Internal waves in 2D domains with ergodic classical dynamics

Analysis of (1.1) goes back to [Sob54]. The spectral properties of the operator $P$ defined in (1.3) were studied in [Ale60] and [Ral73]. The study of internal waves has motivated the mathematical analysis of 0-th order self-adjoint pseudodifferential operators. Such operators on closed surfaces in the presence of attractors were studied in [CdVSR20, CdV20] and then in [DZ19]. The viscosity limits of these 0-th order pseudodifferential operators were recently studied by Galkowski–Zworski [GZ22] and Wang [Wan22]. Spectral properties of 0-th order pseudodifferential operators on the circle were studied by Zhongkai Tao [Tao19] who produced examples of embedded eigenvalues.

Then for the case of 2D planar domains, [DWZ21] showed that when the underlying dynamics has hyperbolic attractors, there can be high concentration of the fluid velocity near these attractors. This phenomenon was predicted in the physics literature by Maas–Lam [ML95] in 1995, and has since been experimentally observed by Maas et al. [MBSL97], Hazewinkel et al. [HTMD10], and Brouzet [Bro16]. We also consider 2D planar domains in our work, but the underlying dynamical assumption is different, thus leading to different conclusions.

From (1.5), it is clear that we need to understand the spectral properties of $P$ at $\lambda^{2}$. We consider the eigenvalue problem

$$P(\lambda)u=0\quad\text{where}\quad P(\lambda):=(P-\lambda^{2})\Delta=(1-\lambda^{2})\partial_{x_{2}}^{2}-\lambda^{2}\partial_{x_{1}}^{2}.$$

(2.1)

Clearly, $P(\lambda):H^{1}_{0}(\Omega)\to H^{-1}(\Omega)$ is invertible if and only if $\lambda^{2}$ is not in the spectrum of the operator $P$ defined in (1.4). The problem has no nontrivial solutions if $\lambda>1$, because then the operator $P(\lambda)$ is elliptic and the result follows from the maximum principle. There is also no solution for $\lambda=0$ or $\lambda=1$ by direct inspection. It is also known that the spectrum of $P$ is precisely $[0,1]$, see [Ral73].

The advantage of working with $P(\lambda)$ is that it is simply a $(1+1)$-dimensional wave operator, and the symbol is given by the quadratic form $-(1-\lambda^{2})\xi_{2}^{2}+\lambda^{2}\xi_{1}^{2}$. The relevant classical dynamics here is given by the Hamiltonian flow of the symbol. The dual of this quadratic form can be factorized as

$$-\frac{x_{1}^{2}}{\lambda^{2}}+\frac{x_{2}^{2}}{1-\lambda^{2}}=\ell^{+}(x,\lambda)\ell^{-}(x,\lambda),\quad\ell^{\pm}(x,\lambda):=\pm\frac{x_{1}}{\lambda}+\frac{x_{2}}{\sqrt{1-\lambda^{2}}}.$$

(2.2)

In particular, the integral curves of the Hamiltonian flow of the symbol projected onto $\Omega$ are precisely the level sets of $\ell^{\pm}(x,\lambda)$.

Under the $\lambda$-simple condition, the dynamics can be reduced to the boundary. In particular, there exist unique smooth orientation-reversing involutions $\gamma^{\pm}(\bullet,\lambda):\partial\Omega\to\partial\Omega$ that satisfy

$$\ell^{\pm}(x)=\ell^{\pm}(\gamma^{\pm}(x)).$$

(2.3)

Composing $\gamma^{+}$ and $\gamma^{-}$, we obtain an orientation preserving diffeomorphism of the boundary $\partial\Omega$. This is known as the chess billiard map $b(\bullet,\lambda):\partial\Omega\to\partial\Omega$, defined by

$$b:=\gamma^{+}\circ\gamma^{-}.$$

(2.4)

See Figure 3. We will often suppress the dependence on $\lambda$ in the notation when there is no ambiguity.

