# Assessing Annotation Accuracy in Ice Sheets Using Quantitative Metrics

###### Abstract

The increasing threat of sea level rise due to climate change necessitates a
deeper understanding of ice sheet structures. This study addresses the need
for accurate ice sheet data interpretation by introducing a suite of
quantitative metrics designed to validate ice sheet annotation techniques.
Focusing on both manual and automated methods, including ARESELP and its
modified version, MARESELP, we assess their accuracy against expert
annotations. Our methodology incorporates several computer vision metrics,
traditionally underutilized in glaciological research, to evaluate the
continuity and connectivity of ice layer annotations. The results demonstrate
that while manual annotations provide invaluable expert insights, automated
methods, particularly MARESELP, improve layer continuity and alignment with
expert labels.

1cm(-2cm,1cm)  This paper is accepted at the IGARSS 2024 - 2024 IEEE
International Geoscience and Remote Sensing Symposium

Index Terms—  Ice sheet annotation, quantitative metrics, automated annotation
techniques, ice sheet structure analysis

##  1 Introduction

The escalating sea level rise, propelled by the ongoing climatic changes,
underscores the vital need to understand the fundamental structure of ice
sheets [1]. Acquiring this understanding is essential for enhancing the
accuracy of future forecasts regarding the rise in sea levels, a topic of
great concern for both coastal communities and climate researchers [2].
However, extracting essential information from ice sheet data is a substantial
problem. The task is intricate and can be accomplished using several
methodologies, such as manual, semi-automated, and fully automated procedures.
Each of these solutions necessitates a substantial dedication of time and
expertise from experts who meticulously annotate and assess complex data sets
[3].

Considering the intricate nature and importance of these interpretations,
validating ice sheet annotation methods becomes a vital element of
glaciological research [4]. Errors or omissions in annotation might result in
significant inaccuracies in comprehending and predicting ice sheet dynamics.
In light of the crucial requirement for precision and dependability, this
study presents a collection of new quantitative measurements. These measures
aim to assess the accuracy of ice sheet annotation techniques thoroughly,
enhancing our comprehension of glacial dynamics and their impact on sea level
fluctuations. This study aims to leverage several metrics from the computer
vision domain that have received limited attention in the existing literature.
In addition, our work is centered around the utilization of automated
annotation approaches, specifically ARESELP [5] and a customized version of
ARESELP (e.g., MARESELP). These techniques are then contrasted with the manual
labeling method conducted by a domain expert.

The evaluation and validation of layer-tracking performance in ice sheets have
been subjects of significant research interest. A common approach to assessing
the performance of layer-tracking algorithms is the use of synthetic age-depth
profiles [6], which involves generating a synthetic age-depth relationship
using a one-dimensional Nye model. This model is used to intersect picked
isochrones with a known age-depth profile, thereby assigning an age to each
pick and propagating this age-depth relationship across the ice sheet. The
concept of isochrone connectivity is proposed by [7]. The metric assesses the
degree of continuity and connectivity between detected ice layers, providing a
quantitative measure of the uncertainty inherent in the tracking process. In
addition, the metric is invaluable in highlighting areas where the
interpretation consists of a high number of disconnected isochrones, which may
indicate sensitivity to low amplitude signal anomalies.

##  2 Material and Method

###  2.1 Radargram and Annotation Products

In this study, we harness the extensive data resources provided by the Centre
for Remote Sensing of Ice Sheets (CReSIS), accessing a publicly available
repository, to develop our annotation products. Our focus is concentrated on a
selected set of 100 radargrams, all sourced from various locations across
North Greenland. These radargrams, carefully chosen for their diverse
characteristics and representativeness, include notable sequences such as
20120330_03_019-028, 20120404_01_004-011, 20120507_07_003-015, 20120508_04
_002-018, 20120508_07_001-012, 20120508_07_015-023, 20120511_01_041-052,
20120511_01_061-067, and 20120516
_01_080-091. Utilizing these specific radargrams, we embark on a detailed
process of generating annotations for the ice sheets.

###  2.2 Layer-Tracking Performance Metrics

####  2.2.1 Isochrones connectivity [7]

This method meticulously evaluates both the connectivity and continuity of the
layers that have been identified. It achieves this by quantifying three key
aspects: the total number of picked layers ($\\#TL$), the number of layers
that are continuous and uninterrupted ($\\#CL$), and the number of layers that
exhibit discontinuities or breaks ($\\#DL$). An effective layer annotator is
characterized by its ability to maximize connectivity while minimizing
discontinuity, thereby ensuring a more accurate and cohesive representation of
the layers.

Require: $mask^{a}$, $mask^{gt}$, window_size

\- Initialize $dip\\_mask^{a}$ and $dip\\_mask^{gt}$ as zero arrays with
dimensions of $mask^{a}$ and $mask^{gt}$, respectively.

for each point in $mask^{a}$ and $mask^{gt}$

\- Select a window_size.

\- Compute transitions in the window.

\- Calculate $y$ and $x$ differences of transitions.

\- Compute angles using $arctan2$ of $y$ and $x$ differences.

\- Calculate average dip as mean of angles.

\- Assign average dip to the corresponding point in dip_results.

end for

\- Calculate Pearson correlation coefficient between $dip\\_mask^{a}$ and
$dip\\_mask^{gt}$.

Return Correlation coefficient $\rho$.

Algorithm 1 Dip Estimation and Comparison

####  2.2.2 Vision-based Metrics

  * •

Pixel accuracy
It quantitatively evaluates the accuracy of two binary masks by comparing them
pixel by pixel. The calculation is the sum of all matching pixels divided by
the total number of pixels in one of the masks. It is formally specified as
$Acc.=\frac{\sum_{i}^{N}(mask^{a}_{i}==mask^{gt}_{i})}{N}$, where
$mask^{a}_{i}$ and $mask^{gt}_{i}$ correspond to the values of the $i^{th}$
pixel in the respective masks (i.e., ARESELP mask and ground truth mask), $N$
is the total number pixels in the mask, and
$\sum_{i}^{N}(mask^{a}_{i}==mask^{gt}_{i})$ represents the sum of pixels where
the two masks have identical values (either both pixels are 1 or both are 0).

  * •

Pearson’s correlation of dip estimation
It measures the similarity between the dip datasets obtained from two binary
masks. The dip of each mask is estimated by calculating the dip angle for each
pixel within a specified window size, and averaging the angles derived from
transitions in the binary data [8]. Algorithm 1 shows a pseudocode of the
metric calculation.

  * •

Structural similarity index (SSIM) [9]
It assesses the similarity between two binary masks. It considers changes in
texture, providing a more perceptually relevant assessment of layer annotation
similarity compared to simpler metrics like mean squared error. Algorithm 2
shows a pseudocode for calculating SSIM.

Require: $mask^{a}$, $mask^{gt}$

\- Define parameters: window size, constants ($C_{1}$ and $C_{2}$) for
stabilizing division with weak denominators.

\- Initialize the SSIM map as a zero array with dimensions based on window
size.

for each overlapping window in $mask^{a}$ and $mask^{gt}$.

\- Extract corresponding windows from $mask^{a}$ and $mask^{gt}$.

\- Calculate mean, variance, and covariance for these windows.

\- Calculate $y$ and $x$ differences of transitions.

\- Compute angles using $arctan2$ of $y$ and $x$ differences.

\- Compute SSIM for the current window:

$\text{SSIM(window)}=\frac{(2\mu_{x}\mu_{y}+C_{1})(2\sigma_{xy}+C_{2})}{(\mu_{x}^{2}+\mu_{y}^{2}+C_{1})(\sigma_{x}^{2}+\sigma_{y}^{2}+C_{2})}$

\- Update SSIM map with computed SSIM value for the current window.

end for

\- Calculate the mean SSIM over the entire image for the final SSIM value.

Return Final SSIM value.

Algorithm 2 Structural Similarity Index Calculation

  * •

Recall IoU ($IoU_{r}$)
The Recall Intersection over Union (IoU) metric is a method used to evaluate
the accuracy of binary masks, specifically focusing on the overlap between a
predicted mask ($mask^{a}$) and a ground truth mask ($mask^{gt}$). It
calculates the ratio of the overlapping area (where both masks agree on
positive pixels) to the total area covered by the ground truth mask, providing
a measure of recall or how well the predicted mask captures the relevant areas
of the ground truth. Algorithm 3 shows a pseudocode implementation of this
metric.

Require: $mask^{a}$, $mask^{gt}$

\- Compute the overlap as the sum of element-wise logical AND between
$mask^{a}$ and $mask^{gt}$.

\- Compute the total number of positive pixels in $mask^{gt}$.

\- Calculate Recall IoU as the ratio of overlap to the total layers in
$mask^{gt}$.

$\text{Recall IoU}=\frac{\text{Overlap}}{\text{Total Layers in $mask^{gt}$}}$.

Return Recall IoU.

Algorithm 3 Recall Intersection over Union Calculation

  * •

Layer-by-layer Recall IoU ($IoU_{r}^{l}$)
It calculates the average recall for layer-by-layer comparison in binary
masks. It first computes the IoU for each pair of layers between two masks.
Then, it selects layer pairs with IoU scores above the average and calculates
the recall for these pairs, which measures the proportion of actual positive
samples (i.e., true layer matches) that are correctly identified. The average
recall across these selected layers provides a metric for the overall accuracy
of the layer identification in the masks. Algorithm 4 shows the outline for
calculating layer-by-layer recall IoU metric.

Require: iou_scores, $mask^{a}$, $mask^{gt}$

\- Initialize an empty list for recalls.

\- Compute average IoU from iou_scores.

\- Select layer pairs from iou_scores with IoU greater than or equal to the
average IoU.

for each selected layer pair ($i$,$j$) in selected_pairs

\- Extract corresponding layers from $mask^{a}$ and $mask^{gt}$

\- Compute recall for the layer pair

\- Append recall to recalls list

end for

\- Compute average recall from the recalls list

Return average recall

Algorithm 4 Layer-by-Layer Recall IoU Calculation

Table 1: The mean quantitative score
($\mathbin{\smash{\raisebox{1.50696pt}{$\underset{\raisebox{2.15277pt}{$\smash{-}$}}{\smash{+}}$}}}$
standard deviation) of all the layer annotation techniques, such as manual
approach by the expert, ARESELP, and MARESELP in terms of isochrones
connectivity.

