Inviscid Damping Near the Couette Flow in a Channel

Abstract

We prove asymptotic stability of the Couette flow for the 2D Euler equations in the domain $\mathbb {T}\times [0,1]$. More precisely we prove that if we start with a small and smooth perturbation (in a suitable Gevrey space) of the Couette flow, then the velocity field converges strongly to a nearby shear flow. Our solutions are defined on the compact set $\mathbb {T}\times [0,1]$ (“the channel”) and therefore have finite energy. The vorticity perturbation, which is initially assumed to be supported in the interior of the channel, will remain supported in the interior of the channel at all times, will be driven to higher frequencies by the linear flow, and will converge weakly to another shear flow as $t\rightarrow \infty$.

Appendix A. Gevrey Spaces and Local Wellposedness of Euler Equations

In this section we review some general properties of the Gevrey spaces of functions and prove the local well-posedness result in Lemma 3.1.

1.1 A.1 The Gevrey spaces

We start with a characterization of the Gevrey spaces on the physical side.

Lemma A.1

(i) Suppose that $0<s<1$, $K>1$, and $f\in C_0^{\infty }({\mathbb {R}}^d)$ satisfies the bounds

$$\begin{aligned} \big |D^{\alpha }f(x)\big |\le K^{m}(m+1)^{m/s}, \end{aligned}$$

(A.1)

for all integers $m\ge 0$ and multi-indeces $\alpha$ with $|\alpha |=m$. Assume that $(\mathrm{supp}\,f)\subseteq [-L,\,L]^d$, $L\ge 1$. Then

$$\begin{aligned} \big |{\widehat{f}}(\xi )\big |\lesssim _{K,s} L^de^{-\mu |\xi |^s}, \end{aligned}$$

(A.2)

for all $\xi \in {\mathbb {R}}^d$ and some $\mu =\mu (K,s)>0$.

Similarly, if $f\in C^{\infty }({\mathbb {T}}\times {\mathbb {R}})$ with $\mathrm{supp}\,f\subseteq {\mathbb {T}}\times [0,1]$ satisfies (A.1), then

$$\begin{aligned} \big |{\widetilde{f}}(k,\xi )\big |\lesssim _{K,s} Le^{-\mu |k,\xi |^s}, \end{aligned}$$

(A.3)

for all $k\in {\mathbb {Z}}, \xi \in {\mathbb {R}}$ and some $\mu =\mu (K,s)>0$.

(ii) Conversely, assume that, for some $\mu >0$ and $s\in (0,1)$,

$$\begin{aligned} \big |{\widehat{f}}(\xi )\big |\le e^{-\mu |\xi |^s}, \end{aligned}$$

(A.4)

for all $\xi \in {\mathbb {R}}^d$. Then there is $K>1$ depending on s and $\mu$ such that

$$\begin{aligned} \left| D^{\alpha }f(x)\right| \lesssim _{\mu ,s} K^m(m+1)^{m/s}, \end{aligned}$$

(A.5)

for all multi-indices $\alpha$ with $|\alpha |=m$.

Similarly, if $f\in C^{\infty }({\mathbb {T}}\times {\mathbb {R}})$ satisfies, for some $\mu >0,s\in (0,1)$,

$$\begin{aligned} \big |{\widetilde{f}}(k,\xi )\big |\le e^{-\mu |k,\xi |^s}, \end{aligned}$$

(A.6)

for all $k\in {\mathbb {Z}}, \xi \in {\mathbb {R}}$, then the bounds (A.5) hold for some $K=K(\mu ,s)>0$ and all $x\in {\mathbb {T}}\times {\mathbb {R}}$.

Proof

(i) We prove only the harder estimates (A.2). We may assume that $|\xi |$ is large. Using the definition of the Fourier transform, integration by parts, and the bounds (A.1), we see that

$$\begin{aligned} \big |{\widehat{f}}(\xi )\big |\le \frac{C^N}{|\xi |^N}K^N(N+1)^{N/s}L^d. \end{aligned}$$

(A.7)

This holds for all integers $N\ge 1$. Choose N to be the largest integer so that $CK(N+1)^{1/s}\le |\xi |/e$, thus

$$\begin{aligned} N=\frac{|\xi |^s}{(CKe)^s}+O(1). \end{aligned}$$

Consequently, using (A.7) we get that

$$\begin{aligned} \big |{\widehat{f}}(\xi )\big |\lesssim _s L^de^{-N}\lesssim _{K,\mu }L^de^{-\mu |\xi |^s}, \end{aligned}$$

for suitable $\mu >0$.

