P}\eta _h + (\mathfrak {P}\eta _l - \eta _l)] + \eta _l$ with

$$\begin{aligned} \left\| \mathfrak {P}\eta _h + (\mathfrak {P}\eta _l - \eta _l)\right\| _{H^{s+1/2}} + \left\| \eta _l\right\| _{\mathcal {H}^{s+1/2}} \lesssim \left\| \eta \right\| _{\mathcal {H}^s}. \end{aligned}$$

(A.23)

Hence, $\mathfrak {P}\eta \in \mathbb {P}^{s+1/2}(\Omega )$ with $\left\| \mathfrak {P}\eta \right\| _{\mathbb {P}^{s+1/2}} \lesssim \left\| \eta \right\| _{\mathcal {H}^s},$ which is the desired bound. $\square$

Finally, we record some results about the normal derivative of the Poisson extension.

Proposition A.8

Let $0 \leqq s \in \mathbb {R}$ and $\eta \in \mathcal {H}^{s+3/2}(\Sigma )$. Then the following hold.

• (1) If $\Gamma = \mathbb {R}^{n-1}$, then $\partial _n \mathfrak {P}\eta (\cdot ,0) - \partial _n \mathfrak {P}\eta (\cdot ,-b) \in \dot{H}^{-1}(\mathbb {R}^{n-1})$ and

  $$\begin{aligned} \left[ \partial _n \mathfrak {P}\eta (\cdot ,0) - \partial _n \mathfrak {P}\eta (\cdot ,-b)\right] _{\dot{H}^{-1}} \leqq b \left\| \nabla \mathfrak {P}\eta \right\| _{L^2} \lesssim \left\| \eta \right\| _{\mathcal {H}^{s+3/2}}. \end{aligned}$$

  (A.24)

• (2) If $\Gamma = \mathbb {T}^{n-1}$, then $\widehat{\partial _n \mathfrak {P}\eta \vert _{\Sigma }}(0) = \widehat{\partial _n \mathfrak {P}\eta \vert _{\Sigma _{-b}}}(0)=0$.

Proof

We’ll only prove the first item, as the second is simpler and similar. Theorem A.7 tells us that $\mathfrak {P}\eta \in \mathbb {P}^{s+2}(\Omega )$, and so Theorem A.4 then implies that $\partial _n \mathfrak {P}\eta \in H^{s+1}(\Omega )$. Note that $\Delta \mathfrak {P}\eta =0$ in $\Omega$. Using this and the absolute continuity of Sobolev functions on lines (see, for instance, Theorem 11.45 in [8]), we may then compute

$$\begin{aligned} \partial _n \mathfrak {P}\eta (x',0) - \partial _n \mathfrak {P}\eta (x',-b)= & {} \int _{-b}^0 \partial _n^2 \mathfrak {P}\eta (x',t) \textrm{d}t = -\int _{-b}^0 \Delta ' \mathfrak {P}\eta (x',t) \textrm{d}t \nonumber \\= & {} -{{\,\textrm{div}\,}}' \int _{-b}^0 \nabla ' \mathfrak {P}\eta (x',t) \textrm{d}t. \end{aligned}$$

(A.25)

Thus, Cauchy–Schwarz, Fubini–Tonelli, and Parseval imply that

$$\begin{aligned}{} & {} \left[ \partial _n \mathfrak {P}\eta (x',0) - \partial _n \mathfrak {P}\eta (x',-b)\right] _{\dot{H}^{-1}}^2 = \int _{\mathbb {R}^{n-1}} \frac{1}{\left| \xi \right| ^2} \left| \xi \cdot \int _{-b}^0 \xi \widehat{\mathfrak {P}\eta }(\xi ,t) \textrm{d}t \right| ^2 \textrm{d}\xi \nonumber \\{} & {} \quad \leqq b^2 \int _{\mathbb {R}^{n-1}} \int _{-b}^0 \left| \xi \widehat{\mathfrak {P}\eta }(\xi ,t)\right| ^2 \textrm{d}t \xi \leqq b^2 \int _{\Omega } \left| \nabla \mathfrak {P}\eta \right| ^2. \end{aligned}$$

(A.26)

The stated inequality follows from this and Theorems A.4 and A.7. $\square$

1.3 A.3 Composition

In this subsection we aim to study some composition operators. We begin by introducing some notation that allows us to extend the flattening maps to full space.

Definition A.9

Let $\chi \in C^\infty _c(\mathbb {R})$ be such that $0 \leqq \chi \leqq 1$, $\chi =1$ on $[-2b,2b]$, and ${{\,\textrm{supp}\,}}(\chi ) \subset (-3b,3b)$. Given $\eta \in \mathcal {H}^{\sigma +1/2}(\Sigma )$ define $\mathfrak {E}_\eta : \Gamma \times \mathbb {R}\rightarrow \Gamma \times \mathbb {R}$ via

$$\begin{aligned} \mathfrak {E}_\eta (x) = x + \chi (x_n)[\eta _l(x') + E(\mathfrak {P}\eta - \eta _l)(x)]\left( 1+\frac{x_n}{b}\right) e_n, \end{aligned}$$

(A.27)

where $E: L^2(\Omega ) \rightarrow L^2(\Gamma \times \mathbb {R})$ is a Stein extension operator, $\mathfrak {P}$ is the Poisson extension as defined in Proposition A.7 and Theorem A.7, and when $\Gamma = \mathbb {R}^{n-1}$ we take $\eta _l = \eta _{l,1} \in \bigcap _{t \geqq 0} \mathcal {H}^{t}(\Sigma ) \cap \bigcap _{k \in \mathbb {N}} C^k_0(\Sigma )$ in the notation of Theorem A.2, while when $\Gamma = \mathbb {T}^{n-1}$ we take $\eta _l =0$. Note that Proposition A.6 and Theorem A.7 show that $\mathfrak {P}\eta - \eta _l \in H^{\sigma +1}(\Omega )$, and since the Stein extension restricts to a bounded map $E: H^{\sigma +1}(\Omega ) \rightarrow H^{\sigma +1}(\Gamma \times \mathbb {R})$ we have that $E(\mathfrak {P}\eta -\eta _l) \in H^{\sigma +1}(\Gamma \times \mathbb {R})$.