Method | $\\#CL\uparrow$ | $\\#DL\downarrow$ | $\\#TL\uparrow$
---|---|---|---
Manual | 7.63$\mathbin{\smash{\raisebox{1.50696pt}{$\underset{\raisebox{2.15277pt}{$\smash{-}$}}{\smash{+}}$}}}$6.75 | 144.03$\mathbin{\smash{\raisebox{1.50696pt}{$\underset{\raisebox{2.15277pt}{$\smash{-}$}}{\smash{+}}$}}}$58.75 | 151.66$\mathbin{\smash{\raisebox{1.50696pt}{$\underset{\raisebox{2.15277pt}{$\smash{-}$}}{\smash{+}}$}}}$59.99
ARESELP [5] | 15.16$\mathbin{\smash{\raisebox{1.50696pt}{$\underset{\raisebox{2.15277pt}{$\smash{-}$}}{\smash{+}}$}}}$8.48 | 39.11$\mathbin{\smash{\raisebox{1.50696pt}{$\underset{\raisebox{2.15277pt}{$\smash{-}$}}{\smash{+}}$}}}$12.74 | 54.27$\mathbin{\smash{\raisebox{1.50696pt}{$\underset{\raisebox{2.15277pt}{$\smash{-}$}}{\smash{+}}$}}}$17.02
MARESELP | 21.89$\mathbin{\smash{\raisebox{1.50696pt}{$\underset{\raisebox{2.15277pt}{$\smash{-}$}}{\smash{+}}$}}}$9.55 | 56.03$\mathbin{\smash{\raisebox{1.50696pt}{$\underset{\raisebox{2.15277pt}{$\smash{-}$}}{\smash{+}}$}}}$20.60 | 77.92$\mathbin{\smash{\raisebox{1.50696pt}{$\underset{\raisebox{2.15277pt}{$\smash{-}$}}{\smash{+}}$}}}$25.57

Table 2: The average quantitative score ($\mathbin{\smash{\raisebox{1.50696pt}{$\underset{\raisebox{2.15277pt}{$\smash{-}$}}{\smash{+}}$}}}$ standard deviation) derived from all pairwise comparisons between the expert labels (ground truth) and the automatic annotation techniques, specifically ARESELP and MARESELP, using vision-based metrics. Method | $\rho\uparrow$ | $SSIM\uparrow$ | $Acc.\uparrow$ | $IoU_{r}\uparrow$ | $IoU^{l}_{r}\uparrow$
---|---|---|---|---|---
ARESELP [5] | 0.527$\mathbin{\smash{\raisebox{1.50696pt}{$\underset{\raisebox{2.15277pt}{$\smash{-}$}}{\smash{+}}$}}}$0.190 | 0.822$\mathbin{\smash{\raisebox{1.50696pt}{$\underset{\raisebox{2.15277pt}{$\smash{-}$}}{\smash{+}}$}}}$0.059 | 0.973$\mathbin{\smash{\raisebox{1.50696pt}{$\underset{\raisebox{2.15277pt}{$\smash{-}$}}{\smash{+}}$}}}$0.010 | 0.633$\mathbin{\smash{\raisebox{1.50696pt}{$\underset{\raisebox{2.15277pt}{$\smash{-}$}}{\smash{+}}$}}}$0.185 | 0.313$\mathbin{\smash{\raisebox{1.50696pt}{$\underset{\raisebox{2.15277pt}{$\smash{-}$}}{\smash{+}}$}}}$0.125
MARESELP | 0.547$\mathbin{\smash{\raisebox{1.50696pt}{$\underset{\raisebox{2.15277pt}{$\smash{-}$}}{\smash{+}}$}}}$0.184 | 0.827$\mathbin{\smash{\raisebox{1.50696pt}{$\underset{\raisebox{2.15277pt}{$\smash{-}$}}{\smash{+}}$}}}$0.056 | 0.974$\mathbin{\smash{\raisebox{1.50696pt}{$\underset{\raisebox{2.15277pt}{$\smash{-}$}}{\smash{+}}$}}}$0.009 | 0.692$\mathbin{\smash{\raisebox{1.50696pt}{$\underset{\raisebox{2.15277pt}{$\smash{-}$}}{\smash{+}}$}}}$0.154 | 0.325$\mathbin{\smash{\raisebox{1.50696pt}{$\underset{\raisebox{2.15277pt}{$\smash{-}$}}{\smash{+}}$}}}$0.120

##  3 Result and Discussion

The implementation of all metrics is readily accessible online to ensure
reproducibility and facilitate further research. Table 1 provides a
comprehensive comparison of different layer annotation techniques - manual
annotation by experts, ARESELP, and MARESELP - focusing on their performance
in terms of isochrone connectivity. The metrics used for this comparison
include the number of continuous layers ($\\#CL$), the number of broken layers
($\\#DL$), and the total number of layers ($\\#TL$), where a higher number of
continuous layers and a higher total number of layers are desirable, while a
lower number of broken layers is preferred. The manual approach, traditionally
considered the gold standard due to its reliance on expert interpretation,
shows a moderate number of continuous layers but a significantly high number
of broken layers, resulting in a total layer count ($\\#TL$) of
151.6$\mathbin{\smash{\raisebox{1.50696pt}{$\underset{\raisebox{2.15277pt}{$\smash{-}$}}{\smash{+}}$}}}$59.99.
This indicates the inherent challenges in manual annotation, where maintaining
continuity across layers can be difficult, leading to a higher incidence of
broken layers. In contrast, the ARESELP method shows a marked improvement in
the number of continuous layers, more than double that of the manual approach.
Notably, it also exhibits a substantial reduction in the number of broken
layers, suggesting that this automated technique is more effective in
maintaining layer continuity. The total number of layers identified by ARESELP
is lower than that identified by the manual method, which may indicate a more
selective layer identification process. MARESELP further advances these
improvements, registering the highest number of continuous layers and a
moderate number of broken layers, resulting in a total layer count ($\\#TL$)
of
77.92$\mathbin{\smash{\raisebox{1.50696pt}{$\underset{\raisebox{2.15277pt}{$\smash{-}$}}{\smash{+}}$}}}$
25.57. This suggests that MARESELP not only excels in identifying continuous
layers but also strikes a balance in total layer detection, possibly offering
a more nuanced and accurate representation of isochrones compared to the other
methods. In summary, while the manual approach provides a substantial number
of total layers, its high number of broken layers highlights the challenges of
manual interpretation. ARESELP and MARESELP, on the other hand, demonstrate
their strengths in automated layer annotation, particularly in maintaining
layer continuity, as evidenced by their higher $\\#CL$ and lower $\\#DL$
scores. This evolution from manual to automated techniques underscores the
potential of automatic approaches to enhance the accuracy and efficiency of
ice sheet annotation.

In addition, Table 2 compares two automated annotation techniques, ARESELP and
MARESELP, based on their performance in several vision-based metrics compared
to expert labels (ground truth). These metrics include the Pearson correlation
coefficient of the dip estimation ($\rho$), Structural Similarity Index
($SSIM$), Pixel Accuracy ($Acc.$), and Recall Intersection over Union for both
global ($IoU_{r}$) and layer-by-layer ($IoU_{r}^{l}$) comparisons. The results
demonstrate that while both ARESELP and MARESELP exhibit high accuracy and a
strong structural resemblance to expert annotations, MARESELP shows a
marginally better performance in aligning with expert labels, particularly in
terms of IoU metrics. This indicates that MARESELP may offer a more refined
and precise annotation than ARESELP, especially in complex layer-by-layer
assessments. The superiority of MARESELP in terms of IoU metrics is
particularly noteworthy, as it reflects a greater efficacy in capturing both
the general and detailed aspects of ice sheet layers as annotated by experts
However, annotation products from automatic approaches need to be further
assessed, particularly when dealing with ”hallucinated layers”- annotations
that do not exist in the underlying radar imagery.

##  4 Conclusion

This study presents various quantitative metrics to validate the performance
of ice sheet annotation techniques, specifically manual, ARESELP, and
MARESELP. Our analysis reveals that while manual annotation provides valuable
expert insight, it struggles with layer continuity, a challenge effectively
mitigated by automated methods. ARESELP shows marked improvement in
maintaining layer continuity, and MARESELP further excels by achieving the
highest number of continuous layers and a balanced total layer count. With
respect to vision-based metrics, MARESELP marginally outperforms ARESELP,
especially in IoU measures, indicating its closer alignment with expert
annotations. These findings highlight the potential of automated annotation
techniques in enhancing the accuracy and efficiency of ice sheet analysis,
which is crucial for understanding ice dynamics and their impact on sea level
changes in the context of climate research.

##  5 Acknowledgment

This work is funded by the National Science Foundation (Institute for
Harnessing Data and Model Revolution in the Polar Regions (iHARP) Award
#2118285).

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# The velocity field and Birkhoff-Rott integral for non-decaying, non-periodic
vortex sheets

David M. Ambrose  Department of Mathematics, Drexel University, Philadelphia,
PA 19104 USA<EMAIL_ADDRESS>
###### Abstract.

The Birkhoff-Rott integral expresses the fluid velocity on a vortex sheet.
This integral converges if certain quantities decay at horizontal infinity,
but can also be summed over periodic images in the horizontally periodic case.
However, non-decaying, non-periodic cases are also of interest, such as the
interaction of periodic wavetrains with non-commensurate periods (i.e.
spatially quasiperiodic solutions), or non-periodic disturbances to periodic
wavetrains. We therefore develop a more general single formula for the
Birkhoff-Rott integral, which unifies and extends the cases of decay and
periodicity. We verify that under some reasonable conditions this new version
of the Birkhoff-Rott integral is the restriction to the vortex sheet of an
incompressible, irrotational velocity field, with continuous normal component
but with a jump in tangential velocity across the vortex sheet. We give a
number of examples of non-decaying, non-periodic sheet positions and sheet
strengths for which our assumptions may be verified. While we develop this in
the case of two-dimensional fluids, the methodology applies equally well to
three-dimensional fluids.

##  1\. Introduction

A vortex sheet is a surface of discontinuity of velocity in a fluid; in an
incompressible fluid, the normal component of velocity across the sheet must
be continuous, but there can be a jump in the tangential component of
velocity. Even with the fluids being irrotational, this velocity
discontinuity, when taking the curl, will lead to vorticity concentrated on
the sheet. If we approach this from the other direction, then the vortex sheet
ansatz (in the two-dimensional case) is that a fluid has its vorticity,
$\omega,$ concentrated along a curve $\mathcal{C},$ parameterized as
$(x(\alpha),y(\alpha)).$ This vorticity has some amplitude, $\gamma(\alpha),$
known as the vortex sheet strength, which is multiplied by the Dirac mass
along the curve. That is,

$\omega=\gamma\delta_{\mathcal{C}}.$

This anszatz for the vorticity can then be used with the Biot-Savart law to
recover velocity from vorticity; this yields the fluid velocity $(u,v)$ at a
point $\mathbf{x}=(x,y)$ away from the curve $\mathcal{C},$

(1)
$(u,v)(\mathbf{x})=\frac{1}{2\pi}\int_{\mathbb{R}}\gamma(\alpha^{\prime})\frac{(-(y-y(\alpha^{\prime})),x-x(\alpha^{\prime}))}{(x-x(\alpha^{\prime}))^{2}+(y-y(\alpha^{\prime}))^{2}}\
\mathrm{d}\alpha^{\prime}.$

We define the Birkhoff-Rott integral, $\mathbf{W}=(W_{1},W_{2}),$ by taking
the point $\mathbf{x}=(x(\alpha),y(\alpha))$ for some value of $\alpha,$ and
interpreting the integral in the principal value sense:

(2)
$\mathbf{W}(\alpha)=\frac{1}{2\pi}\mathrm{PV}\int_{\mathbb{R}}\gamma(\alpha^{\prime})\frac{(-(y(\alpha)-y(\alpha^{\prime})),x(\alpha)-x(\alpha^{\prime}))}{((x(\alpha)-x(\alpha^{\prime}))^{2}+(y(\alpha)-y(\alpha^{\prime}))^{2}}\
\mathrm{d}\alpha^{\prime}.$

The Birkhoff-Rott integral was developed by Birkhoff and Rott in Birkhoff
(1962) and Rott (1956), resepectively, and a detailed development of the
Birkhoff-Rott integral for two-dimensional fluids may be found, for instance,
in the book of Saffman (1992), or in the book of Majda & Bertozzi (2002). Our
primary concern in the present work is the question of how to express the
velocity field associated to a vortex sheet given by a functions $\gamma$ and
a curve $\mathcal{C}$ for which (1) and (2) does not obviously make sense.
These integrals are used either in the spatially decaying case (in which the
curve $\mathcal{C}$ is asymptotic at positive and negative infinity to the
real line, and in which $\gamma$ decays at positive and negative infinity),
and in the spatially periodic case; however, there are other settings of
interest for vortex sheets, beyond the spatially decaying and spatially
periodic cases.