(ii) We consider only the case when $f\in C^\infty ({\mathbb {R}}^d)$. Using (A.4) we have

$$\begin{aligned} \Vert D^\alpha f\Vert _{L^\infty }\le C_0^m\Vert \langle \xi \rangle ^m{\widehat{f}}(\xi )\Vert _{L^1}\le C_1^m\big (1+\sup _{|\xi |\ge 2}(|\xi |^{m+d+1}e^{-\mu |\xi |^s})\big ). \end{aligned}$$

(A.8)

We notice that the function $r\rightarrow r^Ne^{-r}$, $r>0$, $N\ge 1$, has a maximum at $r=N$. Thus

$$\begin{aligned} r^Ne^{-r}\le (N/e)^N \qquad \text { for any }r>0\text { and }N\ge 1, \end{aligned}$$

(A.9)

so the right-hand side of (A.8) is bounded by

$$\begin{aligned} C_1^m\big [1+\sup _{r>0}(r/\mu )^{(1/s)\cdot (m+d+1)}e^{-r}\big ]\le K_1^m(N/e)^N, \end{aligned}$$

where $N=(m+d+1)/s$ and $K_1$ is sufficiently large. The desired bounds (A.5) follow.

$\square$

1.1.1 A.1.1 Gevrey cutoff functions.

Using Lemma A.1, one can construct explicit cutoff functions in Gevrey spaces. For $a>0$ let

$$\begin{aligned} \psi _a(x):={\left\{ \begin{array}{ll} e^{-[1/x^a+1/(1-x)^a]}&{}\quad \text { if }x\in [0,1],\\ 0&{}\quad \text { if }x\notin [0,1]. \end{array}\right. } \end{aligned}$$

(A.10)

Clearly $\psi _a$ are smooth functions on ${\mathbb {R}}$, supported in the interval [0, 1]. Using (A.9) it is easy to verify that $\psi _a$ satisfies the bounds (A.1) for $s:=a/(a+1)$. Thus, for some $\mu =\mu (a)>0$,

$$\begin{aligned} |\widehat{\psi _a}(\xi )|\lesssim e^{-\mu |\xi |^{a/(a+1)}}. \end{aligned}$$

(A.11)

One can also construct compactly supported Gevrey cutoff functions which are equal to 1 in a given interval. Indeed, for any $\rho \in [9/10,1)$, the function

$$\begin{aligned} \psi '_{a,\rho }(x):=\frac{\psi _a(x)}{\psi _a(x)+\psi _a(x-\rho )+\psi _a(x+\rho )} \end{aligned}$$

(A.12)

is smooth, non-negative, supported in [0, 1], and equal to 1 in $[1-\rho ,\rho ]$. Moreover, it follows from Lemma A.1(i) that $|\widehat{\psi '_{a,\rho }}(\xi )|\lesssim e^{-\mu |\xi |^{a/(a+1)}}$ for some $\mu =\mu (a,\rho )>0$.

1.1.2 A.1.2. Compositions of Gevrey functions.

The physical space characterization of Gevrey functions is useful when studying compositions. In our setting, we have the following lemma:

Lemma A.2

(i) Assume $\kappa _1>0$, $s\in (0,1)$, and $f\in {\mathcal {G}}^{\kappa _1,s}({\mathbb {T}}\times {\mathbb {R}})$. Suppose $M\in (0,\infty )$ and $g:{\mathbb {T}}\times {\mathbb {R}}\rightarrow {\mathbb {T}}\times {\mathbb {R}}$ satisfies, for any $m\ge 1$,

$$\begin{aligned} |D^\alpha g(x,y)|\le M^m(m+1)^{m/s}\qquad \text { for any }(x,t)\in \mathbb {T}\times {\mathbb {R}}\text { and }|\alpha |\in [1,m].\nonumber \\ \end{aligned}$$

(A.13)

Suppose that f and $f\circ g$ are supported in ${\mathbb {T}}\times [-2,2]$. Then, for a suitable $\kappa _2>0$ depending on $s,\kappa _1,M$, we have $f\circ g\in {\mathcal {G}}^{\kappa _2,s}$ and

$$\begin{aligned} \left\| f\circ g\right\| _{ {\mathcal {G}}^{\kappa _2,s}}\lesssim _{s,\kappa _1,M} \left\| f\right\| _{ {\mathcal {G}}^{\kappa _1,s}}. \end{aligned}$$