Next we record some properties of these maps.

Proposition A.10

Let $\sigma > n/2$, $\eta \in \mathcal {H}^{\sigma +1/2}(\Sigma )$, and define $\mathfrak {E}_\eta : \Gamma \times \mathbb {R}\rightarrow \Gamma \times \mathbb {R}$ as in Definition A.9. Then the following hold.

• (1) The map $\mathfrak {E}_\eta$ is Lipschitz and $C^1$, and $\left\| \nabla \mathfrak {E}_\eta - I\right\| _{C^0_b} \lesssim \left\| \eta \right\| _{\mathcal {H}^{s+1/2}}$.

• (2) If V is a real finite dimensional inner-product space and $0 \leqq r \leqq \sigma$, then

  $$\begin{aligned} \sup _{1\leqq j,k \leqq n} \left\| \partial _j \mathfrak {E}_\eta \cdot e_k f \right\| _{H^r} \lesssim (1 + \left\| \eta \right\| _{\mathcal {H}^{\sigma +1/2}}) \left\| f\right\| _{H^r} \end{aligned}$$

  (A.28)

  and

  $$\begin{aligned} \sup _{1\leqq j,k \leqq n} \left\| (\partial _j \mathfrak {E}_\eta \cdot e_k - \partial _j \mathfrak {E}_\zeta \cdot e_k ) f \right\| _{H^r} \lesssim \left\| \eta -\zeta \right\| _{\mathcal {H}^{\sigma +1/2}} \left\| f\right\| _{H^r} \end{aligned}$$

  (A.29)

  for every $\eta ,\zeta \in \mathcal {H}^{\sigma +1/2}(\Sigma )$ and $f \in H^r(\Gamma \times \mathbb {R};V)$.

• (3) There exists $0< \delta _*<1$ such that if $\left\| \eta \right\| _{\mathcal {H}^{\sigma +1/2}} < \delta _*$, then $\mathfrak {E}_\eta$ is a bi-Lipschitz homeomorphism and a $C^1$ diffeomorphism, and we have the estimate $\left\| \nabla \mathfrak {E}_\eta - I\right\| _{C^0_b} < 1/2$.

Proof

First note that $\sigma +1 >n/2 +1$, so Proposition A.6, Theorem A.7 and standard Sobolev embeddings show that $E(\mathfrak {P}\eta - \eta _l) \in C^1_b(\Gamma \times \mathbb {R})$. On the other hand, $\eta _l \in \bigcap _{t \geqq 0} \mathcal {H}^t(\Sigma )$, so Theorem A.2 shows that $\eta _l \in C^1_b(\Sigma )$. These observations and their associated bounds then imply the first item. Next we write $\mathfrak {E}_\eta = I + \omega e_n$ so that $\nabla \mathfrak {E}_\eta = I + e_n \otimes \nabla \omega$. To prove the second item it suffices to show that $\left\| \partial _j \omega f\right\| _{H^r} \lesssim \left\| \eta \right\| _{\mathcal {H}^{\sigma +1/2}} \left\| f\right\| _{H^r}$ for $0 \leqq r \leqq \sigma$ and $1 \leqq j \leqq n$. To establish this we observe that on the one hand, thanks to Theorem A.2, $\chi \eta _l \in \bigcap _{k \in \mathbb {N}} C^k_0(\Gamma \times \mathbb {R})$, and on the other $E(\mathfrak {P}\eta - \eta _l) \in H^{\sigma +1}(\Gamma \times \mathbb {R})$. Thus, $\partial _j \omega$ consists of linear combinations of terms in $\bigcap _{k \in \mathbb {N}} C^k_0(\Gamma \times \mathbb {R})$ and in $H^{\sigma }(\mathbb {R}^n)$, and so the sufficient bound follows from standard Sobolev multiplier results (see, for instance, Lemma A.8 in [9]).

To prove the third item we note that if $\omega$ has Lipschitz constant less than unity, then $\omega e_n$ is contractive on $\mathbb {R}^n$, and so the Banach fixed point theorem implies that $\mathfrak {E}_\eta$ is a bi-Lipschitz homeomorphism. To control the Lipschitz constant of $\omega$ we use the supercritical Sobolev embeddings as above to verify that this constant is less than unity provided that $\left\| \eta \right\| _{\mathcal {H}^{\sigma +1/2}} < \delta _*$ for some sufficiently small universal constant $\delta _*\in (0,1)$. $\square$

The next result studies the smoothness properties of composition with the maps from Definition A.9.

Theorem A.11

Let $n/2 < \sigma \in \mathbb {N}$, $0< \delta _*<1$ be as in the third item of Proposition A.10, and V be a real finite dimensional inner-product space. Let $r \in \mathbb {N}$ satisfy $0 \leqq r \leqq \sigma +1$ and let $k \in \{0,1\}$. Consider the map $\Lambda : H^{r+k}(\Gamma \times \mathbb {R};V) \times B_{\mathcal {H}^{\sigma +1/2}(\Sigma )}(0,\delta _*) \rightarrow H^{r}(\Gamma \times \mathbb {R};V)$ given by $\Lambda (f,\eta ) = f\circ \mathfrak {E}_\eta ,$ where $\mathfrak {E}_\eta : \Gamma \times \mathbb {R}\rightarrow \Gamma \times \mathbb {R}$ is as defined in Definition A.9. Then $\Lambda$ is well-defined and $C^k$, and if $k =1$ then $D\Lambda (f,\eta )(g,\zeta ) = \chi \tilde{b} (\eta _l + E(\mathfrak {P}\eta - \eta _l) (\partial _n f \circ \mathfrak {E}_\eta )\zeta + g \circ \mathfrak {E}_\eta ,$ where $\tilde{b}(x) = (1+x_n/b)$.

