Stationary internal waves in a two-dimensional aquarium at low viscosity

TL;DR

The paper addresses the uniform solvability of the stationary internal-wave equation with small viscosity in a bounded 2D domain with real-analytic boundary under Morse–Smale dynamics. It achieves this by constructing a complex deformation of the domain to render the inviscid operator elliptic, performing a boundary-reduction via a fundamental solution and layer potentials, and then applying a robust vanishing-viscosity framework to extend invertibility to small viscosity $\nu>0$. The combination of complex-analytic deformation, microlocal analysis of layer potentials, and Frank's coercive-singular perturbation theory yields a uniform invertibility result for $P_{ω,ν}$ in a complex neighborhood of the forcing frequency $λ$, uniformly in $ν$ as $ν\to0^+$. This work provides a rigorous justification for stationary internal-wave attractors under weak dissipation and connects inviscid limiting absorption concepts with viscous dissipation in domains with boundaries. The methods have potential implications for inertial waves in rotating fluids and geophysical models, where boundary geometry and small viscosity play crucial roles.

Abstract

We prove the uniform solvability of a stationary problem associated to internal waves equation with small viscosity in a two dimensional aquarium with real-analytic boundary, under a Morse--Smale dynamical assumption. This is achieved by using complex deformations of the aquarium, on which the inviscid stationary internal wave operator is invertible.

Figures (3)

• Figure 1: A numerically computed chess billiard trajectory and a chess billiard map. Here we take $\Omega$ to be $\{x\in \mathbb R^2 : x_1^4+x_2^4<1\}$ rotated by $\frac{\pi}{10}$ counter-clock-wisely and $\lambda=\frac{1}{\sqrt{2}}$. Left: A trajectory of the chess billiard starting from a point on $\partial\Omega$. Right: Graph of the map $b^2$ under the parametrization $\theta\mapsto (s_1|\cos(\theta+\frac{\pi}{10})|^{\frac{1}{2}}, s_2 | \sin(\theta+\frac{\pi}{10}) |^{\frac{1}{2}} )$ with $s_1=\mathop{\mathrm{sign}}\nolimits(\cos(\theta+\frac{\pi}{10}))$, $s_2=\mathop{\mathrm{sign}}\nolimits(\sin(\theta+\frac{\pi}{10}))$ and $\theta\in \mathbb R/2\pi\mathbb Z$. The numerical results suggest $\Omega$ and $\lambda$ satisfy the Morse--Smale condition.
• Figure 2: An illustration of the function $h$ trivializing $X=h(\theta)\frac{\partial}{\partial \theta}$ and functions trivializing $(\gamma_\lambda^\pm)^*X$, where we identify $\partial \Omega$ with $\mathbb{S}^1 = \mathbb{R}/ \mathbb{Z}$ using the parametrization $z$. Here $C_{\pm}\coloneqq \{\gamma_\lambda^\pm(\theta)=\theta\}$ are sets of characteristic points and $\Sigma_{\pm}$ are sets of periodic points.
• Figure 3: Maps $\pi_{\pm}^{\textup{up/down}}$ used in the extension of $\Xi$.

Theorems & Definitions (77)

• Theorem 1
• Theorem 2
• Remark 1.1
• Remark 1.2
• Remark 1.3
• Definition 2.1: dwz_internal_waves
• Remark 2.2
• Definition 2.3: dwz_internal_waves
• Lemma 2.4: dwz_internal_waves
• Proposition 2.5