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

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