Proof

With Proposition A.10 established, the result follows from minor and evident modifications of the argument used to prove Theorem 1.1 in [6] (see also Theorem 5.20 in [9]). $\square$

Finally, as a byproduct of this theorem we obtain smoothness properties associated to composition with the flattening maps $\mathfrak {F}_\eta$.

Corollary A.12

Let $n/2 < \sigma \in \mathbb {N}$, $0< \delta _*<1$ be as in the third item of Proposition A.10, and V be a real finite dimensional inner-product space. Let $r \in \mathbb {N}$ satisfy $0 \leqq r \leqq \sigma +1$. For $\eta \in \mathcal {H}^{\sigma +1/2}(\Sigma )$ define $\mathfrak {F}_\eta : \Omega \rightarrow \Omega _\eta$ via (2.1). Then the following hold.

• (1) The map $\Lambda _\Omega : H^{r+1}(\Gamma \times \mathbb {R};V) \times B_{\mathcal {H}^{\sigma +1/2}(\Sigma )}(0,\delta _*) \rightarrow H^{r}(\Omega ;V)$ given by $\Lambda (f,\eta ) = f\circ \mathfrak {F}_\eta$ is well-defined and $C^1$ with $D\Lambda _\Omega (f,\eta )(g,\zeta ) = \tilde{b} \mathfrak {P}\eta (\partial _n f \circ \mathfrak {F}_\eta )\zeta + g \circ \mathfrak {F}_\eta ,$ where $\tilde{b}(x) = (1+x_n/b)$.

• (2) Assume $r \geqq 1$. Then the map $\mathfrak {S}_\Sigma : H^{r+1}(\Gamma \times \mathbb {R};V) \times B_{\mathcal {H}^{\sigma +1/2}(\Sigma )}(0,\delta _*) \rightarrow H^{r-1/2}(\Sigma ;V)$ given by $\mathfrak {S}_\Sigma (f,\eta ) = f \circ \mathfrak {F}_\eta \vert _{\Sigma }$ is well-defined and $C^1$ with $D\mathfrak {S}_\Sigma (f,\eta )(g,\zeta ) = \eta (\partial _n f \circ \mathfrak {F}_\eta )\zeta \vert _{\Sigma } + g \circ \mathfrak {F}_\eta \vert _\Sigma$.

Proof

The first item follows from Theorem A.11 and the observation that $\Lambda _\Omega (f,\eta ) = R_\Omega \Lambda (f,\eta )$, where $R_\Omega : H^{r}(\Gamma \times \mathbb {R};V) \rightarrow H^{r}(\Omega ;V)$ is the bounded linear map given by restriction to $\Omega$. This identity follows directly from the fact that, by construction, $\mathfrak {E}_\eta = \mathfrak {F}_\eta$ in $\Omega$. The second item follows by composing the first item with the bounded linear trace map. $\square$

1.4 A.4 Littlewood–Paley Analysis for the Anisotropic Sobolev Space $\mathcal {H}^s$

In this subsection we develop some Littlewood–Paley theory for the anisotropic spaces.

Definition A.13

Let $\chi \in C^\infty (\mathbb {R}^d)$ be a radial function such that $\chi (\xi )=1$ for $|\xi |\leqq \frac{1}{2}$, $\chi (\xi )=0$ for $|\xi |\geqq 1$. Set

$$\begin{aligned}{} & {} \varphi (\xi )=\chi (\xi )-\chi (2\xi ), \quad \chi _j(\xi )=\chi (2^{-j}\xi ) \text { for } j\in \mathbb {Z}, \quad \varphi _0=\chi , \text { and }\nonumber \\{} & {} \varphi _j(\xi )=\varphi (2^{-j}\xi ) \text { for } j\geqq 1. \end{aligned}$$

(A.30)

The Littlewood–Paley dyadic block $\Delta _j$ is defined by the Fourier multiplier

$$\begin{aligned} \Delta _j=\varphi _j(D_x)\quad \text {for } j\geqq 0,\quad \Delta _j=0\quad \text {for } j\leqq -1. \end{aligned}$$

(A.31)

The low-frequency cut-off operator $S_j$ is defined by

$$\begin{aligned} S_j=\chi _j(D)=\sum _{k=0}^j\Delta _k \text { for } j\geqq 0. \end{aligned}$$

(A.32)

The above Fourier multipliers can act on functions (distributions) defined on $\mathbb {R}^d$ or $\mathbb {T}^d$, and the Fourier transform is defined accordingly. In particular, for $u:\mathbb {T}^d\rightarrow \mathbb {R}$ we have

$$\begin{aligned} \Delta _0(D)u=\frac{1}{(2\pi )^d}\widehat{u}(0)=\frac{1}{(2\pi )^d}\int _{\mathbb {T}^d} u. \end{aligned}$$

(A.33)

Since $\sum _{j=0}^\infty \varphi _j(\xi )=1$ for all $\xi \in \mathbb {R}^d$, we have that $\sum _{j=0}^\infty \Delta _j=\text {Id}$. Moreover, we have ${{\,\textrm{supp}\,}}\varphi _j\subset \{ 2^{j-2}<|\xi |<2^j\}$ for $j\geqq 1$ and $\chi \varphi _j=0$ for $j\geqq 2$.

