Linear Inviscid Damping in Gevrey Spaces

Abstract

We prove linear inviscid damping near a general class of monotone shear flows in a finite channel, in Gevrey spaces. This is an essential step towards proving nonlinear inviscid damping for general shear flows that are not close to the Couette flow, which is a major open problem in 2d Euler equations.

Appendix A: Gevrey Spaces and Gevrey Bounds on the Green’s Function

1.1 Gevrey Spaces

We review first some general properties of the Gevrey spaces of functions.

We start with a characterization of the Gevrey spaces on the physical side. See Lemma A2 in [9] for the elementary proof.

Lemma A.1

(i) Suppose that $0<s<1$, $K>1$, and $g\in C^{\infty }({\mathbb {T}}\times {\mathbb {R}})$ with $\mathrm{supp}\,g\subseteq {\mathbb {T}}\times [-L,L]$ satisfies the bounds

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

(A.1)

for all integers $m\ge 0$ and multi-indeces $\alpha$ with $|\alpha |=m$. Then

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

(A.2)

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

(ii) Conversely, assume that $\mu >0$, $s\in (0,1)$, and $g:{\mathbb {T}}\times {\mathbb {R}}\rightarrow {\mathbb {C}}$ satisfies

$$\begin{aligned} \big \Vert g\big \Vert _{{\mathcal {G}}^{\mu ,s}({\mathbb {T}}\times {\mathbb {R}})}\le 1. \end{aligned}$$

(A.3)

Then there is $K=K(s,\mu )>1$ such that, for any $m\ge 0$ and all multi-indices $\alpha$ with $|\alpha |\le m$,

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

(A.4)

The physical space characterization of Gevrey functions is useful when studying compositions and algebraic operations of functions. For any domain $D\subseteq {\mathbb {T}}\times {\mathbb {R}}$ (or $D\subseteq {\mathbb {R}}$) and parameters $s\in (0,1)$ and $M\ge 1$ we define the spaces

$$\begin{aligned} \widetilde{{\mathcal {G}}}^{s}_M(D):= & {} \big \{g:D\rightarrow {\mathbb {C}}:\,\Vert g\Vert _{\widetilde{{\mathcal {G}}}^{s}_M(D)} \nonumber \\:= & {} \sup _{x\in D,\,m\ge 0,\,|\alpha |\le m}|D^\alpha g(x)|M^{-m}(m+1)^{-m/s}<\infty \big \}. \end{aligned}$$

(A.5)

Lemma A.2

(i) Assume $s\in (0,1)$, $M\ge 1$, and $g_1,g_2\in \widetilde{{\mathcal {G}}}^{s}_M(D)$. Then $g_1g_2\in \widetilde{{\mathcal {G}}}^{s}_{M'}(D)$ and

$$\begin{aligned} \Vert g_1g_2\Vert _{\widetilde{{\mathcal {G}}}^{s}_{M'}(D)}\lesssim \Vert g_1\Vert _{\widetilde{{\mathcal {G}}}^{s}_{M}(D)}\Vert g_2\Vert _{\widetilde{{\mathcal {G}}}^{s}_{M}(D)} \end{aligned}$$

for some $M'=M'(s,M)\ge M$. Similarly, if $g_1\ge 1$ in D then $\Vert (1/g_1)\Vert _{\widetilde{{\mathcal {G}}}^{s}_{M'}(D)}\lesssim 1$.

(ii) Suppose $s\in (0,1)$, $M\ge 1$, $I_1\subseteq {\mathbb {R}}$ is an interval, and $g:{\mathbb {T}}\times I_1\rightarrow {\mathbb {T}}\times I_2$ satisfies

$$\begin{aligned} |D^\alpha g(x)|\le M^m(m+1)^{m/s}\qquad \text { for any }x\in {\mathbb {T}}\times I_1,\,m\ge 1,\text { and }|\alpha |\in [1,m].\nonumber \\ \end{aligned}$$

(A.6)

If $K\ge 1$ and $h\in {\widetilde{G}}^s_K({\mathbb {T}}\times I_2)$ then $h\circ g\in {\widetilde{G}}^s_L({\mathbb {T}}\times I_1)$ for some $L=L(s,K,M)\ge 1$ and

$$\begin{aligned} \left\| h\circ g\right\| _{{\widetilde{G}}^s_L({\mathbb {T}}\times I_1)}\lesssim _{s,K,M} \left\| h\right\| _{{\widetilde{G}}^s_K({\mathbb {T}}\times I_2)}. \end{aligned}$$

(A.7)

(iii) Assume $s\in (0,1)$, $L\in [1,\infty )$, $I,J\subseteq {\mathbb {R}}$ 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.8)

If $|g'(x)|\ge \rho >0$ 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.9)

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

Lemma A.2 can be proved by elementary means using just the definition (A.5). See also Theorem 6.1 and Theorem 3.2 of [20] for more general estimates on functions in Gevrey spaces.