(A.14)

(ii) Assume $s\in (0,1)$, $L\in (0,\infty )$, $I,J\subseteq {\mathbb {R}}$ are open intervals, and $g:I\rightarrow J$ is a smooth bijective map satisfying, for any $m\ge 1$,

$$\begin{aligned} |D^\alpha g(x)|\le L^m(m+1)^{m/s}\qquad \text { for any }x\in I\text { and }|\alpha |\in [1,m]. \end{aligned}$$

(A.15)

If $|g'(x)|\ge 1/10$ for any $x\in I$ then the inverse function $g^{-1}:J\rightarrow I$ satisfies the bounds

$$\begin{aligned} |D^\alpha (g^{-1})(x)|\le M^m(m+1)^{m/s}\qquad \text { for any }x\in J\text { and }|\alpha |\in [1,m], \end{aligned}$$

(A.16)

for some constant $M=M(L,s)\ge L$.

This can be proved using Lemma A.1, and we omit the details. See also Theorem 6.1 and Theorem 3.2 of [25] for more general estimates on compositions of functions in Gevrey spaces.

1.2 A.2. Proof of Lemma 3.1

As we remarked earlier, the lemma can be obtained as a consequence of the more general theory developed in [10, 13, 14, 16]. For the sake of convenience, we provide a complete proof here in our special case, using the Fourier transform.

To prove Gevrey bounds, we have to work with suitable weights. First we define the functions $g:(0,\infty )\rightarrow (0,\infty )$, by

$$\begin{aligned} \begin{aligned} g'(r):= {\left\{ \begin{array}{ll} s\, r^{s-1}-s\rho ^{s-1}&{}\quad \text { if }r\in (0,\rho ],\\ 0&{}\quad \text { if }r\ge \rho , \end{array}\right. } \qquad g(r):=\int _0^r g'(x)\,dx. \end{aligned} \end{aligned}$$

(A.17)

Here $\rho \ge \rho _0$ is a large parameter (which is needed only to guarantee convergence and continuity in time of the energy functionals below), and the desired Gevrey bounds follow by proving estimates independent of $\rho$ and letting $\rho \rightarrow \infty$.

Then we define the main weights $B:[0,T]\times {\mathbb {R}}^2\rightarrow (0,\infty )$,

$$\begin{aligned} B(t,v):=\langle v\rangle ^3\exp [\lambda (t)g(\langle v\rangle )], \end{aligned}$$

(A.18)

where $\lambda (t):[0,T]\rightarrow (0,\infty )$, $\lambda (0)=\lambda _0$, is a positive decreasing function to be chosen below.

With $v=(k,\xi )$, we define the energy functionals

$$\begin{aligned} {\mathcal {E}}(t):=\sum _{k\in {\mathbb {Z}}}\int _{{\mathbb {R}}}B^2(t,k,\xi )\big |{\widetilde{\omega }}(t,k,\xi )\big |^2\,d\xi . \end{aligned}$$

(A.19)

Since $\omega \in C([0,T]:H^{10})$ and $B(t,v)\lesssim _\rho \langle v\rangle ^{3}$, the function ${\mathcal {E}}$ is well-defined and continuous on [0, T]. Moreover, ${\mathcal {E}}(0)\le \Vert \langle \nabla \rangle ^3\omega _0\Vert _{{\mathcal {G}}^{\lambda _0,s}}$.

Step 1. We fix a smooth function $\Psi (y)$ with

$$\begin{aligned} {\mathrm {supp}}\,\Psi \subseteq [\vartheta /8,1-\vartheta /8],\qquad \Psi |_{[\vartheta /4,1-\vartheta /4]}\equiv 1,\qquad |{\widetilde{\Psi }}(\xi )|\lesssim e^{-\langle \xi \rangle ^{7/8}}\text { for all }\xi \in {\mathbb {R}},\nonumber \\ \end{aligned}$$