Bony’s decomposition for product of functions is

$$\begin{aligned} fg=T_fg+T_gf+R(f, g), \end{aligned}$$

(A.34)

where

$$\begin{aligned} T_fg=\sum _{j\geqq 3} S_{j-3}f\Delta _j g \text { and } R(f, g)=\sum _{j, k\geqq 0, |j-k|\leqq 2}\Delta _j f\Delta _kg. \end{aligned}$$

(A.35)

We note that ${{\,\textrm{supp}\,}}\widehat{S_{j-3}f\Delta _j g}\subset \{2^{j-3}<|\xi |<2^{j+1} \}$ for $j\geqq 1$.

We recall the following result from [1].

Lemma A.14

([1, Lemma 2.2]) Let $\mathcal {C}$ be an annulus in $\mathbb {R}^d$, $m\in \mathbb {R}$, and $k=2[1+\frac{d}{2}]$, where [r] denotes the integer part of r. Let $\sigma$ be a k-times differentiable function on $\mathbb {R}^d\setminus \{0\}$ such that for all $\alpha \in \mathbb {R}^d$ with $|\alpha |\leqq k$, there exists a constant $C_\alpha$ such that

$$\begin{aligned} |\partial ^\alpha \sigma (\xi )|\leqq C_\alpha |\xi |^{m-|\alpha |} \text { for all } \xi \in \mathbb {R}^d\setminus \{ 0\}. \end{aligned}$$

(A.36)

There exists a constant C, depending only on the constants $C_\alpha$, such that for any $p\in [1, \infty ]$ and any constant $\lambda >0$, we have, for any function $u\in L^p(M^d)$, $M\in \{\mathbb {R}, \mathbb {T}\}$, with Fourier transform supported in $\lambda \mathcal {C}$,

$$\begin{aligned} \Vert \sigma (D)u\Vert _{L^p}\leqq C\lambda ^m\Vert u\Vert _{L^p}. \end{aligned}$$

(A.37)

Next we recall the definition of the Chemin–Lerner norm.

Definition A.15

Let M be either $\mathbb {R}$ or $\mathbb {T}$. For $I\subset \mathbb {R}$ and $s\in \mathbb {R}$, the Chemin–Lerner norm is defined by

$$\begin{aligned} \Vert u\Vert ^2_{\widetilde{L}^q(I; H^s(M^d))}=\sum _{j=0}^\infty 2^{2sj}\Vert \Delta _j u\Vert ^2_{L^q(I; L^2(M^d))}. \end{aligned}$$

(A.38)

When the low-frequency part is removed, we denote

$$\begin{aligned} \Vert u\Vert ^2_{ H^s_\sharp (M^d)}= & {} \sum _{j= 1}^\infty 2^{2sj}\Vert \Delta _j u\Vert ^2_{L^2(M^d)} \text { and } \Vert u\Vert ^2_{\widetilde{L}^q(I; H^s_\sharp (M^d))}\nonumber \\= & {} \sum _{j= 1}^\infty 2^{2sj}\Vert \Delta _j u\Vert ^2_{L^q(I; L^2(M^d))}. \end{aligned}$$

(A.39)

It what follows, unless otherwise specified, when the set M is omitted in function space notation, it can be either $\mathbb {R}$ or $\mathbb {T}$. We recall another result from [1], this time about products.

Proposition A.16

([1, Corollary 2.54]) For $I\subset \mathbb {R}$, $q\in [1, \infty ]$ and $s>0$, there exists $C=C(d, s)$ such that

$$\begin{aligned}&\Vert f g \Vert _{H^s}\leqq C \Vert f\Vert _{L^\infty }\Vert g\Vert _{H^s}+C\Vert g\Vert _{L^\infty }\Vert f\Vert _{H^s}, \end{aligned}$$

(A.40)

$$\begin{aligned}&\Vert fg \Vert _{\widetilde{L}^q(I; H^s)}\leqq C \Vert f\Vert _{L^\infty (I; L^\infty )}\Vert g\Vert _{\widetilde{L}^q(I; H^s)}+C\Vert g\Vert _{L^\infty (I; L^\infty )}\Vert f\Vert _{\widetilde{L}^q(I; H^s)} \end{aligned}$$

(A.41)

provided that the right-hand sides are finite.

Next we study the boundedness of some key operators in the Chemin–Lerner norm.

Proposition A.17

The following hold.

• (1) There exists an absolute constant C such that for all $1\leqq p\leqq \infty$, $\sigma \in \mathbb {R}$ and $u\in H^\sigma (\mathbb {R}^d)$, we have

  $$\begin{aligned} \left\| \frac{\cosh ((z+b)|D|)}{\cosh (b|D|)}u\right\| _{\widetilde{L}^p_z([-b, 0]; H^{\sigma +\frac{1}{p}})}\leqq \max \{2b^{\frac{1}{p}}, C\}\Vert u\Vert _{H^\sigma }. \end{aligned}$$

  (A.42)

• (2) There exists an absolute constant C such that for all $1\leqq q_2\leqq q_1\leqq \infty$, $\sigma \in \mathbb {R}$ and $f\in \widetilde{L}^{p_2}_z([-b, 0]; H_x^{\sigma -1+\frac{1}{p_2}})$, we have

  $$\begin{aligned}{} & {} \left\| \int _{-b}^z\frac{\cosh ((z'+b)|D|)}{\cosh ((z+b)|D|)}f(x, z')\textrm{d}z'\right\| _{\widetilde{L}^{q_1}_z\left( [-b, 0]; H^{\sigma +\frac{1}{q_1}}\right) }\nonumber \\{} & {} \quad \leqq \max \left\{ b^\frac{q_1+q_2'}{q_1q_2'}, C\right\} \Vert f\Vert _{\widetilde{L}^{q_2}_z\left( [-b, 0]; H^{\sigma -1+\frac{1}{q_2}}\right) }, \end{aligned}$$