One clear application of developing a more general theory would be the ability
to study non-periodic perturbations of periodic wavetrains (and vice versa),
or more generally, to study solutions which have both periodic and decaying
elements. As an example of such a problem, there are known cases of the
existence of solitary water waves with capillary ripples, in which the
solution is neither decaying nor periodic (Beale (1991), Sun (1991)). The
development of a theory which requires neither decay nor periodicity could
allow quite general perturbations of traveling waves to be studied, such as
noisy perturbations in which the perturbation is essentially unstructured.

Considering structure other than periodicity, Wilkening and Zhao have carried
out detailed numerical studies of irrotational water waves (a special case of
the vortex sheet) in the spatially quasiperiodic case (Wilkening & Zhao
(2021a), Wilkening & Zhao (2021b), Wilkening & Zhao (2023a), Wilkening & Zhao
(2023b)). These works use the Dirichlet-to-Neumann operator (Craig & Sulem
(1993)) to describe the fluid velocity rather than the Birkhoff-Rott integral;
this is effective in the case that the free fluid surface remains a graph with
respect to the horizontal, or in the two-dimensional case when a conformal
mapping may be used. See also Dyachenko & Semenova (2023) for further work on
spatially quasiperiodic water waves. We seek instead to be able to formulate
such a problem using a parameterized curve and the Birkhoff-Rott integral, as
this generalizes quite naturally to the three-dimensional setting. Another
non-decaying, non-periodic study of water waves was by Alazard, Burq, and
Zuily, where the initial value problem for irrotational water waves was
studied with initial data which is uniformly locally bounded (Alazard et al.
(2016)). Again, the Dirichlet-to-Neumann operator was used, with a restriction
that the free fluid surface is a graph with respect to the horizontal.

The main thrust of the present work is to write new expressions for the
velocity field $(u,v)$ and the Birkhoff-Rott integral $\mathbf{W};$ beyond
simply rewriting these formulas, we will give conditions on the curve
$\mathcal{C}$ and on the vortex sheet strength $\gamma$ under which the new
integrals converge. We will then show that these conditions are satisfied in
the usual decaying and spatially periodic cases, and we will be able to see
that in these cases the new formulas agree with the familiar formulas.

Before proceeding, it is convenient to introduce some complex notation. If we
denote the Birkhoff-Rott integral as $\mathbf{W}=(W_{1},W_{2}),$ and if we
write $z=x+iy$ and $W=W_{1}+iW_{2},$ and we denote complex conjugation with
$*,$ i.e. $W^{*}=W_{1}-iW_{2},$ then we may write the Birkhoff-Rott integral
more succinctly as

(3) $W^{*}(\alpha)=\frac{1}{2\pi
i}\mathrm{PV}\int_{\mathbb{R}}\frac{\gamma(\alpha^{\prime})}{\xi(\alpha)-\xi(\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime}.$

Similarly, we may write the complexified version of (1) as

(4) $u-iv=\frac{1}{2\pi
i}\int_{\mathbb{R}}\frac{\gamma(\alpha^{\prime})}{z-\xi(\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime}.$

We define the spatially periodic case to be such that

$x(\alpha+2\pi)=x(\alpha)+2\pi,\qquad
y(\alpha+2\pi)=y(\alpha),\qquad\gamma(\alpha+2\pi)=\gamma(\alpha).$

In this case the Birkhoff-Rott integral (3) may be summed over periodic
images:

(5) $W^{*}(\alpha)=\frac{1}{2\pi i}\sum_{k\in\mathbb{Z}}\mathrm{PV}\int_{2\pi
k}^{2\pi
k+2\pi}\frac{\gamma(\alpha^{\prime})}{\xi(\alpha)-\xi(\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime}\\\ =\frac{1}{2\pi
i}\mathrm{PV}\int_{0}^{2\pi}\gamma(\alpha^{\prime})\sum_{k\in\mathbb{Z}}\frac{1}{\xi(\alpha)-\xi(\alpha^{\prime})-2\pi
k}\ \mathrm{d}\alpha^{\prime}.$

So far we have only used complex notation rather than complex analysis. Now,
however, by a complex analysis theorem of Mittag-Leffler (see Ablowitz & Fokas
(1997)), this sum may be evaluated in closed form. This gives the formula

(6) $W^{*}(\alpha)=\frac{1}{4\pi
i}\mathrm{PV}\int_{0}^{2\pi}\gamma(\alpha^{\prime})\mathrm{cot}\left(\frac{1}{2}(\xi(\alpha)-\xi(\alpha^{\prime}))\right)\
\mathrm{d}\alpha^{\prime}.$

We denote the circle, or the one-dimensional torus, as $\mathbb{T}=[0,2\pi]$
with periodic boundary conditions, so (6) holds for $\alpha\in\mathbb{T}.$ We
mention that this summation using the Mittag-Leffler formula must be performed
carefully, as the infinite sum in question does not converge absolutely. The
summation must be performed symmetrically in pairs about the
$\alpha=\alpha^{\prime}$ singularity, which is the same as saying the
principal value must be taken at plus and minus infinity. We will discuss such
principal values at infinity more below. Even though these expressions exist
in the periodic case, a goal of the present work is to be able to formulate
the velocity integral and the Birkhoff-Rott integral in a way that does not
require special treatment for spatially periodic vortex sheets. The formulas
we will develop will unify the decaying and periodic cases, and apply more
generally. These integrals will therefore converge in the periodic case
without needing to carry out the above summation.

The main idea we utilize in generalizing the classical Birkhoff-Rott integral
to more general (non-decaying, non-periodic) vortex sheets comes from the work
of P. Serfati. In Serfati (1995), he gave a reformulation of the two-
dimensional incompressible Euler equations allowing for velocity and vorticity
in the space $L^{\infty}(\mathbb{R}^{2}),$ and a proof of existence of the
solutions of the resulting equations. The reformulation was in rewriting the
Biot-Savart law; specifically, a cutoff function was introduced to separate
out the near-field and far-field contributions of the vorticity to the
velocity. In the far-field, the Biot-Savart integral is non-singular, and
greater decay can be realized by integrating by parts and thus differentiating
the integral kernel. Ideas inspired by this work of Serfati have subsequently
been used by the author and collaborators in works such as Ambrose et al.
(2015) and Ambrose et al. (2023). While the current setting is more
specialized because of the specific form of vorticity arising in a vortex
sheet, we again introduce a cutoff function to separate the velocity integral
into near-field and far-field pieces, and we integrate by parts in the far-
field integral to gain greater decay from the kernel.

After introducing this cutoff function and integrating by parts, we give
fairly general conditions under which the new velocity integrals converge. We
then demonstrate several desirable properties for the new velocity
formulation, such as that the velocity is independent of the particular choice
of the cutoff function. This further includes showing that the velocity field
is divergence-free everywhere and is irrotational away from the vortex sheet,
and that the normal component of velocity is continuous across the sheet. We
also show that there is the expected jump in tangential velocity across the
sheet, and take the limit of the velocity at vertical infinity. In this way we
verify that the velocity field is indeed the velocity field for a vortex
sheet. We also will be able to see that in the decaying case and the
horizontally periodic case, our conditions for the new velocity integrals to
converge do in fact hold, so that we may conclude that the new formulation
does properly generalize the classical setting.

The plan of the paper is as follows. In Section 2.1 we give four motivating
examples of positions of the vortex sheet and vortex sheet strengths which are
not treated by the typical Birkhoff-Rott integral. In Section 2.2 we give an
assumption on $\gamma$ and an assumption on $\mathcal{C}$ which will allow us
to define our new velocity integral. We also verify that these assumptions
hold in the decaying and periodic cases. In Section 3.1 we introduce our
cutoff function and integrate by parts, finding our new expression for the
velocity. In Section 3.2 we then give the corresponding new formula for the
Birkhoff-Rott integral. We show in Section 3.3 that under the assumptions, the
new velocity and new Birkhoff-Rott integral are well-defined. We then state an
additional assumption on approximability of $\xi$ and $\gamma$ in Section 3.4.
We state and prove the main theorem, that this velocity is the velocity
associated to a vortex sheet, in Section 3.5. We discuss the jump in normal
and tangential components of the velocity in Section 3.6, and we take the
limit of the velocity field at vertical infinity in Section 3.7. We revisit
the examples of Section 2.1 in Section 4, verifying that the various
assumptions hold for these four examples. We close with some remarks in
Section 5.

##  2\. Examples and assumptions

We now will begin the work of deriving a new expression for the velocity (1)
and a new expression for the Birkhoff-Rott integral. These new velocity
integrals will agree with the previous forms in both the decaying case and the
periodic case, but will be single formulas (one formula for the velocity in
the plane and one formula for the Birkhoff-Rott integral) which apply not only
in these two cases, but also to non-decaying, non-periodic cases under some
reasonable conditions. We wish to first give some illustrative examples of
such non-decaying, non-periodic curves and vortex sheet strengths to which our
new formulas for the velocity will apply.

###  2.1. Four illustrative examples

It will be helpful to have in mind a few examples of non-decaying, non-
periodic vortex sheets for which the new framework will be useful. We give
here four examples of curves $\mathcal{C}$ and vortex sheet strengths
$\gamma.$ We will in subsequent sections be making assumptions about
$\mathcal{C}$ and $\gamma,$ and in Section 4 below we will verify that these
four examples satisfy all of the assumptions to be made.

(a) A mixed case.:

As a simple example we could consider a mixed decaying and periodic case. For
instance, the curve $\mathcal{C}$ could be taken to be periodic, say
$\xi(\alpha)=\alpha+i\sin(\alpha),$ while the vortex sheet strength could be
decaying, say $\gamma(\alpha)=\left(\frac{1}{1+\alpha^{2}}\right)^{2/5}.$

(b) A quasiperiodic case.:

As a simple spatially quasiperiodic example, for some amplitude parameter
$\mu>0,$ we could take the curve $\mathcal{C}$ to be specified by its
parameterization

$\xi(\alpha)=(\alpha+\mu\sin(\alpha)+\mu\sin(\pi\alpha))+i\mu\sin\left(\sqrt{2}\alpha\right).$

Here, the curve is non-self-intersecting if the parameter $\mu$ is
sufficiently small. The vortex sheet strength could be given by, for instance,
$\gamma(\alpha)=\cos\left(\left(1+4\pi-\sqrt{2}\right)\alpha\right).$

(c) A uniformly local case.:

As a non-periodic, non-decaying example which also does not have the above
quasiperiodic structure, we could take the curve $\mathcal{C}$ to have
parameterization

$\xi(\alpha)=\alpha+i\sin\left(\left(\frac{\alpha^{4}}{1+\alpha^{2}}\right)^{1/4}\right).$

We could take any appropriate $\gamma,$ for example
$\gamma(\alpha)=\frac{\alpha^{2}}{1+\alpha^{2}}.$

(d) A bore.:

We let the curve $\mathcal{C}$ be described by its parameterization
$\xi(\alpha)=\alpha+i\tan^{-1}(\alpha),$ so that the curve $\mathcal{C}$ has
different horizontal limits on the left and on the right. We could take the
vortex sheet strength to be given by $\gamma(\alpha)=1+\sin(\alpha),$ for
instance.