(A.20)

where, in the rest of this section, all implicit constants are allowed to depend on $\vartheta$. By the support property of $\omega (t)$ and the Euler equation, we calculate

$$\begin{aligned} \begin{aligned} \frac{d}{dt}{\mathcal {E}}(t)&=\sum _{k\in {\mathbb {Z}}}\int _{{\mathbb {R}}}2B(t,k,\xi ){\dot{B}}(t,k,\xi )\big |{\widetilde{\omega }}(t,k,\xi )\big |^2\,d\xi \\&\quad +2\mathfrak {R}\,\sum _{k\in {\mathbb {Z}}}\int _{{\mathbb {R}}}B^2(t,k,\xi )\,\overline{{\widetilde{\omega }}(t,k,\xi )}\,\widetilde{\partial _t\omega }(t,k,\xi )\,d\xi \\&=\sum _{k\in {\mathbb {Z}}}\int _{{\mathbb {R}}}2B(t,k,\xi ){\dot{B}}(t,k,\xi )\big |{\widetilde{\omega }}(t,k,\xi )\big |^2\,d\xi +{\mathcal {P}}^1(t)+{\mathcal {P}}^2(t), \end{aligned} \end{aligned}$$

(A.21)

where, using the equations and inserting a factor of $\Psi (y)$,

$$\begin{aligned} \begin{aligned} {\mathcal {P}}^1(t)&:=C\mathfrak {R}\sum _{k\in \mathbb {Z}}\int _{{\mathbb {R}}^2}B^2(t,k,\xi )\,\overline{{\widetilde{\omega }}(t,k,\xi )}\cdot {\widetilde{(y\Psi )}(\xi -\eta )\,ik\,{\widetilde{\omega }}(t,k,\eta )}\,d\xi \,d\eta ,\\&=C\mathfrak {R}\sum _{k\in \mathbb {Z}}\int _{{\mathbb {R}}^2}ik[B^2(t,k,\xi )-B^2(t,k,\eta )]\,\overline{{\widetilde{\omega }}(t,k,\xi )}\\&\quad {\widetilde{\omega }}(t,k,\eta )\widetilde{(y\Psi )}(\xi -\eta )\,d\xi \,d\eta , \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} {\mathcal {P}}^2&(t):=C\mathfrak {R}\sum _{k,\ell \in \mathbb {Z}}\int _{{\mathbb {R}}^2}B^2(t,k,\xi )\,\overline{{\widetilde{\omega }}(t,k,\xi )}\,\Big \{-\widetilde{\partial _y(\Psi \psi )}(t,k-\ell ,\xi -\eta )\,i\ell \,{\widetilde{\omega }}(t,\ell ,\eta )\\&\qquad \qquad +\widetilde{\partial _x(\Psi \psi )}(t,k-\ell ,\xi -\eta )\,i\eta \,{\widetilde{\omega }}(t,\ell ,\eta )\Big \}\,d\xi d\eta \\&=C\mathfrak {R}\sum _{k,\ell \in \mathbb {Z}}\int _{{\mathbb {R}}^2}i[\ell B^2(t,k,\xi )-kB^2(t,\ell ,\eta )]\,\overline{{\widetilde{\omega }}(t,k,\xi )}\\&\quad {\widetilde{\omega }}(t,\ell ,\eta )\widetilde{\partial _y(\Psi \psi )}(t,k-\ell ,\xi -\eta )\,d\xi d\eta \\&+C\mathfrak {R}\sum _{k,\ell \in \mathbb {Z}}\int _{{\mathbb {R}}^2}i[\eta B^2(t,k,\xi )-\xi B^2(t,\ell ,\eta )]\,\overline{{\widetilde{\omega }}(t,k,\xi )}\\&\quad {\widetilde{\omega }}(t,\ell ,\eta )\widetilde{\partial _x(\Psi \psi )}(t,k-\ell ,\xi -\eta )\,d\xi d\eta . \end{aligned} \end{aligned}$$

Step 2. We estimate now $|{\mathcal {P}}^1(t)|$ and $|{\mathcal {P}}^2(t)|$. Using the bounds in Lemma A.3, we see that

$$\begin{aligned} |\ell B^2(t,k,\xi )-&kB^2(t,\ell ,\eta )|+|\eta B^2(t,k,\xi )-\xi B^2(t,\ell ,\eta )|\lesssim B(t,k,\xi )B(t,\ell ,\eta )\nonumber \\&\times B(t,k-\ell ,\xi -\eta )(\min )^{-3}[1+\lambda (t)g(\langle k,\xi \rangle )]\langle k-\ell ,\xi -\eta \rangle , \end{aligned}$$

(A.22)

where $\min :=\min \{\langle k,\xi \rangle ,\langle \ell ,\eta \rangle ,\langle k-\ell ,\xi -\eta \rangle \}$. Indeed, if $|k-\ell ,\xi -\eta |\ge (|k,\xi |+|\ell ,\eta |)/20$ then the bounds follow directly from (A.35), by estimating each term in the left-hand side independently. On the other hand, if $|k-\ell ,\xi -\eta |\le (|k,\xi |+|\ell ,\eta |)/20$ then we use (A.36), which forces us to include the term $\lambda (t)g(\langle k,\xi \rangle )$ in the right-hand side of (A.22).