  (A.43)

  where $\frac{1}{q_2}+\frac{1}{q_2'}=1$. In addition, for any $z\in [-b, 0]$, we have

  $$\begin{aligned}{} & {} \left\| \int _{-b}^z\frac{\cosh ((z'+b)|D|)}{\cosh ((z+b)|D|)}f(x, z')\textrm{d}z'\right\| _{H^\sigma }\nonumber \\{} & {} \quad \leqq \max \{b^\frac{q_1+q_2'}{q_1q_2'}, C\}\Vert f\Vert _{\widetilde{L}^{q_2}_z\left( [-b, 0]; H^{\sigma -1+\frac{1}{p_2}}\right) }, \end{aligned}$$

  (A.44)

Proof

For all $-b\leqq z_1\leqq z_2$, we have $0\leqq z_2-z_1\leqq z_2+b$ and hence

$$\begin{aligned} 1\leqq \frac{\cosh ((z_1+b)c)}{\cosh ((z_2+b)c)}e^{(z_2-z_1)c}=\frac{e^{2(z_2+b)c}+e^{2(z_2-z_1)c}}{e^{2(z_2+b)c}+1}\leqq 2. \end{aligned}$$

(A.45)

for all $c\geqq 0$.

To prove the first item we note that (A.45) implies

$$\begin{aligned} e^{zc} \leqq \frac{\cosh ((z+b)c)}{\cosh (bc)}\leqq 2e^{zc} \end{aligned}$$

(A.46)

for all $z\in [-b, 0]$. Consequently, for $j\geqq 1$ and $u\in \dot{H}^\sigma$, we have

$$\begin{aligned}{} & {} \left\| \Delta _j\frac{\cosh ((z+b)|D|)}{\cosh (b|D|)}u\right\| _{L^2_x} =\left\| \frac{\cosh ((z+b)|D|)}{\cosh (b|D|)}\Delta _ju\right\| _{L^2_x}\nonumber \\{} & {} \quad \leqq \left( \int _{\mathbb {R}^d}4e^{2z|\xi |}|\widehat{\Delta _j u}(\xi )|^2\textrm{d}\xi \right) ^\frac{1}{2}\leqq 2e^{z2^{j-2}}\Vert \Delta _j u\Vert _{L^2} \end{aligned}$$

(A.47)

since $|\xi |\geqq 2^{j-2}$ on the support of $\widehat{\Delta _j}u(\xi )$. It follows that

$$\begin{aligned} \left\| \Delta _j\frac{\cosh ((z+b)|D|)}{\cosh (b|D|)}u\right\| _{L^p_z([-b, 0]; L^2)}\leqq C2^{-\frac{j}{p}}\Vert \Delta _j u\Vert _{L^2}, \end{aligned}$$

(A.48)

where C is an absolute constant. On the other hand, the low frequency part can be bounded as

$$\begin{aligned} \left\| \Delta _0\frac{\cosh ((z+b)|D|)}{\cosh (b|D|)}u\right\| _{L^p([-b, 0]; L^2)}\leqq 2\Vert \Vert \Delta _0u\Vert _{L^2_x}\Vert _{L^p_z([-b, 0])}\leqq 2b^{\frac{1}{p}}\Vert \Delta _0u\Vert _{L^2}. \nonumber \\ \end{aligned}$$

(A.49)

Combining (A.48) and (A.49) yields

$$\begin{aligned} \begin{aligned} \left\| \frac{\cosh ((z+b)|D|)}{\cosh (b|D|)}u\right\| _{\widetilde{L}^p_z([-b, 0]; H^{\sigma +\frac{1}{p}})}^2&\leqq \left( 2b^{\frac{1}{p}}\right) ^2\Vert \Delta _0u\Vert _{L^2}^2+\sum _{j=1}^\infty C^22^{2j\left( \sigma +\frac{1}{p}-\frac{1}{p}\right) }\Vert \Delta _j u\Vert _{L^2}^2\\&\leqq \max \left\{ 2b^{\frac{1}{p}}, C\right\} ^2\Vert u\Vert ^2_{H^\sigma }. \end{aligned} \nonumber \\ \end{aligned}$$

(A.50)

This completes the proof of the first item.

We now turn to the proof of the second item. To prove (A.43), we set

$$\begin{aligned} g(x, z)=\int _{-b}^z\frac{\cosh ((z'+b)|D|)}{\cosh ((z+b)|D|)}f(x, z')\textrm{d}z'. \end{aligned}$$

(A.51)

For $z\in [-b, 0]$ and $j\geqq 1$, using (A.45) we estimate

$$\begin{aligned}{} & {} \left\| \Delta _jg(\cdot , z)\right\| _{L^2} = \left\| \int _{-b}^z\frac{\cosh ((z'+b)|D|)}{\cosh ((z+b)|D|)} \Delta _jf(x, z')\textrm{d}z'\right\| _{L^2_x} \nonumber \\{} & {} \quad \leqq \int _{-b}^z \left\| \frac{\cosh ((z'+b)|D|)}{\cosh ((z+b)|D|)} \Delta _jf(\cdot , z')\right\| _{L^2_x}\textrm{d}z' \nonumber \\{} & {} \quad \leqq \int _{-b}^z \left( \int _{\mathbb {R}^d}e^{-2(z-z')|\xi |} |\varphi (2^{-j}\xi )\widehat{f}(\xi , z')|^2\textrm{d}\xi \right) ^\frac{1}{2}\textrm{d}z'\nonumber \\{} & {} \quad \leqq \int _{-b}^z \left( \int _{\mathbb {R}^d}e^{-(z-z')2^{j-1}} |\widehat{\Delta _j f}(\xi , z')|^2\textrm{d}\xi \right) ^\frac{1}{2}\textrm{d}z' \nonumber \\{} & {} \quad \leqq \int _{-b}^z e^{-(z-z')2^{j-2}}\Vert \Delta _j f(\cdot , z')\Vert _{L^2} \textrm{d}z'. \end{aligned}$$