###  2.2. Assumptions on $\gamma$ and $\mathcal{C}$

We will now state an assumption on $\gamma$ and an assumption on the curve
$\mathcal{C}$ which will allow the generalized formulas to be developed.

######  Assumption A.

We say that $\gamma$ satisfies Assumption A if there exists a constant $c$ and
an antiderivative $\Gamma$ of $\gamma-c$ such that for some $\varepsilon>0,$
there exist constants $C>0$ and $0\leq\beta<1$ such that for all $\alpha$ with
$|\alpha|>\varepsilon,$

(7) $\frac{|\Gamma(\alpha)|}{|\alpha|^{\beta}}\leq C.$

Assumption A states that the antiderivative of $\gamma$ grows sublinearly,
although a constant may be subtracted first. The periodic case demonstrates
the necessity of subtracting a constant; constant functions have linearly
growing antiderivative, so if we want sublinear growth in the antiderivative
we must eliminate this constant first. The need for a sublinearly growing
antiderivative will be seen below in (13), our new expression for the velocity
induced by a vortex sheet.

We next see that the constant, $c,$ to be subtracted from $\gamma$ is uniquely
determined.

######  Lemma 1.

Let $\gamma$ satisfy Assumption A. The constant, $c,$ associated to $\gamma$
and $\Gamma$ is uniquely defined. We denote the constant as $c=c[\gamma].$

The proof of this lemma is straightforward, since attempting to use a
different choice of constant would cause linear growth in the antiderivative.

We next see that Assumption A is satisfied in both the decaying and periodic
cases.

######  Lemma 2.

If $\gamma\in L^{2}(\mathbb{R})$ or $\gamma\in L^{2}(\mathbb{T}),$ then
Assumption A is satisfied.

We prove this lemma in the appendix.

We will be manipulating the classical Birkhoff-Rott integral to give our new
version of it, and in so doing, as suggested in the statement of Assumption A,
we will be adding and subtracting $c[\gamma];$ this means one term we must
deal with is

$\frac{1}{2\pi
i}\mathrm{PV}\int_{-\infty}^{\infty}\frac{c[\gamma]}{\xi(\alpha)-\xi(\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime}.$

We now state an assumption; when we make this assumption, we will be assuming
that this integral makes sense. We verify in Lemma 3 that this assumption
holds in the usual decaying or periodic cases.

######  Assumption B.

We say that the curve $\mathcal{C}$ satisfies Assumption B if the following
hold:

  1. (1)

for all $\alpha\in\mathbb{R},$ the integral

(8)
$\mathrm{PV}\int_{-\infty}^{\infty}\frac{1}{\xi(\alpha)-\xi(\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime}$

converges in the principal value sense, with the principal value being taken
both at $\alpha^{\prime}=\alpha$ and at $\alpha^{\prime}=\pm\infty,$

  2. (2)

for all $z\in\mathbb{C}$ such that $\mathrm{dist}(z,\mathcal{C})>0,$ the
principal value integral

(9)
$\mathrm{PV}_{\infty}\int_{-\infty}^{\infty}\frac{1}{z-\xi(\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime}$

converges.

######  Remark 1.

We clarify what we mean by the principal value at infinity. If we are only
taking the principal value at infinity, when we write

$\mathrm{PV}_{\infty}\int_{-\infty}^{\infty}g(s)\ \mathrm{d}s,$

this indicates that there exists $a\in\mathbb{R}$ and $M_{n}\nearrow\infty$
such that

(10) $\lim_{n\rightarrow\infty}\int_{a-M_{n}}^{a+M_{n}}g(s)\ \mathrm{d}s$

exists, and the principal value integral is equal to the value of this limit;
furthermore, if there are multiple choices of $a$ and the sequence $M_{n},$
then the principal value at infinity is only well-defined if all such choices
will give the same limit in (10). We will sometimes identify the particular
$a\in\mathbb{R}$ by writing, for example,

$\mathrm{PV}_{\infty}\left(\int_{-\infty}^{a}g(s)\
\mathrm{d}s+\int_{a}^{\infty}g(s)\ \mathrm{d}s\right).$

We also could exclude a portion of the domain and write

$\mathrm{PV}\left(\int_{-\infty}^{a}g(s)\ \mathrm{d}s+\int_{b}^{\infty}g(s)\
\mathrm{d}s\right).$

This indicates that we take the similar limit,

$\lim_{n\rightarrow\infty}\int_{a-M_{n}}^{a}g(s)\
\mathrm{d}s+\int_{b}^{b+M_{n}}g(s)\ \mathrm{d}s.$

If we take the principal value at both a finite value and also at infinity, we
simply write $\mathrm{PV}$ (as in (8)).

It is frequently the case when considering the curve $\mathcal{C}$ that we
need it to be non-self-intersecting. We typically accomplish this by imposing
the chord-arc condition, as in Wu (1997). This condition is

(11) $\exists K>0\ \mathrm{such}\ \mathrm{that}\
\forall\alpha,\alpha^{\prime},\quad\left|\frac{\xi(\alpha)-\xi(\alpha^{\prime})}{\alpha-\alpha^{\prime}}\right|\geq
K.$

######  Lemma 3.

If $\mathcal{C}$ satisfies the chord-arc condition (11) and
$\xi(\alpha)-\alpha$ is twice continuously differentiable and bounded, and is
either in $L^{2}(\mathbb{R})$ or $L^{2}(\mathbb{T}),$ and if there exists
$M>0$ such that for all $\alpha\in\mathbb{R},$
$\frac{1}{M}\leq|\xi_{\alpha}(\alpha)|\leq M,$ then Assumption B is satisfied.

Again, we postpone the proof of this lemma until the appendix.

##  3\. The new expression for the velocity

In this section we will form our new expression for the velocity field
associated to a vortex sheet, and we will also give our new expression for the
Birkhoff-Rott integral. We will then show that, under some assumptions, this
velocity is divergence-free, is curl-free in the interior of the two fluid
regions, and satisfies the desired jump conditions at the fluid interface.
These formulas will use a cutoff function which we will introduce, and we will
also show that under the same assumptions, the velocity is independent of the
choice of cutoff function.

###  3.1. Rewriting the velocity

We introduce a cutoff function, $a:\mathbb{R}\rightarrow\mathbb{R}.$ We will
take $a$ to be of class $C^{\infty},$ with $a(s)=1$ for $|s|\leq 1$ and
$a(s)=0$ for $|s|\geq 2.$ On the interval $s\in[-2,-1]$ we take $a$ to be
monotone increasing, and on the interval $s\in[1,2]$ we take $a$ to be
monotone decreasing.

We assume Assumption A and Assumption B. With $c[\gamma]$ as in Assumption A,
we denote $\tilde{\gamma}=\gamma-c[\gamma].$ Then, we first decompose the
velocity (4) for $z\notin\mathcal{C}$ as

$(u-iv)(z)=\frac{1}{2\pi
i}\mathrm{PV}_{\infty}\int_{-\infty}^{\infty}\frac{c[\gamma]}{z-\xi(\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime}+\frac{1}{2\pi}\mathrm{PV}_{\infty}\int_{-\infty}^{\infty}\frac{\tilde{\gamma}(\alpha^{\prime})}{z-\xi(\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime}.$

We have written these as principal value integrals at infinity in case this is
necessary for convergence (as in the spatially periodic case, for example). We
next introduce the cutoff function into the second integral on the right-hand
side:

(12) $(u-iv)(z)=\frac{1}{2\pi
i}\mathrm{PV}_{\infty}\int_{-\infty}^{\infty}\frac{c[\gamma]}{z-\xi(\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime}+\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{a(|z-\xi(\alpha^{\prime})|^{2})\tilde{\gamma}(\alpha^{\prime})}{z-\xi(\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime}\\\
+\frac{1}{2\pi}\mathrm{PV}_{\infty}\int_{-\infty}^{\infty}\frac{(1-a(|z-\xi(\alpha^{\prime})|^{2}))\tilde{\gamma}(\alpha^{\prime})}{z-\xi(\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime}.$

Notice that the second integral on the right-hand side is no longer a
principal value, as the cutoff function localizes the integral. As we
discussed in the introduction, the point of introducing the cutoff function is
to be able to realize greater decay from the kernel when integrating by parts
in the far-field piece. We therefore integrate by parts in the third integral
on the right-hand side, using the identity $\Gamma_{\alpha}=\tilde{\gamma},$
finding

(13) $(u-iv)(z)=\frac{1}{2\pi
i}\mathrm{PV}_{\infty}\int_{-\infty}^{\infty}\frac{c[\gamma]}{z-\xi(\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime}+\frac{1}{2\pi
i}\int_{-\infty}^{\infty}\frac{a(|z-\xi(\alpha^{\prime})|^{2})\tilde{\gamma}(\alpha^{\prime})}{z-\xi(\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime}\\\ -\frac{1}{2\pi
i}\int_{-\infty}^{\infty}\frac{(a^{\prime}(|z-\xi(\alpha^{\prime})|^{2}))\left(\mathrm{Re}(\xi_{\alpha}(\alpha^{\prime})(\overline{z-\xi(\alpha^{\prime})}))\right)\Gamma(\alpha^{\prime})}{z-\xi(\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime}.\\\ -\frac{1}{2\pi
i}\int_{-\infty}^{\infty}\frac{(1-a(|z-\xi(\alpha^{\prime})|^{2})\xi_{\alpha}(\alpha^{\prime})\Gamma(\alpha^{\prime})}{(z-\xi(\alpha^{\prime}))^{2}}\
\mathrm{d}\alpha^{\prime}.$

Notice that the final two integrals on the right-hand side are again no longer
principal value integrals; the first of these has now been localized, while
the final integral converges without the principal value. We introduce further
notation following this decomposition,

$u-iv=I_{0}+I_{1}+I_{2}+I_{3}.$

###  3.2. The new expression for the Birkhoff-Rott integral

By replacing $z$ in the velocity integrals with a point $\xi(\alpha)$ on
$\mathcal{C},$ we define our extension of the Birkhoff-Rott integral to be

(14) $W=W_{0}+W_{1}+W_{2}+W_{3},$

where each of these integrals are defined as

$W_{0}^{*}=\frac{1}{2\pi
i}\mathrm{PV}\int_{-\infty}^{\infty}\frac{c[\gamma]}{\xi(\alpha)-\xi(\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime},$ $W_{1}^{*}=\frac{1}{2\pi
i}\int_{-\infty}^{\infty}\frac{a(|\xi(\alpha)-\xi(\alpha^{\prime})|^{2})\tilde{\gamma}(\alpha^{\prime})}{\xi(\alpha)-\xi(\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime},$ $W_{2}^{*}=-\frac{1}{2\pi
i}\int_{-\infty}^{\infty}\frac{(a^{\prime}(|\xi(\alpha)-\xi(\alpha^{\prime})|^{2}))\left(\mathrm{Re}(\xi_{\alpha}(\alpha^{\prime})(\overline{\xi(\alpha)-\xi(\alpha^{\prime})}))\right)\Gamma(\alpha^{\prime})}{\xi(\alpha)-\xi(\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime},$ $W_{3}^{*}=-\frac{1}{2\pi
i}\int_{-\infty}^{\infty}\frac{(1-a(|\xi(\alpha)-\xi(\alpha^{\prime})|^{2})\xi_{\alpha}(\alpha^{\prime})\Gamma(\alpha^{\prime})}{(\xi(\alpha)-\xi(\alpha^{\prime}))^{2}}\
\mathrm{d}\alpha^{\prime}.$

###  3.3. The new velocity is well-defined

Under some assumptions, we can ensures that the new expressions for the
velocity integral and the Birkhoff-Rott integral are well-defined. We now
detail the assumptions, and then we discuss how this gives convergence of the
relevant integrals.