Let

$$\begin{aligned} \begin{aligned} H(t,k,\xi )&:=\langle k,\xi \rangle ^2|\widetilde{\Psi \psi }(t,k,\xi )|+|{\widetilde{\omega }}(t,k,\xi )|,\\ H'(\xi )&:=\langle \xi \rangle |\widetilde{y\Psi }(t,k,\xi )|. \end{aligned} \end{aligned}$$

(A.23)

It follows from (A.22), changes of variables, and the identities above that

$$\begin{aligned} |{\mathcal {P}}^1(t)|&\lesssim \sum _{k\in \mathbb {Z}}\int _{{\mathbb {R}}^2}|{\widetilde{\omega }}(t,k,\xi )||{\widetilde{\omega }}(t,k,\eta )||H'(\xi -\eta )|\cdot B(t,k,\xi )B(t,k,\eta )\nonumber \\&\quad \times B(t,0,\xi -\eta )[1+\lambda (t)g(\langle k,\xi \rangle )]\,d\xi d\eta , \end{aligned}$$

(A.24)

$$\begin{aligned} |{\mathcal {P}}^2(t)|&\lesssim \sum _{k,\ell \in \mathbb {Z}}\int _{{\mathbb {R}}^2}|H(t,k,\xi )||H(t,\ell ,\eta )||H(t,k-\ell ,\xi -\eta )|\nonumber \\&\quad \times B(t,k,\xi )B(t,\ell ,\eta )B(t,k-\ell ,\xi -\eta ){\mathbf {1}}_{{\mathcal {R}}}((k,\xi ),(\ell ,\eta ))\nonumber \\&\quad \times \langle k-\ell ,\xi -\eta \rangle ^{-3}[1+\lambda (t)g(\langle k,\xi \rangle )]\,d\xi d\eta , \end{aligned}$$

(A.25)

where ${\mathcal {R}}:=\big \{\langle k-\ell ,\xi -\eta \rangle \le \min \{\langle k,\xi \rangle ,\langle \ell ,\eta \rangle \}\big \}$.

To estimate $|{\mathcal {P}}^1(t)|$ and $|{\mathcal {P}}^2(t)|$ we use the following elementary bounds (compare with Lemma 8.1): for any functions $F_1,F_2,F_3:\mathbb {Z}\times {\mathbb {R}}\rightarrow [0,\infty )$ we have

$$\begin{aligned} \sum _{k,\ell \in \mathbb {Z}}\int _{{\mathbb {R}}^2}F_1(k,\xi )F_2(\ell ,\eta )F_3(k-\ell ,\xi -\eta )\,d\xi d\eta \lesssim \Vert F_1\Vert _{L^2}\Vert F_2\Vert _{L^2}\Vert F_3\Vert _{L^1}.\nonumber \\ \end{aligned}$$

(A.26)

For $t\in [0,T]$ let

$$\begin{aligned} Y(t):=\big \Vert B(t,k,\xi )\,{\widetilde{\omega }}(t,k,\xi )\big \Vert _{L^2_{k,\xi }},\qquad Y'(t):=\big \Vert \sqrt{g(\langle k,\xi \rangle )}B(t,k,\xi )\,{\widetilde{\omega }}(t,k,\xi )\big \Vert _{L^2_{k,\xi }}.\nonumber \\ \end{aligned}$$

(A.27)

Using Lemma A.4, we have

$$\begin{aligned} \big \Vert W(t,k,\xi )H(t,k,\xi )\big \Vert _{L^2_{k,\xi }}\lesssim \big \Vert W(t,k,\xi )\,{\widetilde{\omega }}(t,k,\xi )\big \Vert _{L^2_{k,\xi }}, \end{aligned}$$

(A.28)

provided that $W(t,k,\xi )=B(t,k,\xi )$ or $W(t,k,\xi )=\sqrt{g(\langle k,\xi \rangle )}B(t,k,\xi )$.