(A.52)

Applying Young’s inequality in z we deduce

$$\begin{aligned} \Vert \Delta _jg\Vert _{L^{q_1}_z([-b, 0]; L^2)}\leqq & {} \Vert e^{-z 2^{j-2}}\Vert _{L^q_z(\mathbb {R}_+)}\Vert \Delta _j f\Vert _{L^{q_2}_z([-b, 0]; L^2)} \nonumber \\= & {} \frac{1}{q^\frac{1}{q}}2^{-\frac{j-2}{q}}\Vert \Delta _j f\Vert _{L^{q_2}_z([-b, 0]; L^2)}\nonumber \\\leqq & {} C2^{-\frac{j}{q}}\Vert \Delta _j f\Vert _{L^{q_2}_z([-b, 0]; L^2)}, \end{aligned}$$

(A.53)

where $\frac{1}{q}=1+\frac{1}{q_1}-\frac{1}{q_2}$ and C is an absolute constant. On the other hand, it is readily seen that

$$\begin{aligned} \left\| \Delta _0g(\cdot , z)\right\| _{L^2} \leqq 2\int _{-b}^z \Vert \Delta _0 f(\cdot , z')\Vert _{L^2} \textrm{d}z' \leqq 2b^\frac{1}{q_2'} \Vert \Delta _0 f\Vert _{L^{q_2}_z([-b, 0]; L^2)}, \nonumber \\ \end{aligned}$$

(A.54)

and hence

$$\begin{aligned} \left\| \Delta _0g\right\| _{L^{q_1}_z([-b, 0]; L^2)} \leqq 2b^\frac{q_1+q_2'}{q_1q_2'} \Vert \Delta _0 f\Vert _{L^{q_2}_z([-b, 0]; L^2)}. \end{aligned}$$

(A.55)

A combination of (A.53) and (A.55) leads to (A.43).

Finally, the proof of (A.44) is similar to the case $q_1=\infty$ of (A.43). $\square$

Next we consider some more product estimates.

Proposition A.18

Let $s>0$, $p\in [1, \infty ]$, and $I\subset \mathbb {R}$. Then, there exists $C=C(d, s)$ such that the estimate

$$\begin{aligned} \begin{aligned} \Vert fg\Vert _{\widetilde{L}^p(I; H^s(\mathbb {R}^d))}&\leqq C \left( \Vert f\Vert _{L^\infty (I; L^\infty (\mathbb {R}^d))} +\Vert \chi \widehat{f}\Vert _{L^\infty (I; L^1(\mathbb {R}^d))}\right) \Vert g\Vert _{\widetilde{L}^p(I; H^s(\mathbb {R}^d))}\\&\quad +C\Vert g\Vert _{L^\infty (I; L^\infty (\mathbb {R}^d))} \Vert f\Vert _{\widetilde{L}^p(I; H^s_\sharp (\mathbb {R}^d))} \end{aligned} \nonumber \\ \end{aligned}$$

(A.56)

holds provided that the right-hand side is finite. Consequently, for $s>0$ and $s_0>\frac{d}{2}$, there exists $C=C(d, s, s_0)$ such that

$$\begin{aligned} \Vert fg\Vert _{\widetilde{L}^p(I; H^s)}\leqq C\Vert f\Vert _{L^\infty (I; \mathcal {H}^{s_0}(\mathbb {R}^d))}\Vert g\Vert _{\widetilde{L}^p(I; H^s)}+C\Vert g\Vert _{L^\infty (I; L^\infty )} \Vert f\Vert _{\widetilde{L}^p(I; H^s_\sharp )}. \nonumber \\ \end{aligned}$$

(A.57)

Proof

We first note that for $M=\mathbb {T}$, (A.57) is a consequence of (A.41) and the continuous embedding $H^{s_0}(\mathbb {T}^d)\subset L^\infty (\mathbb {T}^d)$ for $s_0>\frac{d}{2}$.

To prove (A.56) and (A.57) for $M=\mathbb {R}$, we shall consider functions f(x, z) and g(x, z) defined on $\mathbb {R}^d\times I$. For fixed $z\in I$, we use Bony’s decomposition (A.34): $fg=T_fg+T_gf+R(f, g)$, where $T_fg=\sum _{j\geqq 3}S_{j-3}f\Delta _j g$. For $j\geqq 3$ we have ${{\,\textrm{supp}\,}}\widehat{S_{j-3}f\Delta _j g}\subset \{2^{j-3}<|\xi |<2^{j+1}\}$ and hence $\Delta _k (S_{j-3}f\Delta _j g)=0$ for all $k\geqq 0$ satisfying $|j-k|\geqq 3$. Thus, for $k\geqq 0$ using Bernstein’s inequality we obtain

$$\begin{aligned} 2^{sk}\Vert \Delta _k T_fg\Vert _{L^2_x}{} & {} =2^{sk}\Vert \sum _{j\geqq 3, |j-k|\leqq 2} \Delta _k(S_{j-3}f\Delta _j g)\Vert _{L^2_x}\nonumber \\{} & {} \leqq C\sum _{j\geqq 3, |j-k|\leqq 2} 2^{sj}\Vert S_{j-3}f\Delta _j g\Vert _{L^2_x}\nonumber \\{} & {} \leqq C\Vert f\Vert _{L^\infty _x} \sum _{j\geqq 3, |j-k|\leqq 2}2^{sj}\Vert \Delta _j g\Vert _{L^2_x}, \end{aligned}$$

(A.58)

where $C=C(d, s)$. Since $f\in \widetilde{L}^p(I; H^s_\sharp )$, we have $\Delta _j f\in L^2_x$ a.e. $z\in I$ for $j\geqq 3$. Consequently, the preceding estimate for $T_fg$ also holds for $T_gf$; that is,

$$\begin{aligned} 2^{sk}\Vert \Delta _k T_gf\Vert _{L^2_x}\leqq C\Vert g\Vert _{L^\infty _x} \sum _{j\geqq 3, |j-k|\leqq 2}2^{sj}\Vert \Delta _j f\Vert _{L^2_x}. \end{aligned}$$

