ON THE SMOOTHNESS AND REGULARITY OF THE CHESS BILLIARD FLOW AND THE POINCARÉ PROBLEM SALLY ZHU

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ON THE SMOOTHNESS AND REGULARITY OF THE CHESS

BILLIARD FLOW AND THE POINCARÉ PROBLEM

SALLY ZHU

Under the direction of

Zhenhao Li

Graduate Student

Massachusetts Institute of Technology

Abstract. The Poincaré 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(λ)is Diophan-

tine, 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 fis smooth, then

uis 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(λ)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.

arXiv:2210.13384v1 [math.AP] 12 Oct 2022

2 SALLY ZHU

1. Introduction

Internal waves are important to the study of oceanography and to the theory of rotating

ﬂuids. The waves describe how an originally unmoving ﬂuid can move and evolve under a

periodic forcing function. We study the behavior of a particular two-dimensional model for

these internal waves, called the Poincaré problem, which forms patterns called billiard ﬂows.

Billiard ﬂows are a type of mapping that maps points on a boundary of a given shape to

another point on that boundary, with some sort of reﬂection or bouncing step. For example,

the most familiar billiard ﬂow is the pool billiard mapping in a game of pool, in which the

ball is bounced oﬀ one side of the table and follows a reﬂective trajectory in which the angle

of incidence equals the angle of exit.

We consider a variation of that billiard ﬂow, called the chess billiard ﬂow. Rather than

preserving the angles of reﬂection, this map instead preserves the slopes of the trajectories.

Each mapping bconsists of traveling ﬁrst on a line of slope ρ, and then bouncing oﬀ the

boundary at a slope of −ρ. We visualize this in Figure 1 from Dyatlov et al. [1], in which

we start at x, travel on the blue line of some slope ρ, then bounce oﬀ the side and traveling

on the red line of slope −ρ, to b(x).

Figure 1. Depiction of a series of chess billiard mappings on a trapezoid,

traveling across the parallel red and blue lines, from xto b(x)to b2(x)and so

on, from Dyatlov et al. [1]

The ﬁgure shows the mapping from xto b(x)to b2(x), and so on, recursively traveling on

the blue and red lines. Note that each mapping consists of two movements: one blue line,

then one red line.

SMOOTHNESS AND REGULARITY IN THE POINCARÉ PROBLEM 3

Another way to describe the chess billiard map is as a rotation of points along the bound-

ary. Figure 2a shows one mapping on the square, from xto b(x). We can imagine xas being

rotated counterclockwise to b(x), following the arrow in the ﬁgure. Figure 2b further depicts

this, where we can visualize b(x)being rotated to b2(x)following the arrow. This rotation

allows us to more easily understand how the chess billiard map maps points.

(a) Mapping from xto b(x).

(b) Mapping from b(x)to x=

b2(x).

Figure 2. A depiction of how the chess billiard map can be viewed as a

rotation of points along the boundary.

We quantify this rotation using the rotation number r, which can be thought of as the

average rotation per mapping over time. For example, in Figure 2b, since it takes two

mappings of bto complete a full rotation (as x=b2(x)), then, on average, each mapping

travels 1/2of the boundary. Hence, the rotation number is r=1

Now we discuss rational and irrational rotation, which correspond directly to the ra-

tionality or irrationality of r. When ris rational, as in Figure 2, we can see a periodic

trajectory—the same lines are traveled along again and again. However, in an irrational

rotation (i.e. ris irrational), a point can never be mapped to itself again in some integer

number of rotations, so we don’t expect any periodic trajectories. The chess billiard ﬂow

has been previously studied in depth for rational rotation, and we study irrational rotation.

We look at the wave problem known as the Poincaré problem (see Equation (3) in Section

2), which is a two-dimensional model that describes how an originally unmoving stable-

stratiﬁed ﬂuid can move and evolve. Speciﬁcally, the solution udescribes the behavior of

the waves formed, which follows the chess billiard mappings. The direct connection between

4 SALLY ZHU

the Poincaré problem and the chess billiard ﬂow has been shown and veriﬁed mathematically

and experimentally [2, 3].

We examine the diﬀerentiability of ufor diﬀerent types of trajectories of the billiard ﬂow.

Previously, Dyatlov et al [1] showed that when ris rational (i.e. it is a periodic trajectory),

then uis not smooth (i.e. uis not highly diﬀerentiable). This is because singularity points

(points where the derivative does not exist) form along the trajectory of the billiard ﬂow.

For example, in the illustrations by Dyatlov et al. [1] in Figure 3, distinct lines form in the

rational rotation in Figure 3a, where uis not smooth. We instead investigate diﬀerentiation

in cases of irrational rotation, in which case we don’t expect the formation of singularities

(since no obvious trajectory exists), and instead expect the waves to eventually smooth out.

For example, in the near-irrational rotation in Figure 3b, the solution seems much smoother.

(a) An illustration of rational rotation

with singularity points.

(b) An illustration of nearly irrational

rotation, which appears smooth.

Figure 3. Illustrations of rational vs. irrational rotations, from Dyatlov et

al. [1]. Note that we assume much stronger irrationality conditions than the

irrational rotation in panel (b).

We show that suﬃciently irrational rotation results in highly diﬀerentiable and smooth

solutions u.

This paper is organized as follows. In Section 2, we establish the necessary deﬁnitions and

prove supporting lemmas. In Section 3, we present the proof of our main result. Finally, in

Section 4, we present numerical simulations and conjecture an extension of our result.

2. Preliminaries

2.1. The Chess Billiard Map and Rotation in the Square. We more formally describe

the chess billiard mapping, as in Dyatlov et al [1]. Given a domain Ωand its boundary ∂Ω,

we write the slopes in terms of λ, as ρ=√1−λ2

λand −ρ=−√1−λ2

λ, following Dyatlov

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ONTHESMOOTHNESSANDREGULARITYOFTHECHESSBILLIARDFLOWANDTHEPOINCARÉPROBLEMSALLYZHUUnderthedirectionofZhenhaoLiGraduateStudentMassachusettsInstituteofTechnologyAbstract.ThePoincaréproblemisamodeloftwo-dimensionalinternalwavesinstable-stratieduid.Thechessbilliardow,avariationofatypicalbilliardow,driv...

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ON THE SMOOTHNESS AND REGULARITY OF THE CHESS BILLIARD FLOW AND THE POINCARÉ PROBLEM SALLY ZHU

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