Using (A.24), (A.20), and (A.26), we have, for any $t\in [0,T]$,

$$\begin{aligned} |{\mathcal {P}}^1(t)|\lesssim (Y(t))^2+\lambda (t)(Y'(t))^2 \end{aligned}$$

(A.29)

To estimate ${\mathcal {P}}^2$, we first observe that $B(t,v)\le \langle v\rangle ^3+\lambda (t)g(\langle v\rangle )B(t,v)$ with $v=(k,\xi )$. As a consequence, we have

$$\begin{aligned} B(t,v) [1+\lambda (t)g(\langle v\rangle )]\lesssim \langle v\rangle ^3+\lambda (t)g(\langle v\rangle )B(t,v). \end{aligned}$$

(A.30)

Using (A.25)–(A.28) and (A.30), we estimate

$$\begin{aligned} |{\mathcal {P}}^2(t)|\lesssim \lambda (t)Y(t)(Y'(t))^2+\Vert \omega (t)\Vert _{H^6}(Y(t))^2. \end{aligned}$$

(A.31)

Step 3. We reexamine now the identities (A.21). Notice that

$$\begin{aligned} {\dot{B}}(t,k,\xi )=\lambda '(t)g(\langle k,\xi \rangle )B(t,k,\xi ). \end{aligned}$$

We use (A.29)–(A.31) to estimate

$$\begin{aligned} \begin{aligned} \frac{d}{dt}{\mathcal {E}}(t)&\le \lambda '(t)(Y'(t))^2+{\mathcal {P}}^1(t)+{\mathcal {P}}^2(t)\\&\le \lambda '(t)(Y'(t))^2+C_\vartheta \big \{(Y(t))^2+\lambda (t)(Y'(t))^2\\&\quad +\lambda (t)Y(t)(Y'(t))^2+\Vert \omega (t)\Vert _{H^6}(Y(t))^2\big \}. \end{aligned} \end{aligned}$$

Now suppose $\lambda (t)$ satisfies

$$\begin{aligned} \lambda '(t)+C_\vartheta (Y(t)+1)\lambda (t)\le 0, \end{aligned}$$

(A.32)

then we would obtain

$$\begin{aligned} \frac{d}{dt}{\mathcal {E}}(t)\le C_{\vartheta }(1+\Vert \omega (t)\Vert _{H^6}){\mathcal {E}}(t), \end{aligned}$$

which gives

$$\begin{aligned} {\mathcal {E}}(t)\le {\mathcal {E}}(0)\exp {\left[ C_{\vartheta }\int _0^t(1+\Vert \omega (s)\Vert _{H^6})ds\right] }. \end{aligned}$$

(A.33)

From (A.33), we see that we can guarantee (A.32) if $\lambda (t)$ is chosen so that

$$\begin{aligned} \lambda '(t)+4C_{\vartheta }\lambda (t)+4C_{\vartheta }Y(0)\exp {\left[ C_{\vartheta }\int _0^t(1+\Vert \omega (s)\Vert _{H^6})ds\right] }\lambda (t)\le 0. \end{aligned}$$

(A.34)

The desired bounds (3.3) and (3.4) follow easily from (A.33), (A.34), by sending $\rho \rightarrow \infty$.

We prove now suitable bounds on the weights B which are used for (A.22).

Lemma A.3

For $v,w\in \mathbb {Z}\times {\mathbb {R}}$ we have

$$\begin{aligned} B(t,v+w)\lesssim B(t,v)B(t,w)\min (\langle v\rangle ,\langle w\rangle )^{-3}. \end{aligned}$$

(A.35)

Moreover, if $|v|\le |w|$ and $|w-v|\le |v|/4$, then

$$\begin{aligned} \big |B(t,w)-B(t,v)\big |&\lesssim [1+\lambda (t) g(\langle v\rangle )]\langle v\rangle ^{-1}B(t,v)\cdot B(t,v-w)\langle v-w\rangle ^{-2}.\nonumber \\ \end{aligned}$$

(A.36)

Proof

For $\lambda \in [0,1]$, $s\in [1/4,3/4]$, $\rho \ge 2^{40}$, and g as in (A.17), we define the functions $f_\lambda :(0,\infty )\rightarrow (0,\infty )$ as

$$\begin{aligned} f_\lambda (r):=\exp [\lambda g(r)]. \end{aligned}$$

(A.37)

Step 1. We show first that for $x,y\ge 1$ we have

$$\begin{aligned} \frac{f_\lambda (x+y)}{f_\lambda (x)f_\lambda (y)}\le 1. \end{aligned}$$

(A.38)

Indeed, we may assume $x\le y$ and calculate

$$\begin{aligned} \frac{f_\lambda (x+y)}{f_\lambda (x)f_\lambda (y)}= & {} \exp [\lambda (g(x+y)-g(x)-g(y))]\\= & {} \exp \Big [-\lambda \int _0^xg'(r)\,dr+\lambda \int _y^{y+x}g'(r)\,dr\Big ]. \end{aligned}$$

The bounds (A.38) follow since $g':(0,\infty )\rightarrow (0,\infty )$ is a continuous and decreasing function.