We take $\gamma$ to be locally $L^{2}$ integrable, and we take the curve
$\mathcal{C}$ to have twice continuously differentiable parameterization
$\xi.$ We assume Assumption A and we either assume Assumption B or that
$c[\gamma]=0.$ We assume that $\xi$ satisfies the chord-arc condition (11). We
take the parameterization $\xi$ to be such that there exists $M>0$ such that
for all $\alpha\in\mathbb{R},$ $\frac{1}{M}\leq|\xi_{\alpha}(\alpha)|\leq M.$
Finally, we assume that the function $\xi(\alpha)-\alpha$ may be decomposed as
$f_{1}+f_{2},$ with $f_{1}\in L^{2}(\mathbb{R})$ and $f_{2}\in
L^{\infty}(\mathbb{R}).$

We mention the reason that we take $\xi(\alpha)-\alpha$ to be able to be
written as the sum of a function in $L^{2}$ plus a bounded function. It is
because either of these assumption on their own is helpful; in the decaying
case we have $\xi(\alpha)-\alpha$ in $L^{2},$ and the velocity is well-defined
in this case. In the periodic case we typically have enough regularity that
$\xi(\alpha)-\alpha$ is bounded, and the velocity is well-defined in this
case. We can work with either of these, and to be more general we make the
assumption that there is a part, $f_{1},$ which is in $L^{2},$ and a part,
$f_{2},$ which is bounded.

We now discuss how these assumptions give convergence of the integrals. By
Assumption A, the constant $c[\gamma]$ exists. Either $c[\gamma]=0,$ in which
case $I_{0}$ and $W_{0}$ exist, or we may rely upon Assumption B which tells
us that $I_{0}$ and $W_{0}$ exist.

We next consider $I_{1}$ and $I_{2}.$ By the chord-arc condition (11) we see
that $|z(\alpha)|\rightarrow\infty$ as $\alpha\rightarrow\pm\infty.$ With the
presence of the cutoff function in $I_{1}$ and $I_{2},$ we infer that for
fixed $z,$ these could be written as integrals over only a finite interval of
values of $\alpha.$ Because the distance from $z$ to $\mathcal{C}$ is non-
zero, the denominators in these integrals are not singular. The assumption
that $\gamma$ is locally $L^{2}$ integrable is now enough to ensure that
$I_{1}$ and $I_{2}$ converge.

For $I_{3},$ we add and subtract to see

$\frac{1}{z-\xi(\alpha^{\prime})}=\frac{1}{z-\alpha^{\prime}}+\frac{\xi(\alpha^{\prime})-\alpha^{\prime}}{(z-\alpha^{\prime})(z-\xi(\alpha^{\prime}))}.$

If the final numerator is bounded, then since $z$ is a positive distance from
$\mathcal{C},$ this behaves like $\frac{1}{\alpha^{\prime}}$ as
$|\alpha^{\prime}|\rightarrow\infty.$ Together with the sublinear growth of
$\Gamma$ guaranteed by Assumption A, this is enough to ensure that this
integral converges. If instead the final numerator is in $L^{2}(\mathbb{R}),$
then we may instead use that the final fraction is a product of two $L^{2}$
functions and is thus integrable. (Note that we can without loss of generality
assume $z\neq\alpha^{\prime}$ for any $\alpha^{\prime}$ in the relevant domain
of integration; this is because if we did have $z=\alpha^{\prime}$ then we
could decompose the integral into a piece with a finite domain of integration
and a new improper integral excluding such value of $\alpha^{\prime}.$ The
integral with finite domain of integration then becomes similar to $I_{2}.$)

For $W_{1},$ we consider the decomposition

$\frac{1}{\xi(\alpha)-\xi(\alpha^{\prime})}=\frac{1}{\xi_{\alpha}(\alpha^{\prime})(\alpha-\alpha^{\prime})}+\left[\frac{1}{\xi(\alpha)-\xi(\alpha^{\prime})}-\frac{1}{\xi_{\alpha}(\alpha^{\prime})(\alpha-\alpha^{\prime})}\right].$

The first term on the right-hand side gives rise to a localized Hilbert
transform which converges since we have assumed $\gamma$ is locally $L^{2}$
integrable. The second term on the right-hand side (in brackets) is
expressible in terms of divided differences, as in the proof of Lemma 3 (see
Appendix A). This term in non-singular (i.e. the singularity has been
subtracted), and the integral is finite.

For $W_{2}$ and $W_{3},$ we use the chord-arc condition (11), since

$\left|\frac{1}{\xi(\alpha)-\xi(\alpha^{\prime})}\right|=\left|\frac{\alpha-\alpha^{\prime}}{\xi(\alpha)-\xi(\alpha^{\prime})}\right|\left|\frac{1}{\alpha-\alpha^{\prime}}\right|\leq\frac{1}{K}\left|\frac{1}{\alpha-\alpha^{\prime}}\right|.$

These integrals are non-singular because of the presence of the cutoff
function. Specifically, these integrals are only supported at values of
$\alpha^{\prime}$ such that $|\xi(\alpha)-\xi(\alpha^{\prime})|\geq 1,$ and
thus for any given $\alpha$ there is a neighborhood of $\alpha$ for which the
integrand is zero. Now, because Assumption A implies that $\Gamma$ grows
sublinearly at positive and negative infinity, these integrals can be seen to
converge.

###  3.4. Approximation by decaying vortex sheets and uniform convergence on
compact sets

We next wish to conclude that the velocity field given by (13) is indeed the
velocity field for a vortex sheet. However, it is difficult to see directly
that the divergence and curl of the velocity in this form are zero in the
interior of the fluid regions. Instead, we will use an approximation scheme,
then compute the divergence and curl for the approximate solutions, study
convergence of the approximations, and then take the limit of the divergence
and curl in the sense of distributions. We assume that the curve $\xi$ and the
modified vortex sheet strength $\tilde{\gamma}$ can be well-approximated by
decaying curves $\xi^{n}$ and and decaying functions $\tilde{\gamma}^{n}$ in a
certain sense.

Before discussing this approximation scheme, we make some remarks on the
parameterization $\xi$ of the curve $\mathcal{C}.$ First, we have assumed that
the parameterization is such that $|\xi_{\alpha}|$ is bounded above and below
(away from zero); it is always possible to find such a parameterization, for
instance by parameterizing by arclength. Next, the chord-arc condition (11)
implies that $|\xi(\alpha)|\rightarrow\infty$ as $|\alpha|\rightarrow\infty,$
and $\xi$ and $\alpha$ go to infinity at comparable rates. As a consequence of
these facts, for any compact set, $K_{0},$ for large enough $|\alpha|,$ we
have that $\xi(\alpha)\notin K_{0}.$ These are some features which we seek to
maintain in making an approximation by decaying vortex sheets. Furthermore, we
wish to have the property that for some interval of values
$[\alpha_{-n},\alpha_{n}],$ the approximation $\xi^{n}$ and the original
parameterization $\xi$ will agree, with $\alpha_{-n}\rightarrow-\infty$ and
$\alpha_{n}\rightarrow\infty$ as $n\rightarrow\infty.$

Specifying such an approximation is complicated in general, as the interface
may have complicated overturning features. The approximation of $\mathcal{C}$
by decaying vortex sheets is most easily achievable in the graph case, when
$\xi(\alpha)=\alpha+i\eta(\alpha).$ For any $n\in\mathbb{N},$ we introduce a
cutoff function $\phi_{n}.$ This satisfies $\phi_{n}(\alpha)=1$ for
$\alpha\in[-n,n],$ and $\phi_{n}(\alpha)=0$ for $|\alpha|>n+1.$ Otherwise,
$\phi_{n}$ is smooth and monotone. Then we let
$\xi^{n}(\alpha)=\alpha+i\eta(\alpha)\phi_{n}(\alpha)=\alpha+(\xi(\alpha)-\alpha))\phi_{n}(\alpha).$
We may take $\alpha_{\pm n}=\pm n,$ then. We also need to discuss the
approximate vortex sheet strength, and we let $\tilde{\gamma}^{n}$ be given by
$\tilde{\gamma}^{n}(\alpha)=\tilde{\gamma}(\alpha)\phi_{n}(\alpha).$

In general, for $n\in\mathbb{N}$ we let

$\alpha_{n}=\min\\{\alpha\in\mathbb{R}:\mathrm{Re}(z(\alpha))\geq
n\\},\qquad\alpha_{-n}=\max\\{\alpha\in\mathbb{R}:\mathrm{Re}(z(\alpha))\leq-n\\}.$

Then for $\alpha\in[\alpha_{-n},\alpha_{n}],$ we define
$\xi^{n}(\alpha)=\xi(\alpha).$ The approximate curve can then be further
specified by smoothly extending this curve to the remaining values of
$\alpha\in\mathbb{R},$ such that the curve decays at horizontal infinity.

Beyond needing to define such an approximation, we also need convergence of a
certain integral. We reiterate our assumptions on the approximation sequence
and state the assumption on the convergence of the integral in the following
assumption.

######  Assumption C.

We say that the curve $\mathcal{C}$ and vortex sheet strength $\gamma$ satisfy
Assumption C if there exist sequences $\alpha_{n},$ $\alpha_{-n},$ $\xi^{n},$
and $\tilde{\gamma}^{n}$ such that

  * •

$\alpha_{n}\rightarrow\infty$ and $\alpha_{-n}\rightarrow-\infty$ as
$n\rightarrow\infty,$

  * •

for all $n\in\mathbb{N},$ for all $\alpha\in[\alpha_{-n},\alpha_{n}],$
$\xi^{n}(\alpha)=\xi(\alpha)$ and
$\tilde{\gamma}^{n}(\alpha)=\tilde{\gamma}(\alpha),$

  * •

for all $n\in\mathbb{N},$ $\xi^{n}(\alpha)-\alpha\in L^{2}(\mathbb{R}),$
$\tilde{\gamma}^{n}\in L^{2}(\mathbb{R}),$

  * •

for all $n\in\mathbb{N},$ $\xi^{n}$ is twice continuously differentiable,

  * •

for all $n\in\mathbb{N},$ $\xi^{n}$ satisfies the chord-arc assumption (11),

  * •

for any compact set $K_{0}\subset\mathbb{C},$ there exists $N\in\mathbb{N}$
and $A_{\pm}\in\mathbb{R}$ such that for all $n\geq N,$ for all
$\alpha\notin[A_{-},A_{+}],$ $\xi^{n}(\alpha)\notin K_{0},$ and

  * •

letting
$\Gamma^{n}(\alpha)=\int_{0}^{\alpha}\tilde{\gamma}^{n}(\alpha^{\prime})\
\mathrm{d}\alpha^{\prime},$ and
$\Omega_{n}=(-\infty,\alpha_{-n}]\cup[\alpha_{n},\infty),$ for any compact set
$K_{0},$ the sequence of integrals

(15)
$\int_{\Omega_{n}}\frac{|\xi^{n}_{\alpha}(\alpha^{\prime})||\Gamma^{n}(\alpha^{\prime})|}{|z-\xi^{n}(\alpha^{\prime})|^{2}}\
\mathrm{d}\alpha^{\prime}$

converges to zero as $n$ goes to infinity, uniformly with respect to $z\in
K_{0}.$

Note that this assumption appears to have many facets, but the truth is
simpler than the assumption may make it appear. For each of our examples
listed in Section 2.1, we will be verifying that Assumption C holds in Section
4 below. All of the items in the assumption are consequences of cutting off
non-decaying functions by multiplying by a function with compact support. We
leave the situation as general as it is given in Assumption C so as to not
limit ourselves unnecessarily.

