On the Smoothness and Regularity of the Chess Billiard Flow and the Poincar\'e Problem

TL;DR

This paper investigates the regularity of solutions in the chess billiard flow model for internal waves, establishing conditions under which solutions are smooth and exploring implications for oceanic wave modeling.

Contribution

It extends previous studies by linking the regularity of solutions to the irrationality and Diophantine properties of rotation numbers in the chess billiard flow.

Findings

Smooth forcing functions lead to smooth solutions.

Regularity depends on the irrationality of rotation numbers.

Numerical simulations reveal plateau formation and fractal dimensions.

Abstract

The Poincar\'e problem is a model of two-dimensional internal waves in stable-stratified fluid. The chess billiard flow, a variation of a typical billiard flow, drives the formation behind and describes the evolution of these internal waves, and its trajectories can be represented as rotations around the boundary of a given domain. We find that for sufficiently irrational rotation in the square, or when the rotation number $r(\lambda)$ is Diophantine, the regularity of the solution $u(t)$ of the evolution problem correlates directly to the regularity of the forcing function $f(x)$. Additionally, we show that when $f$ is smooth, then $u$ is also smooth. These results extend studies that have examined singularity points, or the lack of regularity, in rational rotations of the chess billiard flow. We also present numerical simulations in various geometries that analyze plateau formation and fractal dimension in $r(\lambda)$ and conjecture an extension of our results. Our results can be applied in modeling two dimensional oceanic waves, and they also relate the classical quantum correspondence to fluid study.