Let $\mathbf{x}:\mathbb{S}^{1}=\mathbb{R}/\mathbb{Z}\to\partial\Omega$ be a positively oriented parametrization on $\partial\Omega$. This gives rise to a covering map $\tilde{\pi}:\mathbb{R}\to\partial\Omega$ given by $\tilde{\pi}(x)=\mathbf{x}(x\bmod 1)$. Let $\mathbf{b}(\bullet,\lambda):\mathbb{R}\to\mathbb{R}$ be a lift of $b(\bullet,\lambda)$, i.e. $\mathbf{b}(\bullet,\lambda)$ satisfies

$$\pi(\mathbf{b}(x,\lambda))=b(\pi(x),\lambda)$$

for all $x\in\mathbb{R}$. Fix an initial point $x_{0}\in\mathbb{R}$. The rotation number of $b(\bullet,\lambda)$ is then defined as

$$\mathbf{r}(\lambda):=\lim_{k\to\infty}\frac{\mathbf{b}^{k}(x_{0},\lambda)-x_{0}}{k}\in\mathbb{R}/\mathbb{Z}.$$

(2.5)

The rotation number is simply the averaged rotation along an orbit, and it is a well-defined quantity:

See e.g. [Wal99] for a proof of Lemma 2.2. We refer the reader to [dMvS93, Chapter 1] for a thorough discussion of circle homeomorphisms.

We wish to understand to what extent a circle diffeomorphism $b$ can be conjugated to a rotation. This is clearly impossible for rational rotations, and it was proved by Denjoy in [Den32] that when the rotation number is irrational, the circle diffeomorphism is topologically conjugate to a rotation. We will need more regularity than just topologically conjugate, which is known when the rotation number satisfies the following Diophantine condition:

Roughly speaking, this is a “sufficiently irrational” condition; $\alpha$ is Diophantine if it is sufficiently far away from all rational numbers. Diophantine numbers exist and form a full measure set. Indeed, take some $\beta>0$ and a small $c>0$, and consider the set of numbers in (0,1) that are Diophantine with respect to $\beta$ and $c$:

$$E=\left\{\alpha\in(0,1):\left|\alpha-\frac{p}{q}\right|\geq\frac{c}{q^{2+\beta}}\,\text{ for all $p\in\mathbb{Z}$ and $q\in\mathbb{N}$}\right\}.$$

We can think of $E$ as the set remaining after stripping away small neighborhoods of rational numbers, so the measure of $E$ is at least

$$|E|\geq 1-\sum_{q=1}^{\infty}2q\cdot\frac{c}{q^{2+\beta}}.$$

The right hand side becomes arbitrarily close to $1$ as $c\to 0$.

If the rotation number is irrational, then the topological conjugation can be upgraded to smooth conjugation to a rotation. This upgrade is due to Herman [Her79] and Yoccoz [Yoc84]. We collect the conjugation results from [Den32, Her79, Yoc84] in the following proposition.

A standard consequence of the topological conjugacy result is the unique ergodicity of circle diffeomorphisms with irrational rotation number.

Eventually, we will need to solve cohomological equations of the form

$$v(\theta)-v(\theta+\alpha)=g(\theta)$$

(2.6)

given $g\in C^{\infty}_{\mathrm{c}}(\mathbb{S}^{1})$ with vanishing integral. If $\alpha$ satisfies the Diophantine condition, then we can have explicit high frequency control of $v$. In particular, if $g$ is smooth, then we have smooth solutions.

We first show that solutions to the eigenvalue problem take a very convenient form.

We first construct the right inverse on smooth functions to $P(\lambda)$ when the rotation number defined in (2.5) is Diophantine.

The obstruction to energy boundedness in the functional calculus solution (1.5) is the singularity at $z=\lambda^{2}$ that appears as $t\to\infty$. This can be cancelled out using the right inverse to $P(\lambda)$ constructed in the previous proposition.

We now use the right inverse $R(\lambda)$ in Proposition 3.2 to obtain bounds on the spectral measure near $\lambda$.

We consider the square domain $\Omega=[0,1]\times[0,1]$. Clearly the Fourier modes provide a basis of eigenfunctions and it is easy to see that the eigenvalues are dense in $[0,1]$, so Theorem 2 for the square is clear.

The square thus has the advantage that (1.1) can be solved directly in Fourier series, so we can verify the contents of Theorem 1 for the square domain, despite it having corners.