###  3.5. The main result

We now show that under our assumptions, the new expression for the velocity
field is indeed the velocity field associated to a vortex sheet in an
incompressible flow.

######  Theorem 1.

If Assumption A and Assumption C are satisfied, and if either Assumption B is
satisfied or $c[\gamma]=0,$ and if the parameterization $\xi$ satisfies the
following conditions:

  * •

$\xi$ satisfies the chord-arc condition (11),

  * •

there exists $M>0$ such that for all $\alpha\in\mathbb{R},$
$\frac{1}{M}\leq|\xi_{\alpha}(\alpha)|\leq M,$ and

  * •

the function $\xi(\alpha)-\alpha$ may be decomposed as $f_{1}+f_{2}$ with
$f_{1}\in L^{2}(\mathbb{R})$ and $f_{2}\in L^{\infty}(\mathbb{R}),$

then the velocity field as defined in (13) is incompressible and has vorticity
equal to $\gamma\delta_{C}.$ Furthermore, this velocity is independent of the
choice of cutoff function.

###### Proof.

Given the family of curves $\xi^{n}$ we define an approximate velocity
$(u^{n},v^{n}):$

(16) $u^{n}-iv^{n}=\frac{1}{2\pi
i}\mathrm{PV}_{\infty}\int_{-\infty}^{\infty}\frac{c[\gamma]}{z-\xi(\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime}\\\ +\frac{1}{2\pi
i}\int_{-\infty}^{\infty}\frac{a(|z-\xi^{n}(\alpha^{\prime})|^{2})\tilde{\gamma}^{n}(\alpha^{\prime})}{z-\xi^{n}(\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime}\\\ -\frac{1}{2\pi
i}\int_{-\infty}^{\infty}\frac{(a^{\prime}(|z-\xi^{n}(\alpha^{\prime})|^{2}))\left(\mathrm{Re}(\xi^{n}_{\alpha}(\alpha^{\prime})(\overline{z-\xi^{n}(\alpha^{\prime})}))\right)\Gamma^{n}(\alpha^{\prime})}{z-\xi^{n}(\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime}.\\\ -\frac{1}{2\pi
i}\int_{-\infty}^{\infty}\frac{(1-a(|z-\xi^{n}(\alpha^{\prime})|^{2})\xi^{n}_{\alpha}(\alpha^{\prime})\Gamma^{n}(\alpha^{\prime})}{(z-\xi^{n}(\alpha^{\prime}))^{2}}\
\mathrm{d}\alpha^{\prime}.$

We again decompose this, as

$u^{n}-iv^{n}=I_{0}+I_{1}^{n}+I_{2}^{n}+I_{3}^{n}.$

Note that $I_{0}$ does not have a superscript here because it is unmodified
from the corresponding expression in (13).

We will study the convergence of $(u^{n},v^{n})$ to $(u,v),$ but first we want
to consider $(u^{n},v^{n})$ in its own right. Considering for the moment only
$I_{2}^{n}+I_{3}^{n},$ we can write

(17) $I_{2}^{n}+I_{3}^{n}=\frac{1}{2\pi
i}\int_{-\infty}^{\infty}\partial_{\alpha^{\prime}}\left(\frac{(1-a(|z-\xi^{n}(\alpha^{\prime})|^{2}))}{z-\xi^{n}(\alpha^{\prime})}\right)\tilde{\Gamma}^{n}(\alpha^{\prime})\
\mathrm{d}\alpha^{\prime}.$

Since $\tilde{\gamma}^{n}\in L^{2}(\mathbb{R}),$ by Lemma 2 and its proof, we
have that $\tilde{\Gamma}^{n}(\alpha^{\prime})$ grows at infinity only like
$\sqrt{\alpha^{\prime}}$ at worst. Thus, we may integrate (17) by parts with
zero boundary contributions. When combining this with the remaining parts of
the velocity, we find

(18) $u_{n}-iv_{n}=I_{0}+\frac{1}{2\pi
i}\mathrm{PV}\int_{-\infty}^{\infty}\frac{\tilde{\gamma}^{n}(\alpha^{\prime})}{z-\xi^{n}(\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime}.$

This is the velocity field for a decaying vortex sheet with vortex sheet
strength equal to $\tilde{\gamma}^{n}+c[\gamma],$ with the position of the
sheet given by the curve $\mathcal{C}^{n}$ whose position in the complex plane
is given by $\xi^{n}.$ Thus $(u_{n},v_{n})$ is an incompressible velocity
field, with vorticity equal to
$c[\gamma]\delta_{\mathcal{C}}+\tilde{\gamma}^{n}\delta_{\mathcal{C}_{n}}.$

Let $K\subset\mathbb{R}^{2}$ be a compact set. We now wish to show that
$u^{n}-iv^{n}$ converges uniformly to $u-iv$ on $K.$ We will accomplish this
by showing that $I_{1}^{n}$ converges uniformly to $I_{1}$ on $K,$ $I_{2}^{n}$
converges uniformly to $I_{2}$ on $K,$ and $I_{3}^{n}$ converges uniformly to
$I_{3}$ on $K.$

We define a second compact set, $K_{3},$ to be the set of all points
$p\in\mathbb{R}^{2}$ such that $d(p,K)\leq 3.$ Then there exists
$A_{\pm}\in\mathbb{R}$ such that for $\alpha\geq A_{+}$ or for $\alpha<A_{-},$
$\xi(\alpha)\notin K_{3}.$ Furthermore, by our assumptions on the
approximations $\xi^{n},$ there exists $N\in\mathbb{N}$ such that for all
$n\geq N,$ for all $\alpha\in[A_{-},A_{+}],$ $\xi^{n}(\alpha)=\xi(\alpha)$ and
$\tilde{\gamma}^{n}(\alpha)=\tilde{\gamma}(\alpha),$ and for all
$\alpha\notin[A_{-},A_{+}],$ $\xi^{n}(\alpha)\notin K_{3}.$ Let $z\in K$ be
given. Then $I_{1}$ and $I_{1}^{n}$ can be rewritten as

$I_{1}=\frac{1}{2\pi
i}\int_{A_{-}}^{A_{+}}\frac{a(|z-\xi(\alpha^{\prime})|^{2})\tilde{\gamma}(\alpha^{\prime})}{z-\xi(\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime},$ $I_{1}^{n}=\frac{1}{2\pi
i}\int_{A_{-}}^{A_{+}}\frac{a(|z-\xi^{n}(\alpha^{\prime})|^{2})\tilde{\gamma}^{n}(\alpha^{\prime})}{z-\xi^{n}(\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime},$

for $n\geq N.$ But then we see immediately that for all $n\geq N,$ we have
$I_{1}=I_{1}^{n}.$ So, indeed, $I_{1}^{n}$ converges uniformly on $K$ to
$I_{1},$ The same situation holds for $I_{2}^{n}$ converging uniformly on $K$
to $I_{2}^{n}.$

It only remains to show that $I_{3}^{n}$ converges uniformly on $K$ to
$I_{3}.$ Note that we only need to consider the domain of integration to be
$(-\infty,\alpha_{-n}]\cup[\alpha_{n},\infty),$ as the integrands for $I_{3}$
and $I_{3}^{n}$ agree on the complement of this domain. We will only consider
$[\alpha_{n},\infty)$ in detail, as the other case is entirely similar. We
define

$J^{n}=-\frac{1}{2\pi
i}\int_{\alpha_{n}}^{\infty}\frac{(1-a(|z-\xi(\alpha^{\prime})|^{2})\xi_{\alpha}(\alpha^{\prime})\Gamma(\alpha^{\prime})}{(z-\xi(\alpha^{\prime}))^{2}}\
\mathrm{d}\alpha^{\prime},$ $I_{3}^{n,+}=-\frac{1}{2\pi
i}\int_{\alpha_{n}}^{\infty}\frac{(1-a(|z-\xi^{n}(\alpha^{\prime})|^{2})\xi^{n}_{\alpha}(\alpha^{\prime})\Gamma^{n}(\alpha^{\prime})}{(z-\xi^{n}(\alpha^{\prime}))^{2}}\
\mathrm{d}\alpha^{\prime}.$

We need to show that $J^{n}-I_{3}^{n,+}$ can be made small by taking $n$
large, uniformly with respect to $z\in K.$ We will accomplish this by showing
that the two terms may be made individually small by taking $n$ large,
uniformly with respect to $z\in K.$

For $n$ sufficiently large, for $\alpha^{\prime}>\alpha_{n},$ it will be the
case that $a(|z-\xi(\alpha^{\prime})|^{2})=0.$ Thus we may bound $|J_{n}|$ as

$|J_{n}|\leq\int_{\alpha_{n}}^{\infty}\frac{|\xi_{\alpha}(\alpha^{\prime})||\Gamma(\alpha^{\prime})|}{|z-\xi(\alpha^{\prime})|^{2}}\
\mathrm{d}\alpha^{\prime}.$

Using the assumption that $|\xi_{\alpha}|$ is bounded and that $\gamma$
satisfies Assumption A, we may further bound this as

$|J_{n}|\leq
c\int_{\alpha_{n}}^{\infty}\frac{|\alpha^{\prime}|^{\beta}}{|z-\xi(\alpha^{\prime})|^{2}}\
\mathrm{d}\alpha^{\prime},$

where the constant $c$ and the power $\beta\in[0,1)$ are taken to be
independent of $n.$ We let $\alpha_{*}$ be such that $\xi(\alpha_{*})$ is the
nearest point on $\mathcal{C}$ to $z.$ Then we have

$|J_{n}|\leq
c\int_{\alpha_{n}}^{\infty}\frac{|\alpha^{\prime}|^{\beta}}{|\xi(\alpha_{*})-\xi(\alpha^{\prime})|^{2}}\
\mathrm{d}\alpha^{\prime}.$

Using the chord-arc condition (11), we may then bound this as

$|J_{n}|\leq
c\int_{\alpha_{n}}^{\infty}\frac{|\alpha^{\prime}|^{\beta}}{(\alpha^{\prime}-\alpha_{*})^{2}}\
\mathrm{d}\alpha^{\prime}.$

Since $z$ is chosen from a compact set, the set of all such $\alpha_{*}$ may
also be taken from a compact set; we use the name $\alpha_{**}$ for the
largest possible such $\alpha_{*}.$ Then we have

(19) $|J_{n}|\leq
c\int_{\alpha_{n}}^{\infty}\frac{|\alpha^{\prime}|^{\beta}}{(\alpha^{\prime}-\alpha_{**})^{2}}\
\mathrm{d}\alpha^{\prime}.$

Since $\alpha_{n}\rightarrow\infty$ as $n\rightarrow\infty,$ we conclude that
$J_{n}\rightarrow 0$ as $n\rightarrow\infty.$ Furthermore, since (19) no
longer has any $z$ dependence on the right-hand side, this convergence is seen
to be uniform with respect to the choice of $z\in K.$