1.1.1 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] and independent of the periodic variable. It is easy to verify that $\psi _a$ satisfies the bounds (A.1) for $s:=a/(a+1)$. Thus

$$\begin{aligned} |\widetilde{\psi _a}(\xi )|\lesssim e^{-\mu |\xi |^{a/(a+1)}}\qquad \text { for some }\mu =\mu (a)>0. \end{aligned}$$

(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 $|\widetilde{\psi '_{a,\rho }}(\xi )|\lesssim e^{-\mu |\xi |^{a/(a+1)}}$ for some $\mu =\mu (a,\rho )>0$.

1.2 Gevrey Bounds on the Localized Green’s Function

We now provide the proof of Lemma 4.2 (which we recall below).

Lemma A.3

Define the localized Green’s function ${\mathcal {G}}_k$ as

$$\begin{aligned} {\mathcal {G}}^{*}_k(v,w):=\Psi (v){\mathcal {G}}_k(v,w)\Psi (w),\qquad \mathrm{for}\,\,v,w\in {\mathbb {R}}. \end{aligned}$$

(A.13)

Then, for some $\delta _0:=\delta _0(\vartheta _1)>0$, we have the bounds

$$\begin{aligned} \big |\widetilde{{\mathcal {G}}_k^{*}}(\xi ,\eta )\big |\lesssim \frac{e^{-\delta _0\langle \xi +\eta \rangle ^{(s+1)/2}}}{k^2+|\eta |^2},\qquad \mathrm{for}\,\,\xi ,\eta \in {\mathbb {R}}. \end{aligned}$$

(A.14)

Proof

In view of the definitions (3.2), for $y,z\in [0,1]$,

$$\begin{aligned}&G_k(y,z) =\frac{1}{4|k|\sinh {|k|}}\nonumber \\&\quad \times \left[ e^{|k|}e^{-|k||y-z|}+e^{-|k|}e^{|k||y-z|}-e^{|k|}e^{-|k|(y+z)}-e^{-|k|}e^{|k|(y+z)}\right] .\nonumber \\ \end{aligned}$$

(A.15)

Therefore, by direct computation, we conclude that for any $\delta >0$ the following bounds hold:

$$\begin{aligned} \left| \partial _y^m\partial _z^lh_k(y,z)\right| \lesssim (\delta /2)^{-m-l}\big [(m+l)!\big ]e^{-(\delta /20)|k|},\qquad \mathrm{for}\,\,k,l\in {\mathbb {Z}}\cap [0,\infty ),\nonumber \\ \end{aligned}$$

(A.16)

where either $h_k\in \big \{e^{-|k||y-z|}, e^{-2|k|}e^{|k||y-z|}\big \}$ and $y,z\in [0,1], |y-z|>\delta$; or $h_k\in \big \{e^{-|k|(y+z)}$, $e^{-2|k|}e^{|k|(y+z)}\big \}$ and $y,z\in [\delta ,1-\delta ]$; or $h_k=|k|G_k$ and $y,z\in [\delta ,1-\delta ]$, $|y-z|>\delta$.

Notice that

$$\begin{aligned} b^{-1}(v)-b^{-1}(w)=(v-w)\int _0^1\big (b^{-1}\big )'(w+s(v-w))\,ds. \end{aligned}$$

(A.17)

Denote

$$\begin{aligned} F(v,w):=\int _0^1\big (b^{-1}\big )'(w+s(v-w))\,ds,\qquad F^{*}(v,w):=b^{-1}(v)+b^{-1}(w).\nonumber \\ \end{aligned}$$

(A.18)

Then by (1.6)–(1.7), the physical space characterization of Gevrey spaces (see Lemma A.1 and Lemma A.2) $g\in \{F,F^{*}\}$ satisfies, for $v,w\in [{\underline{v}},{\overline{v}}]$,

$$\begin{aligned} \left| \partial _v^m\partial _w^lg(v,w)\right| \lesssim C^{m+l}((m+l)!)^{2/(s+1)},\qquad \mathrm{for}\,\,m,l\in {\mathbb {Z}}\cap [0,\infty ). \end{aligned}$$

(A.19)

In view of the change of variables (3.23) and the definitions (A.18), we can write for $v,w\in [{\underline{v}},{\overline{v}}]$ that

$$\begin{aligned} \begin{aligned} {\mathcal {G}}_k(v,w)&=\frac{1}{4|k|\sinh {|k|}}\Big [e^{-|k|}e^{|k||b^{-1}(v)-b^{-1}(w)|}+e^{|k|}e^{-|k||b^{-1}(v)-b^{-1}(w)|}\\&\quad -e^{|k|}e^{-|k|(b^{-1}(v)+b^{-1}(w))}-e^{-|k|}e^{|k|(b^{-1}(v)+b^{-1}(w))}\Big ]\\&=\frac{1}{4|k|\sinh {|k|}}\Big [e^{-|k|}e^{|k|F(v,w)|v-w|}+e^{|k|}e^{-|k|F(v,w)|v-w|}\\&\quad -e^{|k|}e^{-|k|F^{*}(v,w)}-e^{-|k|}e^{|k|F^{*}(v,w)}\Big ]. \end{aligned} \end{aligned}$$