The chess billiard flow on the square is the same as the standard billiard flow, so the rotation number function $\mathbf{r}(\lambda)$ defined in (2.5) is smooth and can be written down explicitly. It is given by

$$\mathbf{r}(\lambda)=\frac{\lambda}{\sqrt{1-\lambda^{2}}+\lambda}.$$

See [Zhu22] for the full derivation. We can formally write

$$u(t,x)=\sum_{\mathbf{k}\in\mathbb{N}^{2}}\hat{u}(t,\mathbf{k})\sin(\pi k_{1}x_{1})\sin(\pi k_{2}x_{2})$$

where $\mathbf{k}=(k_{1},k_{2})\in\mathbb{N}^{2}$. Only in this subsection, we use the hat to denote Fourier transform with respect to the Dirichlet sine basis. If $u(t,x)$ is a solution to (1.1), the coefficients must satisfy the periodically driven harmonic oscillator equation

$$\begin{gathered}-\pi^{2}(k_{1}^{2}+k_{2}^{2})\partial_{t}^{2}\hat{u}(t,\mathbf{k})+\pi^{2}k_{2}^{2}\hat{u}(t,\mathbf{k})=\hat{f}(\mathbf{k})\cos\lambda t,\\ \text{where}\quad\hat{f}(\mathbf{k})=\int_{[0,1]^{2}}f(x)\sin(\pi k_{1}x_{1})\sin(\pi k_{2}x_{2})\,dx_{1}dx_{2}\end{gathered}$$

(4.1)

This has solution

$$\hat{u}(t,\mathbf{k})=\frac{\hat{f}(\mathbf{k})}{\lambda^{2}k_{1}^{2}-(1-\lambda^{2})k_{2}^{2}}\left[\cos(\lambda t)-\cos\left(\frac{k_{2}}{|\mathbf{k}|}t\right)\right]$$

(4.2)

If $\mathbf{r}(\lambda)$ is Diophantine, then the result of Theorem 1 holds. In fact, in this case, we get that $u(t)\in C^{\infty}(\Omega)$ uniformly in all the seminorms for all $t\in\mathbb{R}$. Indeed, if $\mathbf{r}(\lambda)$ is Diophantine, then there exist constants $c,\,\beta>0$ such that

$$|q\cdot\mathbf{r}(\lambda)-p|\geq\frac{c}{q^{1+\beta}}$$

for any $p\in\mathbb{Z}$ and $q\in\mathbb{N}$. Rewriting this condition in terms of $\lambda$, we find that

$$|(q-p)\lambda-p\sqrt{1-\lambda^{2}}|\geq\frac{c}{q^{1+\beta}}$$

where $c$ is a possibly different constant. Put $q=k_{1}+k_{2}$ and $p=k_{2}$ to find that

$$|k_{1}\lambda-k_{2}\sqrt{1-\lambda^{2}}|\geq\frac{c}{q^{1+\beta}}\gtrsim\frac{1}{|\mathbf{k}|^{1+\beta}}.$$

(4.3)

where the hidden constant is independent of $\mathbf{k}$. We clearly have

$$|k_{1}\lambda+k_{2}\sqrt{1-\lambda^{2}}|\geq 1,$$

(4.4)

so combining (4.3) and (4.4) yields

$$|\lambda^{2}k_{1}^{2}-(1-\lambda^{2})k_{2}^{2}|\geq\frac{c}{|\mathbf{k}|^{1+\beta}}.$$

(4.5)

for a possibly different constant $c>0$. Since $f\in C^{\infty}_{\mathrm{c}}(\Omega;\mathbb{R})$, $\hat{f}(\mathbf{k})$ is rapidly decreasing in $\mathbf{k}$, which tempers the denominator in (4.2) to give uniform smoothness of $u(t)$ in time.

The above analysis also exhibits the spectral result part (c) of Theorem 1. Note that $\kappa(\mathbf{k})\sin(\pi k_{1}x_{1})\sin(\pi k_{2}x_{2})$, $\mathbf{k}\in\mathbb{N}^{2}$ form a complete orthonormal basis for $H^{-1}(\Omega)$, where $\kappa(\mathbf{k})=\mathcal{O}(|k|)$ are simply normalizing constants. This basis consists of eigenfunctions with eigenvalues $\frac{k_{1}}{k_{1}+k_{2}}$ for the operator $P$ defined in (1.4). In particular, $\sin(\pi k_{1}x_{1})\sin(\pi k_{2}x_{2})$ has eigenvalue $\frac{k_{2}^{2}}{k_{1}^{2}+k_{2}^{2}}$. If $\mathbf{r}(\lambda)$ is Diophantine, then (4.5) gives a characterization of the eigenvalues near $\lambda^{2}$:

$$\left|\frac{k_{2}^{2}}{k_{1}^{2}+k_{2}^{2}}-\lambda^{2}\right|\leq\varepsilon\implies\frac{1}{|\mathbf{k}|^{3+\beta}}\gtrsim\varepsilon^{-1}$$