That the same conclusion holds for $I_{3}^{n,+}$ follows analogously, relying
upon Assumption C. This concludes the proof that $(u_{n},v_{n})$ converges to
$(u,v)$ on compact sets. We next must demonstrate the remaining claims, namely
that $(u,v)$ has the desired divergence and curl, and is independent of the
choice of cutoff function, $a.$

With the velocity $(u_{n},v_{n})$ converging to $(u,v)$ uniformly on compact
sets, we conclude that $(u_{n},v_{n})$ converges to $(u,v)$ in the sense of
distributions. That is, for any pair of test functions $(\phi_{1},\phi_{2}),$
which is a pair of $C^{\infty}$ functions with compact support, we have

$((u_{n},v_{n}),(\phi_{1},\phi_{2}))=\int_{X}u_{n}\phi_{1}+v_{n}\phi_{2}\rightarrow\int_{X}u\phi_{1}+v\phi_{2}=((u,v),(\phi_{1},\phi_{2})).$

Here, $X$ is a compact set containing the support of $\phi_{1}$ and
$\phi_{2}.$ Furthermore, for any test function $\phi,$ this implies

$(u_{n},\phi_{x})\rightarrow(u,\phi_{x}),\qquad(u_{n},\phi_{y})\rightarrow(u,\phi_{y}),$
$(v_{n},\phi_{x})\rightarrow(v,\phi_{x}),\qquad(v_{n},\phi_{y})\rightarrow(v,\phi_{y}).$

This implies that $\mathrm{curl}(u_{n},v_{n})$ converges to
$\mathrm{curl}(u,v)$ in the sense of distributions, and
$\mathrm{div}(u_{n},v_{n})$ converges to $\mathrm{div}(u,v)$ in the sense of
distributions. But $\mathrm{div}(u_{n},v_{n})=0.$ We conclude that $(u,v)$ is
divergence-free, in the sense of distributions. Note that it is not possible
that the velocity field would be divergence-free in a classical sense, as the
velocity is not continuous and therefore not differentiable in the classical
sense. As we have said, the vorticity associated to $(u_{n},v_{n})$ is
$c[\gamma]\delta_{\mathcal{C}}+\tilde{\gamma}^{n}\delta_{C_{n}},$ by our
construction. This means that $(u_{n},v_{n})$ is irrotational in the interior
of the two fluid regions. Since the boundary curves $\xi_{n}$ converge
uniformly (eventually identically, in fact) to the curves $\xi$ on compact
sets, and similarly for $\tilde{\gamma}^{n},$ we may conclude that the
vorticity associated to the velocity field $(u,v)$ is $\gamma\delta_{C},$ and
again this implies the the fluid is irrotational in the interior of either
fluid region.

Finally, we discuss independence of the choice of cutoff function. As can be
seen in (18), the approximate velocities $(u_{n},v_{n})$ do not depend on the
choice of the cutoff function, $a.$ Since these converge to the velocity
$(u,v),$ it is not possible for the velocity $(u,v)$ to depend on this choice.
∎

###  3.6. Jump conditions

We may express the limiting values of the velocity $(u,v)$ as the point
$\mathbf{x}$ approaches $\mathcal{C}$ in terms of $\mathbf{W}.$ We let
$\mathbf{\hat{t}}$ and $\mathbf{\hat{n}}$ be the usual frame of unit normal
and tangent vectors to the curve,

$\mathbf{\hat{t}}=\frac{(x_{\alpha},y_{\alpha})}{s_{\alpha}},\qquad\mathbf{\hat{n}}=\frac{(-y_{\alpha},x_{\alpha})}{s_{\alpha}},$

where the arclength element $s_{\alpha}$ is given by
$s_{\alpha}=\sqrt{x_{\alpha}^{2}+y_{\alpha}^{2}}.$ We wish to take the limit
of (13) as $z$ approaches $\mathcal{C},$ and so we take the limit of each of
$I_{0},$ $I_{1},$ $I_{2},$ and $I_{3}.$ Since the integrals $I_{2}$ and
$I_{3}$ are non-singular, the limit of these is just $W_{2}$ and $W_{3},$ when
$z\rightarrow\xi(\alpha).$ For $I_{0}$ and $I_{1}$ we are able to apply the
Plemelj formulas (see Muskhelishvili (1992), and see also Baker et al. (1982))
when taking the limit as $z$ approaches $\mathcal{C}.$ This is just as in the
usual decaying case of vortex sheets, and we find

(20) $(u,v)\rightarrow W\pm\frac{\gamma}{2s_{\alpha}}\mathbf{\hat{t}},$

where the plus or minus depends on whether the interface is approached from
above or below. We see from (20) that there is no jump in normal velocity
across the vortex sheet, while the jump in tangential velocity is given by the
vortex sheet strength, $\gamma.$

###  3.7. Limit at vertical infinity

It is of interest for a vortex sheet to compute the limiting velocity at
vertical infinity, i.e. to take the limit of (13) as $y\rightarrow\pm\infty.$
It is immediate to see that as $y\rightarrow\pm\infty,$ $I_{1}$ and $I_{2}$
become identically zero because of the presence of the cutoff function. Thus
we only must work to take the limit of $I_{0}$ and $I_{3}.$ In fact, only
$I_{0}$ makes a nonzero contribution to the limit. We may write $I_{0}$ as

$I_{0}=\frac{c[\gamma]}{2\pi
i}\mathrm{PV}_{\infty}\int_{-\infty}^{\infty}\frac{1}{z-\alpha^{\prime}}\
\mathrm{d}\alpha^{\prime}+\frac{c[\gamma]}{2\pi
i}\mathrm{PV}_{\infty}\int_{-\infty}^{\infty}\frac{\xi(\alpha^{\prime})-\alpha^{\prime}}{(z-\alpha^{\prime})(z-\xi(\alpha^{\prime}))}\
\mathrm{d}\alpha^{\prime}.$

The first term on the right-hand side has limit $\mp\frac{c[\gamma]}{2}$ as
$y\rightarrow\pm\infty;$ notice this integral is now in the decaying case, and
the calculation follows from being able to exactly evaluate the integral. The
second integral on the right-hand side has limit zero at vertical infinity,
which can be seen because the denominator is now quadratic with respect to the
$z$ variable. Similarly, the limit of $I_{3}$ as $y\rightarrow\pm\infty$ is
zero, because the denominator is quadratic with respect to $z.$ We conclude
that

$\lim_{y\rightarrow\pm\infty}(u,v)(x,y)=\mp\frac{c[\gamma]}{2},$

which is the same limit as in the periodic and decaying cases.

##  4\. Revisiting the examples

In this section we consider again the four illustrative examples of curves
$\mathcal{C}$ and vortex sheet strengths $\gamma$ which we introduced in
Section 2.1 above. We will see that for all of these examples, all the needed
assumptions are satisfied.

In each of the four examples of Section 2.1, we let the approximate curve
$\mathcal{C}_{n}$ be given by its parameterization
$\xi^{n}(\alpha)=\alpha+(\xi(\alpha)-\alpha))\phi_{n}(\alpha).$ To see that
Theorem 1 applies in each of the four examples, we must verify Assumption A
and Assumption C, and either we must have that $c[\gamma]=0$ or we must verify
Assumption B. Furthermore, the bullet-point list of conditions on the
parameterization $\xi$ which appears in the theorem must be verified. Most of
these points to verify are obvious and will not be commented on explicitly.
Below we give some further details.

(a) A mixed case.:

We remark that the vortex sheet strength
$\gamma(\alpha)=\left(\frac{1}{1+\alpha^{2}}\right)^{2/5}$ satisfies
Assumption A with $c[\gamma]=0$ and $\beta=1/5.$ Since $c[\gamma]=0,$ we do
not need to consider Assumption B. We only need to verify the condition on the
integral in (15). If we call the integral $\mathcal{I}_{n}(z),$ we have the
estimate, for some $c$ independent of $n$ and $z,$

$|\mathcal{I}_{n}(z)|\leq c\int_{\Omega_{n}}\frac{\alpha^{\prime
1/5}}{(x-\xi^{n}_{1}(\alpha^{\prime}))^{2}+(y-\xi^{n}_{2}(\alpha^{\prime}))^{2}}\
\mathrm{d}\alpha^{\prime}.$

We have written here $z=x+iy.$ We let $x_{*+}$ be the largest $x$-value in the
compact set $K_{0},$ and we let $x_{*-}$ be the smallest $x$-value in $K_{0}.$
We can use our specific parameterization, namely $\xi^{n}_{1}(\alpha)=\alpha,$
and we neglect the term $(y-\xi^{n}_{2}(\alpha^{\prime}))^{2}$ in the
denominator. We identify the values $\alpha_{n}=n$ and $\alpha_{-n}=-n.$ These
considerations lead to the bound

$|\mathcal{I}_{n}(z)|\leq c\int_{-\infty}^{-n}\frac{\alpha^{\prime
1/5}}{(x_{*-}-\alpha^{\prime})^{2}}\
\mathrm{d}\alpha^{\prime}+c\int_{n}^{\infty}\frac{\alpha^{\prime
1/5}}{(x_{*+}-\alpha^{\prime})^{2}}\ \mathrm{d}\alpha^{\prime}.$

Note that for sufficiently large $n,$ it is not possible for the denominators
of the integrands to equal zero. These integrals do converge to zero as $n$
goes to infinity, and this is clearly uniform with respect to $z.$

(b) A quasiperiodic case.:

The vortex sheet strength
$\gamma(\alpha)=\cos\left(\left(1+4\pi-\sqrt{2}\right)\alpha\right)$ satisfies
Assumption A with $c[\gamma]=0$ and $\beta=0.$ Since $c[\gamma]=0,$ we do not
need to consider Assumption B. In this case we need to pay special attention
to the chord-arc condition (11), for $\alpha$ and $\alpha^{\prime}$ close to
each other. It can be seen that the chord-arc quantity (i.e. the divided
difference $q_{1}[\xi]$) is equal to $1+O(\mu),$ for $\alpha-\alpha^{\prime}$
small. Thus, for $\mu$ sufficiently small, the chord-arc condition is
satisfied and the curve is non-self-intersecting. The other properties are
similar to those in example (a).

(c) A uniformly local case.:

The vortex sheet strength $\gamma(\alpha)=\frac{\alpha^{2}}{1+\alpha^{2}}$
satsifes Assumption A with $c[\gamma]=1$ and $\beta=0.$ We next consider
Assumption B, and we argue as in the proof of Lemma 3. We follow the proof of
the lemma until (25); then, it is clear that the integral on the right-hand
side of (25) converges with
$\xi(\alpha)=\alpha+i\sin\left(\left(\frac{\alpha^{4}}{1+\alpha^{2}}\right)^{1/4}\right),$
since the numerator of the second factor is bounded and the denominator is of
the order $\alpha^{\prime 2}.$ To bound this, we must also use the fact that
the parameterization $\xi$ satisfies the chord-arc condition (11), which it
does. We then continue to follow the proof of the lemma until (28), and we
again note that the integral on the right-hand side converges since the
numerator is bounded and the denominator is of the order $\alpha^{\prime 2}.$
The other properties are again similar to those in example (a).

(d) A bore.:

The vortex sheet strength $\gamma(\alpha)=1+\sin(\alpha)$ satisfies Assumption
A with $c[\gamma]=1$ and $\beta=0.$ That Assumption B holds is verified in the
same manner as in example (c), and the remaining properties are similar to
those in example (a).

