Dispersive Estimates for Linearized Water Wave-Type Equations in $\mathbb {R}^d$

Abstract

We derive a $L^1_x (\mathbb {R}^d)-L^{\infty }_x (\mathbb {R}^d)$ decay estimate of order $\mathcal O \left( t^{-d/2}\right)$ for the linear propagators

$$\begin{aligned} \exp \left( {\pm it \sqrt{ |D|\left( 1+ \beta |D|^2\right) \tanh |D | } }\right) , \qquad \beta \in \{0, 1\}. \quad D= -i\nabla , \end{aligned}$$

with a loss of 3d/4 or d/4–derivatives in the case $\beta =0$ or $\beta =1$, respectively. These linear propagators are known to be associated with the linearized water wave equations, where the parameter $\beta$ measures surface tension effects. As an application, we prove low regularity well-posedness for a Whitham–Boussinesq-type system in $\mathbb {R}^d$, $d\ge 2$. This generalizes a recent result by Dinvay, Selberg and the third author where they proved low regularity well-posedness in $\mathbb {R}$ and $\mathbb {R}^2$.

Appendix

In this appendix, we derive some useful estimates on the derivatives of all order for the function

$$\begin{aligned} m_\beta (r)=\sqrt{ r\left( 1+ \beta r^2\right) \tanh ( r ) }, \qquad \beta \in \{0, 1\}. \end{aligned}$$

Estimates for the first- and second-order derivatives of this function are derived recently in [11].

Clearly,

$$\begin{aligned} m_\beta (r) \sim r \langle \sqrt{\beta } r \rangle \langle r \rangle ^{-1/2}. \end{aligned}$$

(6.1)

Lemma 5

Let $\beta \in \{0, 1\}$ and $r>0$. Then,

$$\begin{aligned} m_\beta '(r)&\sim \langle \sqrt{\beta } r \rangle \langle r \rangle ^{-1/2}, \end{aligned}$$

(6.2)

$$\begin{aligned} |m_\beta ''(r)|&\sim r \langle \sqrt{\beta } r \rangle \langle r \rangle ^{-5/2}. \end{aligned}$$

(6.3)

Moreover,

$$\begin{aligned} |m_\beta ^{(k)}(r)|\underset{k}{\lesssim }r^{1- k} \langle \sqrt{\beta } r \rangle \langle r \rangle ^{-1/2} \qquad ( k \ge 3 ). \end{aligned}$$

(6.4)

Proof

The estimates (6.2) and (6.3) are proved in [11, Lemma 3.2, see its proof in Section 5]. So we only prove (6.4).

Let

$$\begin{aligned} T(r)=\tanh r, \qquad S(r)={{\,\textrm{sech}\,}}r. \end{aligned}$$

Then,

$$\begin{aligned} T'&=S^2, \qquad S'=-T S, \qquad T''=-2TS^2. \end{aligned}$$

In general, we have

$$\begin{aligned} T^{(j)} (r)= S^2 \cdot P_{j-1}( S, T) \qquad (j\ge 1) \end{aligned}$$

for some polynomial $P_{j-1}$ of degree $j-1$.

Clearly,

$$\begin{aligned} T(r) \sim r\langle r \rangle ^{-1} \qquad \text {and} \qquad S(r) \sim e^{-r}. \end{aligned}$$

So $|P_{j-1}( S, T) |\lesssim 1$, and hence

$$\begin{aligned} |T^{(j)}(r)| \lesssim e^{-2r} \qquad (j\ge 1). \end{aligned}$$

(6.5)

Write

$$\begin{aligned} m_\beta (r)= f_\beta (r) \cdot T_0(r), \end{aligned}$$

where $f_\beta (r)= \sqrt{ r} \langle \sqrt{\beta } r \rangle$ and $T_0(r)=\sqrt{ T(r) }.$ One can show that

$$\begin{aligned} \Bigl \vert f_\beta ^{(j)}(r) \Bigr \vert \lesssim r^{\frac{1}{2} -j} \langle \sqrt{\beta } r \rangle \qquad ( j\ge 0). \end{aligned}$$

(6.6)

Combining (6.5) with $T(r) \sim r\langle r \rangle ^{-1}$ we obtain

$$\begin{aligned} T_0(r) \sim r^\frac{1}{2} \langle r \rangle ^{-\frac{1}{2}} , \qquad \Bigl \vert T^{(j)}_0(r) \Bigr \vert \lesssim r^{\frac{1}{2} -j } \langle r \rangle ^{j-\frac{1}{2} } e^{-2 r} \qquad ( j\ge 1). \end{aligned}$$

(6.7)

Finally, we use (6.6) and (6.7) to obtain for all $k \ge 3$,

$$\begin{aligned} \Bigl \vert m_\beta ^{(k)}(r) \Bigr \vert&= \Bigl \vert f_\beta ^{(k)}(r) T_0(r) + \sum _{j=1}^k \begin{pmatrix} k \\ j \end{pmatrix} f_\beta ^{(k-j)}(r) T^{(j)}_0(r) \Bigr \vert \\ {}&\lesssim r^{1 -k} \langle \sqrt{\beta } r \rangle \langle r \rangle ^{-\frac{1}{2}} + \sum _{j=1}^k \begin{pmatrix} k \\ j \end{pmatrix} r^{\frac{1}{2} -(k-j)} \langle \sqrt{\beta } r \rangle \cdot r^{\frac{1}{2} -j } \langle r \rangle ^{j-\frac{1}{2} } e^{-2r} \\ {}&\lesssim r^{1 -k} \langle \sqrt{\beta } r \rangle \langle r \rangle ^{-\frac{1}{2}} . \end{aligned}$$

$\square$

Corollary 2

For $\lambda , r>0$, define $m_{\beta , \lambda }(r)= m_\beta (\lambda r)$. Then,

$$\begin{aligned} \max _{ r\sim 1}\Bigl \vert \partial _r ^k \left( \frac{1}{ m'_{\beta , \lambda }(r)} \right) \Bigr \vert \underset{k}{\lesssim }\lambda ^{-1} \langle \sqrt{\beta } \lambda \rangle ^{-1}\langle \lambda \rangle ^{\frac{1}{2}} \qquad (k \ge 0). \end{aligned}$$

(6.8)

Proof

Observe that

$$\begin{aligned} m^{(k)}_{\beta , \lambda }(r) = \lambda ^k m_\beta ^{(k)}(\lambda r) \qquad (k\ge 1). \end{aligned}$$

By (6.2) we have, for $r\sim 1$,

$$\begin{aligned} | m'_{\beta , \lambda }(r) |\sim \lambda \langle \sqrt{\beta } \lambda \rangle \langle \lambda \rangle ^{-\frac{1}{2}} \end{aligned}$$

and by (6.3)–(6.4)

$$\begin{aligned} |m^{(k)}_{\beta , \lambda }(r) | \underset{k}{\lesssim }\lambda \langle \sqrt{\beta } \lambda \rangle \langle \lambda \rangle ^{-\frac{1}{2}} \qquad (k\ge 2). \end{aligned}$$

Finally, one can combine these two estimates with the differentiation formula

$$\begin{aligned} \partial _r^k \left( \frac{1}{f} \right) =\sum _{p=1}^k \sum _{ \begin{array}{c} k_1, \cdots , k_p \in \mathbb {N}\\ k_1+ \cdots + k_p =k \end{array}} c_{p, k_1, \cdots , k_p } \frac{ \partial _r^{k_1} f \cdots \partial _r^{k_p} f }{f^{p+1}} \end{aligned}$$

to obtain the desired estimate (6.8). $\square$