(A.59)

It follows that

$$\begin{aligned}{} & {} \Vert T_fg\Vert _{\widetilde{L}^p(I; H^s)}^2 \leqq C\Vert f\Vert _{L^\infty (I; L^\infty )} ^2 \sum _{k=0}^\infty \Big (\sum _{j\geqq 3, |j-k|\leqq 2}2^{sk}\Vert \Delta _j g\Vert _{L^p(I; L^2)}\Big )^2\nonumber \\{} & {} \quad \leqq C\Vert f\Vert _{L^\infty (I; L^\infty )} ^2 \sum _{j=3}^\infty 2^{2sj}\Vert \Delta _j g\Vert _{L^p(I; L^2)}^2 \leqq C\Vert f\Vert _{L^\infty (I; L^\infty )}^2\Vert g\Vert _{\widetilde{L}^p(I; H_\sharp ^s)}^2, \nonumber \\ \end{aligned}$$

(A.60)

and similarly we have

$$\begin{aligned} \Vert T_gf\Vert _{\widetilde{L}^p(I; H^s)}\leqq C\Vert g\Vert _{L^\infty (I; L^\infty )}\Vert f\Vert _{\widetilde{L}^p(I; H_\sharp ^s)}. \end{aligned}$$

(A.61)

As for the remainder $R(f, g)=\sum _{j\geqq 0} \sum _{|\nu |\leqq 2} \Delta _j f\Delta _{j+\nu }g$, we note that ${{\,\textrm{supp}\,}} \widehat{ \Delta _j f\Delta _{j+\nu }g}\subset \{|\xi |<2^{j+3}\}$. Thus $\Delta _k(\Delta _j f\Delta _{j+\nu }g)=0$ for $k\geqq j+5$ and

$$\begin{aligned} \Vert \Delta _k \sum _{|\nu |\leqq 2} \Delta _j f\Delta _{j+\nu }g\Vert _{L^2_x}\leqq \sum _{j\geqq k-4}\sum _{|\nu |\leqq 2} \Vert \Delta _k (\Delta _j f\Delta _{j+\nu }g)\Vert _{L^2_x}, \end{aligned}$$

(A.62)

where

$$\begin{aligned} \Vert \Delta _k (\Delta _j f\Delta _{j+\nu }g)\Vert _{L^2_x} \lesssim \Vert \Delta _j f\Vert _{L^2_x}\Vert g\Vert _{L^\infty _x }\quad \text {if } j\geqq 1 \end{aligned}$$

(A.63)

and

$$\begin{aligned} \Vert \Delta _k (\Delta _j f\Delta _{j+\nu }g)\Vert _{L^2_x} \lesssim \Vert \Delta _0 f\Vert _{L^\infty _x}\Vert \Delta _{\nu } g\Vert _{L^2_x}\lesssim \Vert \chi \widehat{f}\Vert _{L^1}\Vert \Delta _{\nu } g\Vert _{L^2_x} \quad \text {if } j=0.\nonumber \\ \end{aligned}$$

(A.64)

It follows that

$$\begin{aligned}{} & {} 2^{ks}\Vert \Delta _k \sum _{|\nu |\leqq 2} \Delta _j f\Delta _{j+\nu }g\Vert _{L^p(I; L^2)}\nonumber \\{} & {} \quad \lesssim {\left\{ \begin{array}{ll} 2^{ks}\sum _{j\geqq 1, j\geqq k-4}\Vert \Delta _j f\Vert _{L^p(I; L^2)}\Vert g\Vert _{L^\infty (I; L^\infty )}\quad \text {if } k\geqq 5,\\ 2^{ks}\Vert \chi \widehat{f}\Vert _{L^\infty (I; L^1)}\Vert \Delta _{\nu } g\Vert _{L^p(I; L^2)}\quad \text {if } k\leqq 4 \end{array}\right. }\nonumber \\{} & {} \quad \lesssim {\left\{ \begin{array}{ll} \sum _{j\geqq 1, j\geqq k-4}2^{js}\Vert \Delta _j f\Vert _{L^p(I; L^2)}\Vert g\Vert _{L^\infty (I; L^\infty )}2^{(k-j)s}\quad \text {if } k\geqq 5,\\ \Vert \chi \widehat{f}\Vert _{L^\infty (I; L^1)}\Vert g\Vert _{\widetilde{L}^p(I; L^2)}\quad \text {if } k\leqq 4. \end{array}\right. } \end{aligned}$$

(A.65)

By Young’s inequality for series, we deduce

$$\begin{aligned} \Big \Vert 2^{ks}\Vert \Delta _k \sum _{|\nu |\leqq 2} \Delta _j f\Delta _{j+\nu }g\Vert _{L^p(I; L^2)}\Big \Vert _{\ell ^2(\{k\geqq 5\})}\lesssim \Vert g\Vert _{L^\infty (I; L^\infty )}\Vert f\Vert _{\widetilde{L}^p(I; H_\sharp ^s)}.\nonumber \\ \end{aligned}$$

(A.66)

We thus obtain

$$\begin{aligned} \Vert R(f, g)\Vert _{\widetilde{L}^p(I; H^s)}\lesssim \Vert \chi \widehat{f}\Vert _{L^\infty (I; L^1)}\Vert g\Vert _{\widetilde{L}^p(I; H^s)}+ \Vert g\Vert _{L^\infty (I; L^\infty )}\Vert f\Vert _{\widetilde{L}^p(I; H_\sharp ^s)}.\nonumber \\ \end{aligned}$$

(A.67)

Combining (A.60), (A.61) and (A.67) we obtain (A.56). Finally, (A.57) follows from (A.56) and (A.6). $\square$

Our next result records some estimates for nonlinear maps of the form $(f,g) \mapsto g(1+f)^{-1}$.