Therefore the spectral measure $\mu_{f,f}$ satisfies part (c) of Theorem 1 near $\lambda^{2}$. Indeed, using the fact that the coefficients $\hat{f}(k)$ of $f\in C^{\infty}_{\mathrm{c}}(\Omega)$ defined in (4.1) are rapidly decreasing, we have

$$\mu_{f,f}((\lambda^{2}-\varepsilon,\lambda^{2}+\varepsilon))\lesssim\sum_{\mathbf{k}:\big{|}\frac{k_{2}}{k_{1}^{2}+k_{2}^{2}}-\lambda^{2}\big{|}\leq\varepsilon^{-1}}\hat{f}(\mathbf{k})^{2}\kappa(\mathbf{k})^{-2}<C_{d}\varepsilon^{d}$$

for any $d\in\mathbb{N}$. See Figure 2.

Finally, we mention that by tilting the square by $\eta$, the set of $\lambda$ for which $\mathbf{\lambda}$ is Diophantine is no longer a full measure set. More specifically, we consider the square domain specified by the vertices

$$\big{\{}(0,0),\,(\cos\eta,\sin\eta),\,(-\sin\eta,\cos\eta),\,\sqrt{2}(\cos(\eta+\tfrac{\pi}{4}),\sin(\eta+\tfrac{\pi}{4}))\big{\}}.$$

Then graph of $\mathbf{r}(\lambda)$ is constant near values of $\lambda$ for which $\mathbf{r}(\lambda)$ is rational. See Figure 4 for an illustration and [DWZ21, §2.5] for details.

We finally consider the case when $\Omega=\mathbb{D}$ is the unit disk. This has previously been studied in [Ale60, Section 9]. The 3-dimensional case of a triaxial ellipsoid has been studied in [CdVV23], following the physics work of [IJW15, BR17].

We may assume $\mathbb{D}$ is centered at origin. Parameterize the boundary $\partial\mathbb{D}$ counterclockwise by arclength with $\phi=0$ being the point $(1,0)$. The boundary is then identified with the circle $\mathbb{R}/2\pi\mathbb{Z}$. This is a different convention from the previous sections, and we switch conventions to avoid carrying factors of $2\pi$. The mod 1 rotation number as defined in 2.5 is given by

$$\mathbf{r}(\lambda)=1-\frac{2}{\pi}\arccos(\lambda),$$

(4.6)

see [Li23, §2.2.2] for details and Figure 4. It will be convenient to rescale the rotation number and define

$$\alpha(\lambda)=\frac{\pi}{2}\mathbf{r}(\lambda).$$

(4.7)

We henceforth drop $\lambda$ from the notation when there is no ambiguity. The terms of the factorization (2.2) and the chess billiard map take the explicit forms

$$\ell^{\pm}(x_{1},x_{2})=x_{1}\cos\alpha\pm x_{2}\sin\alpha,\qquad b(\phi)=\phi+4\alpha.$$

(4.8)

We will provide an explicit complete basis of eigenfunctions. Recall from (2.2) that $\ell^{+}(x,\lambda)\ell^{-}(x,\lambda)$ is dual to the symbol of $P(\lambda)$, so $u\in H^{1}_{0}(\Omega)$ is a solution to the eigenvalue problem if and only if

$$\begin{gathered}u(x)=u_{+}(\ell^{+}(x))+u_{-}(\ell^{-}(x))\quad\text{and}\\ U_{+}+U_{-}=0\quad\text{where}\quad U_{+}:=(u_{+}\circ\ell^{+})|_{\partial\Omega},\quad U_{-}:=(u_{-}\circ\ell^{-})|_{\partial\Omega}.\end{gathered}$$

(4.9)

for some $u_{\pm}\in C^{0}$ and $u_{\pm}\circ\ell^{\pm}\in H^{1}(\Omega)$. See Lemma 3.1 for discussion of the regularity of $u_{\pm}$. Using the explicit formulas (4.8), we can compute

$$\begin{gathered}U_{\pm}(\phi)=u_{\pm}(\cos\phi\cos\alpha\pm\sin\phi\sin\alpha)=u_{\pm}(\cos(\phi\mp\alpha)),\\ U_{\pm}(\phi)=U_{\pm}(\phi+4\alpha)\end{gathered}$$

(4.10)