(A.20)

Therefore, for any $v,w\in {\mathbb {R}}$,

$$\begin{aligned} \begin{aligned}&\Psi (v){\mathcal {G}}_k(v,v+w)\Psi (v+w)\\&\quad =\frac{\Psi (v)\Psi (v+w)}{4|k|\sinh {|k|}}\Big [e^{-|k|}e^{|k|F(v,v+w)|w|}+e^{|k|}e^{-|k|F(v,v+w)|w|}\\&\qquad -e^{|k|}e^{-|k|F^{*}(v,v+w)}-e^{-|k|}e^{|k|F^{*}(v,v+w)}\Big ]. \end{aligned} \end{aligned}$$

(A.21)

We claim the following bounds for $H_1=e^{-|k||w|F(v,v+w)}, H_2=e^{-2|k|+|k||w|F(v,v+w)},$ and $H_3\in \big \{-e^{-|k|F^{*}(v,v+w)},-e^{-2|k|}e^{|k|F^{*}(v,v+w)}\big \}$ with $(v,w)\in \mathrm{supp}\,\Psi (v)\Psi (v+w)$:

$$\begin{aligned} \left| \partial _v^lH_a(v,w)\right| \lesssim e^{-\mu |kw|}C^l(l!)^{2/(1+s)},\qquad \mathrm{for}\,\,l\in {\mathbb {Z}}\cap [0,\infty ), a\in \{1,2,3\},\nonumber \\ \end{aligned}$$

(A.22)

where $\mu >0$ is a sufficiently small number (depending on $\vartheta _1$).

The bounds (A.22) for $H_3$ follow from the bounds (A.16) for the functions $e^{-|k|(y+z)}$ and $e^{|k|(y+z-2)}$ where $y,z\in [\delta ,1-\delta ]$ for a sufficiently small $\delta >0$, and the property of Gevrey regular functions under compositions, see Lemma A.2.

To prove the bounds (A.22) for $H_1$ we consider separately the cases $|kw|<1$ and $|kw|>1$, and use (A.15)–(A.19). More precisely, if $|kw|<1$, the bounds (A.22) for $H_1$ follow direclty from the property of Gevrey regular functions under composition, since $|kw|F(v,v+w)$ is Gevrey regular in v with uniform bounds in k and w satisfying $|kw|<1$. For the case of $|kw|>1$, we first note the function $a\rightarrow e^{-\kappa a}$ is uniformly Gevrey regular with respect to $\kappa >1$ in $a\in [\delta ,\infty )$ for any fixed $\delta >0$. More precisely, we have for $\kappa >1$,

$$\begin{aligned} \left| \partial _a^me^{-\kappa a}\right| \lesssim 4^m \delta ^{-m}(m!) e^{-\kappa \delta /2},\qquad \mathrm{for \,\,}m\in {\mathbb {Z}}\cap [1,\infty ). \end{aligned}$$

(A.23)

Now we view $H_1$ as the composition of $e^{-\kappa a}$ and $F(v,v+w)$ with $\kappa =|kw|$, notice from (A.18) that $F(v,v+w)\approx _{\vartheta _1} 1$ on the support of $\Psi (v)\Psi (v+w)$. Then the bounds (A.22) for $H_1$ follow from (A.23) and the property of Gevrey spaces under compositions, see Lemma A.2.

The bounds (A.22) for $H_2$ follow from similar arguments as the case of $H_1$. The case $|kw|<1$ follows from the same argument so focus on the case $|kw|>1$. We view $H_2$ as the composition of the function $a\rightarrow e^{-|k|a}$ and $2-F(v,v+w)|w|$ and notice that $F(v,v+w)|w|<2-\delta$ on the support of $\Psi (v)\Psi (v+w)$ for a small $\delta$ depending on $\vartheta _1$, see (A.17)–(A.18). The bounds (A.22) for $H_2$ then follow from analogous arguments as in the case of $H_1$. This completes the proof of (A.22).

To finish the proof of Lemma A.3, we make the observation that for $\xi ,\eta \in {\mathbb {R}}$,

$$\begin{aligned} \begin{aligned}&\int _{{\mathbb {R}}^2}\Psi (v){\mathcal {G}}_k(v,w)\Psi (w)e^{-i v\xi -iw\eta }\mathrm{{d}}v \mathrm{{d}}w \\&\quad =\int _{{\mathbb {R}}^2}\Psi (v){\mathcal {G}}_k(v,v+w)\Psi (v+w)e^{-iv(\xi +\eta )-iw\eta }\,\mathrm{{d}}v\mathrm{{d}}w. \end{aligned} \end{aligned}$$

(A.24)

The claimed bounds (A.14) follow from integration by parts in (A.24) in the variable w (twice) and then apply Lemma A.1 in the variable v, using (A.21)–(A.22). $\quad \square$