Proposition A.19

Let $I\subset \mathbb {R}$, $p\in [1, \infty ]$, $s>0$, and $s_0>\frac{d}{2}$. There exists a positive constant $C=C(d, s, s_0)$ such that if $\Vert f\Vert _{L^\infty (I; \mathcal {H}^{s_0})}<\frac{1}{2C}$ then

$$\begin{aligned} \left\| \frac{g}{1+f}\right\| _{\widetilde{L}^p(I; H^s)}{} & {} \leqq \Vert g\Vert _{\widetilde{L}^p(I; H^s)}+C\Vert f\Vert _{L^\infty (I; \mathcal {H}^{s_0})}\Vert g\Vert _{\widetilde{L}^p(I; H^s)}\nonumber \\{} & {} \quad +\Vert g\Vert _{L^\infty (I; L^\infty )} \Vert f\Vert _{\widetilde{L}^p(I; H^s_\sharp )}. \end{aligned}$$

(A.68)

Proof

By virtue of (A.6) we have $\Vert f\Vert _{L^\infty (I; L^\infty )}\leqq C_1\Vert f\Vert _{L^\infty (I; \mathcal {H}^{s_0})}$, $C_1=C_1 (d, s, s_0)$, and hence $|f|\leqq \frac{1}{2}$ a.e. if $\Vert f\Vert _{L^\infty (I; \mathcal {H}^{s_0})}\leqq \frac{1}{2C_1}$. Then the expansion

$$\begin{aligned} \frac{g}{1+f}=\sum _{j\geqq 0}(-1)^jgf^j \end{aligned}$$

(A.69)

holds a.e on $\mathbb {R}^d$. We claim that with $C_2=\max \{C_1, C\}$, where C is given in (A.57), we have

$$\begin{aligned} \Vert gf^j\Vert _{\widetilde{L}^p(I; H^s)}\leqq & {} C_2(C_2\Vert f\Vert _{L^\infty (I; \mathcal {H}^{s_0})})^{j-1}\nonumber \\{} & {} \quad \left\{ \Vert f\Vert _{L^\infty (I; \mathcal {H}^{s_0})}\Vert g\Vert _{\widetilde{L}^p(I; H^s)}+j\Vert g\Vert _{L^\infty (I; L^\infty )} \Vert f\Vert _{\widetilde{L}^p(I; H^s_\sharp )}\right\} \nonumber \\ \end{aligned}$$

(A.70)

for all $j\geqq 1$. Indeed, the case $j=1$ follows at once from (A.57). Assume that (A.70) holds for some $j\geqq 1$. Applying (A.57) once again, we deduce

$$\begin{aligned} \begin{aligned} \Vert gf^{j+1}\Vert _{\widetilde{L}^p(I; H^s)}&=\Vert f(gf^j)\Vert _{\widetilde{L}^p(I; H^s)}\\&\leqq C_2\Vert f\Vert _{L^\infty (I; \mathcal {H}^{s_0})}\Vert gf^j\Vert _{\widetilde{L}^p(I; H^s)}+C_2\Vert gf^j\Vert _{L^\infty (I; L^\infty )} \Vert f\Vert _{\widetilde{L}^p(I; H^s_\sharp )}\\&\leqq C_2(C_2\Vert f\Vert _{L^\infty (I; \mathcal {H}^{s_0})})^j\Big \{\Vert f\Vert _{L^\infty (I; \mathcal {H}^{s_0})}\Vert g\Vert _{\widetilde{L}^p(I; H^s)}\\&\quad +j\Vert g\Vert _{L^\infty (I; L^\infty )} \Vert f\Vert _{\widetilde{L}^p(I; H^s_\sharp )}\Big \}\\&\quad +C_2\Vert gf^j\Vert _{L^\infty (I; L^\infty )} \Vert f\Vert _{\widetilde{L}^p(I; H^s_\sharp )}. \end{aligned} \end{aligned}$$

(A.71)

Combining this with the estimate

$$\begin{aligned} \Vert gf^j\Vert _{L^\infty (I; L^\infty )} \leqq \Vert g\Vert _{L^\infty (I; L^\infty )} \Vert f\Vert ^j_{L^\infty (I; L^\infty )}\leqq \Vert g\Vert _{L^\infty (I; L^\infty )}(C_1 \Vert f\Vert _{L^\infty (I; \mathcal {H}^{s_0})})^j \nonumber \\ \end{aligned}$$

(A.72)

we obtain (A.70) for $j+1$.

Finally, for $\Vert f\Vert _{L^\infty (I; \mathcal {H}^{s_0})}\leqq \frac{1}{2C_2}$ we can sum (A.70) over $j\geqq 1$ to obtain

$$\begin{aligned}{} & {} \left\| \frac{g}{1+f} \right\| _{\widetilde{L}^p(I; H^s)}\nonumber \\{} & {} \quad \leqq \Vert g\Vert _{\widetilde{L}^p(I; H^s)} +\frac{C}{1-C\Vert f\Vert _{L^\infty (I; \mathcal {H}^{s_0})}}\Vert f\Vert _{L^\infty (I; \mathcal {H}^{s_0}(\mathbb {R}^d))}\Vert g\Vert _{\widetilde{L}^p(I; H^s)} \nonumber \\{} & {} q\quad +\frac{C}{(1-C\Vert f\Vert _{L^\infty (I; \mathcal {H}^{s_0})})^2}\Vert g\Vert _{L^\infty (I; L^\infty )} \Vert f\Vert _{\widetilde{L}^p(I; H^s_\sharp )}. \end{aligned}$$

(A.73)

This implies (A.68). $\square$