Step 2. We show now that if $y\ge 1$ and $x\in [0,y]$ then

$$\begin{aligned} |f_\lambda (y+x)-f_\lambda (y)|\le f_\lambda (x)f_\lambda (y)\lambda g(y)(x/y). \end{aligned}$$

(A.39)

Indeed, using the monotonicity of $g'$ we write

$$\begin{aligned} |f_\lambda (y+x)-f_\lambda (y)|=\int _y^{x+y}f_\lambda (r)\lambda g'(r)\,dr\le \lambda \int _0^{x}f_\lambda (y+r)g'(y)\,dr. \end{aligned}$$

(A.40)

Using now (A.38) we estimate the right-hand side of (A.40) by

$$\begin{aligned} \begin{aligned} \lambda g'(y)\int _0^{x}f_\lambda (y+r)\,dr\le \lambda g'(y)xf_\lambda (y+x)\le f_\lambda (x)f_\lambda (y)\lambda g'(y)x. \end{aligned} \end{aligned}$$

Since $yg'(y)\le g(y)$, this completes the proof of (A.39).

Step 3. Recall that $B(t,v)=\langle v\rangle ^3f_{\lambda (t)}(\langle v\rangle )$. The bounds (A.35) follow from (A.38) and the monotonicity of the functions $f_\lambda$. The bounds (A.36) follow from (A.39) and monotonicity. This completes the proof of the lemma. $\quad \square$

Finally, we prove weighted elliptic estimates for the functions $\Psi \,\psi$.

Lemma A.4

If $W(k,\xi ):=e^{\lambda g(\langle k,\xi \rangle )}\langle k,\xi \rangle ^dg(\langle k,\xi \rangle )^p$, $\lambda ,p\in [0,2]$, $d\in [0,6]$, and $t\in [0,T]$, then

$$\begin{aligned} \big \Vert \langle k,\xi \rangle ^2 W(k,\xi )\widetilde{\big (\Psi \,\psi \big )}(t,k,\xi )\big \Vert _{L^2_{k,\xi }}\lesssim \big \Vert W(k,\xi )\,{\widetilde{\omega }}(t,k,\xi )\big \Vert _{L^2_{k,\xi }}, \end{aligned}$$

(A.41)

uniformly in $\lambda ,p,d$, and $\rho$ (the parameter in the definition of the function g).

Proof

We only need to use the equation $\Delta \psi =\omega$, $\psi (x,0)=\psi (x,1)=0$, see (1.6). For simplicity of notation, we also drop the parameter t.

Step 1. As in Sect. 5.2.1, we take the partial Fourier transform along $\mathbb {T}$, thus

$$\begin{aligned} \partial _y^2\psi ^*(k,y)-k^2\psi ^*(k,y)=\omega ^*(k,y),\qquad y\in [0,1],\,k\in \mathbb {Z}. \end{aligned}$$

(A.42)

In particular, since $\omega ^*(k,y)=0$ if $y\in [0,\vartheta /2]$ or $y\in [1-\vartheta /2,1]$, we have

$$\begin{aligned} \begin{aligned} \psi ^*(k,y)= {\left\{ \begin{array}{ll} b_k^0\sinh (ky)/k&{}\qquad \text { if }k\ne 0\text { and }y\in [0,\vartheta /2],\\ b_k^1\sinh (k(y-1))/k&{}\qquad \text { if }k\ne 0\text { and }y\in [1-\vartheta /2,1],\\ b_k^0y&{}\qquad \text { if }k=0\text { and }y\in [0,\vartheta /2],\\ b_k^1(y-1)&{}\qquad \text { if }k=0\text { and }y\in [1-\vartheta /2,1], \end{array}\right. } \end{aligned} \end{aligned}$$

(A.43)

where $b_k^0:=(\partial _y\psi ^*)(k,0)$ and $b_k^1:=(\partial _y\psi ^*)(k,1)$.

We calculate the coefficients $b_k^0$ and $b_k^1$ using Green functions, as in Sect. 5.2.1. Indeed, it follows from (A.42) that

$$\begin{aligned} \psi ^*(k,y)=-\int _0^1\omega ^*(k,z)G_k(y,z)\,dz, \end{aligned}$$

(A.44)

where

$$\begin{aligned} G_k(y,z)=\frac{1}{k\sinh k} {\left\{ \begin{array}{ll} \sinh (k(1-z))\sinh (ky)&{}\qquad \text { if }y\le z,\\ \sinh (kz)\sinh (k(1-y))&{}\qquad \text { if }y\ge z, \end{array}\right. } \end{aligned}$$

(A.45)

if $k\ne 0$, and

$$\begin{aligned} G_0(y,z)= {\left\{ \begin{array}{ll} (1-z)y&{}\qquad \text { if }y\le z,\\ z(1-y)&{}\qquad \text { if }y\ge z. \end{array}\right. } \end{aligned}$$

(A.46)

In particular, for $\iota \in \{0,1\}$,

$$\begin{aligned} b_k^\iota =-\int _0^1\omega ^*(k,z)G_k^\iota (z)\,dz, \end{aligned}$$

(A.47)

where

$$\begin{aligned} \begin{aligned}&G_k^0(z)=\frac{\sinh (k(1-z))}{\sinh k}\,\,\text { if }\,\,k\ne 0,\qquad G_0^0(t)=1-z,\\&G_k^1(z)=\frac{-\sinh (k z)}{\sinh k}\,\,\text { if }\,\,k\ne 0,\quad \qquad \,\,\,\, G_0^1(t)=-z. \end{aligned} \end{aligned}$$

(A.48)

Step 2. We return now to the proof of (A.41). We start from the equation

$$\begin{aligned} \Delta (\Psi \psi )(x,y)=\Psi (y)\omega (x,y)+2\Psi '(y)\partial _y\psi (x,y)+\Psi ''(y)\psi (x,y). \end{aligned}$$

(A.49)

As in Lemma 5.3, see (5.34), using (A.47) and the support restriction on $\omega$, it follows that

$$\begin{aligned} |b_k^0|^2+|b_k^1|^2\lesssim \int _{{\mathbb {R}}}|{\widetilde{\omega }}(k,\xi )|^2e^{-|k|\vartheta /3}e^{-|\xi |^{7/8}}\,d\xi , \end{aligned}$$

(A.50)

for any $k\in {\mathbb {Z}}$. Let $\phi (x,y):=2\Psi '(y)\partial _y\psi (x,y)+\Psi ''(y)\psi (x,y)$ denote the last two terms in (A.49). It follows from (A.43) and (A.50) that

$$\begin{aligned} |{\widetilde{\phi }}(k,\xi )|\lesssim \Vert {\widetilde{\omega }}\Vert _{L^2_{k,\xi }}e^{-\vartheta |k|/20}e^{-|\xi |^{5/6}}, \end{aligned}$$

compare with (5.41). Moreover, since the decay of ${\widetilde{\Psi }}$ is faster than the variation of the weights W (compare with the definitions (A.18)), we have

$$\begin{aligned} \big \Vert W(k,\xi )\widetilde{\big (\Psi \,\omega \big )}(k,\xi )\big \Vert _{L^2_{k,\xi }}\lesssim \big \Vert W(k,\xi )\,{\widetilde{\omega }}(k,\xi )\big \Vert _{L^2_{k,\xi }}. \end{aligned}$$

The last two inequalities and the identity (A.49) show that

$$\begin{aligned} \big \Vert (k^2+\xi ^2) W(k,\xi )\widetilde{\big (\Psi \,\psi \big )}(k,\xi )\big \Vert _{L^2_{k,\xi }}\lesssim \big \Vert W(k,\xi )\,{\widetilde{\omega }}(k,\xi )\big \Vert _{L^2_{k,\xi }}. \end{aligned}$$

(A.51)

Finally, in the case $k=0$ we can use the formulas for $\psi ^*(0,y)$ in (A.43) and the bounds (A.50). It follows that $|\widetilde{\Psi \psi }(0,\xi )|\lesssim \Vert {\widetilde{\omega }}\Vert _{L^2_{k,\xi }}e^{-|\xi |^{5/6}}$. The desired conclusion (A.41) follows. $\quad \square$