While in the case of the bore we took the location of the vortex sheet to have
different limits at positive and negative infinity, it is worth noting that
the same cannot be done for the vortex sheet strength. If, say, $\gamma$
approached $-1$ at negative infinity and $1$ at positive infinity, then we
would not be able to satisfy Assumption A.

##  5\. Discussion

In this work, we have developed a more general expression for the velocity
field associated to a vortex sheet than the one found by directly using the
vortex sheet ansatz for vorticity in the Biot-Savart law. Taking the limit of
this velocity field as the vortex sheet is approached yields a new, more
general expression for the Birkhoff-Rott integral. The Birkhoff-Rott integral
has the advantages that it is useful for analytical and numerical studies of
the motion of vortex sheets (including water waves) in situations in which the
sheet is overturned, i.e. has multi-valued height, and also that can be used
in this way for both two-dimensional and three-dimensional fluid flows. While
we have only treated the case of two-dimensional fluids here, we expect that
this work can be carried over to the three-dimensional case. A development of
the usual Birkhoff-Rott integral for three-dimensional fluids may be found in
Caflisch & Li (1992).

In future work, we expect to categorize the set of interface locations for
which Assumption C on approximability by decaying sheets holds. We also expect
to use the new expression for the Birkhoff-Rott integral to prove well-
posedness theory for vortex sheets and water waves in the non-decaying, non-
periodic context. The author and Masmoudi have previously made such studies in
two-dimensional fluids in the periodic setting (Ambrose (2003), Ambrose &
Masmoudi (2005)), and for three-dimensional fluids in the decaying setting
(Ambrose & Masmoudi (2007), Ambrose & Masmoudi (2009)). In these works a key
ingredient has been approximation of the Birkhoff-Rott integral by the Hilbert
transform or by Riesz transforms; this approach has also been employed in
similar works such as Beale et al. (1993) or Córdoba et al. (2010). If a
function $f$ satisfies Assumption A, we may define its Hilbert transform as

(21)
$H(f)(\alpha)=\frac{1}{\pi}\mathrm{PV}\int_{-\infty}^{\infty}\frac{a(\alpha-\alpha^{\prime})f(\alpha^{\prime})}{\alpha-\alpha^{\prime}}\
\mathrm{d}\alpha^{\prime}-\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{a^{\prime}(\alpha-\alpha^{\prime})F(\alpha^{\prime})}{\alpha-\alpha^{\prime}}\
\mathrm{d}\alpha^{\prime}\\\
-\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{(1-a(\alpha-\alpha^{\prime}))F(\alpha^{\prime})}{(\alpha-\alpha^{\prime})^{2}}\
\mathrm{d}\alpha^{\prime}.$

Here, $F$ is the antiderivative of $f-c[f]$ discussed in Assumption A. The
formula (21) is developed analogously to the development of the new Birkhoff-
Rott integral. Just as in that case, this is a single new formula for the
Hilbert transform which unifies the decaying and periodic settings while
extending its reach to additional functions. Of course, the Hilbert transform
has additional properties such as connections with analytic functions, and we
will leave it to future work to develop such theory for this new version of
the Hilbert transform. This future work will also prioritize demonstrating the
necessary estimates for approximation of the new Birkhoff-Rott integral by the
new Hilbert transform.

##  Appendix A Proofs of lemmas

In this appendix. we give the proofs of two lemmas. First we have the proof of
Lemma 2, about Assumption A.

###### Proof of Lemma 2.

Let $\gamma\in L^{2}(\mathbb{R}).$ We define $\Gamma$ by

$\Gamma(\alpha)=\int_{0}^{\alpha}\gamma(\alpha^{\prime})\
\mathrm{d}\alpha^{\prime}.$

Then by Hölder’s inequality (Kreyszig (1989)), we have for any $\alpha,$

$|\Gamma(\alpha)|\leq\left(\int_{0}^{\alpha}1^{2}\
\mathrm{d}\alpha^{\prime}\right)^{1/2}\left(\int_{0}^{\alpha}\gamma^{2}(\alpha^{\prime})\
\mathrm{d}\alpha^{\prime}\right)^{1/2}\leq\sqrt{\alpha}\|\gamma\|_{L^{2}(\mathbb{R})}.$

We conclude that Assumption A is satisfied with
$C=\|\gamma\|_{L^{2}(\mathbb{R})}$ and $\beta=1/2,$ and with constant
$c[\gamma]=0.$

Now assume instead that $\gamma\in L^{2}(\mathbb{T}).$ Then we may express
$\gamma$ by its Fourier series,

$\gamma(\alpha)=\sum_{k\in\mathbb{Z}}\hat{\gamma}(k)e^{ik\alpha}.$

(Here we have again set the periodicity length to be $2\pi,$ but this is not
necessary.) Then we may identify the constant $c[\gamma]=\hat{\gamma}(0).$ The
antiderivative $\Gamma$ may be taken to be

$\Gamma(\alpha)=\sum_{k\in\mathbb{Z}\setminus\\{0\\}}\frac{\hat{\gamma}(k)}{ik}e^{ik\alpha}.$

Since $\gamma$ is in $L^{2}(\mathbb{T}),$ by Plancherel’s theorem we may
conclude $\Gamma\in L^{2}(\mathbb{T})$ as well:

(22)
$\|\Gamma\|_{L^{2}(\mathbb{T})}=\sum_{k\in\mathbb{Z}\setminus\\{0\\}}\frac{|\hat{\gamma}(k)|^{2}}{k^{2}}\leq\sum_{k\in\mathbb{Z}}|\hat{\gamma}(k)|^{2}=\|\gamma\|_{L^{2}(\mathbb{T})}.$

By Sobolev embedding (Adams & Fournier (2003)), this implies $\Gamma\in
L^{\infty}.$ We conclude that Assumption A is satisfied, with $\beta=0.$ ∎

Finally, we have the proof of Lemma 3, about Assumption B.

###### Proof of Lemma 3.

We begin by considering item (i) of Assumption B. We decompose the integral as
$N+F,$ (standing for the “near” part and the “far” part), with

$N=\mathrm{PV}\int_{\alpha-1}^{\alpha+1}\frac{1}{\xi(\alpha)-\xi(\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime},$
$F=\mathrm{PV}_{\infty}\left(\int_{-\infty}^{\alpha-1}\frac{1}{\xi(\alpha)-\xi(\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime}+\int_{\alpha+1}^{\infty}\frac{1}{\xi(\alpha)-\xi(\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime}\right).$

We first deal with $N.$ Adding and subtracting, we can write this as

(23)
$N=\mathrm{PV}\int_{\alpha-1}^{\alpha+1}\frac{1}{\xi_{\alpha}(\alpha^{\prime})(\alpha-\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime}+\int_{\alpha-1}^{\alpha+1}\left[\frac{1}{\xi(\alpha)-\xi(\alpha^{\prime})}-\frac{1}{\xi_{\alpha}(\alpha^{\prime})(\alpha-\alpha^{\prime})}\right]\
\mathrm{d}\alpha^{\prime}.$

The first integral on the right-hand side converges, as this is a localized
Hilbert transform. For the second integral on the right-hand side, it is the
integral of a ratio of divided differences. That is, we let

$q_{1}[\xi](\alpha,\alpha^{\prime})=\frac{\xi(\alpha)-\xi(\alpha^{\prime})}{\alpha-\alpha},\qquad
q_{2}[\xi](\alpha,\alpha^{\prime})=\frac{\xi(\alpha)-\xi(\alpha^{\prime})-\xi_{\alpha}(\alpha^{\prime})(\alpha-\alpha^{\prime})}{(\alpha-\alpha^{\prime})^{2}};$

these are known as the first and second divided differences for $z.$ Then the
second integral on the right-hand side of (23) is the integral of
$\frac{q_{2}[\xi](\alpha,\alpha^{\prime})}{\xi_{\alpha}(\alpha^{\prime})q_{1}[\xi](\alpha,\alpha^{\prime})}.$
By standard results on divided differences (as in e.g. Beale et al. (1993)),
if $\xi$ is twice differentiable then $q_{2}$ is locally integrable. Using
together our assumption that $\xi$ is twice differentiable, the chord-arc
condition (11) and the stated assumption on the parameterization that
$\xi_{\alpha}$ is bounded away from zero, we may conclude that the integral
$N$ converges.

We next consider $F.$ In each of the integrands in $F,$ we subtract
$\frac{1}{\alpha-\alpha^{\prime}}:$

(24)
$F=\mathrm{PV}_{\infty}\Bigg{(}\int_{-\infty}^{\alpha-1}\frac{1}{\alpha-\alpha^{\prime}}\
\mathrm{d}\alpha^{\prime}+\int_{-\infty}^{\alpha-1}\frac{1}{\xi(\alpha)-\xi(\alpha^{\prime})}-\frac{1}{\alpha-\alpha^{\prime}}\
\mathrm{d}\alpha^{\prime}\\\
+\int_{\alpha+1}^{\infty}\frac{1}{\alpha-\alpha^{\prime}}\
\mathrm{d}\alpha^{\prime}+\int_{\alpha+1}^{\infty}\frac{1}{\xi(\alpha)-\xi(\alpha^{\prime})}-\frac{1}{\alpha-\alpha^{\prime}}\
\mathrm{d}\alpha^{\prime}\Bigg{)}.$

We can clearly compute part of this principal value:

$\mathrm{PV}_{\infty}\left(\int_{-\infty}^{\alpha-1}\frac{1}{\alpha-\alpha^{\prime}}\
\mathrm{d}\alpha^{\prime}+\int_{\alpha+1}^{\infty}\frac{1}{\alpha-\alpha^{\prime}}\
\mathrm{d}\alpha^{\prime}\right)=0.$

This leaves us with the following formula for $F:$

$F=\mathrm{PV}_{\infty}\Bigg{(}\int_{-\infty}^{\alpha-1}\frac{1}{\xi(\alpha)-\xi(\alpha^{\prime})}-\frac{1}{\alpha-\alpha^{\prime}}\
\mathrm{d}\alpha^{\prime}+\int_{\alpha+1}^{\infty}\frac{1}{\xi(\alpha)-\xi(\alpha^{\prime})}-\frac{1}{\alpha-\alpha^{\prime}}\
\mathrm{d}\alpha^{\prime}\Bigg{)}.$

This is no longer, in fact, a principal value integral at infinity, however,
since these integrals converge individually. To see this, we arrange the
second integral on the right-hand side as

(25)
$\int_{\alpha+1}^{\infty}\frac{1}{\xi(\alpha)-\xi(\alpha^{\prime})}-\frac{1}{\alpha-\alpha^{\prime}}\
\mathrm{d}\alpha^{\prime}=-\int_{\alpha+1}^{\infty}\frac{\alpha-\alpha^{\prime}}{\xi(\alpha)-\xi(\alpha^{\prime})}\frac{(\xi(\alpha)-\alpha)-(\xi(\alpha^{\prime})-\alpha^{\prime})}{(\alpha-\alpha^{\prime})^{2}}\
\mathrm{d}\alpha^{\prime}.$

We then decompose this into two integrals, as

(26)
$\int_{\alpha+1}^{\infty}\frac{1}{\xi(\alpha)-\xi(\alpha^{\prime})}-\frac{1}{\alpha-\alpha^{\prime}}\
\mathrm{d}\alpha^{\prime}=-(\xi(\alpha)-\alpha)\int_{\alpha+1}^{\infty}\frac{\alpha-\alpha^{\prime}}{\xi(\alpha)-\xi(\alpha^{\prime})}\frac{1}{(\alpha-\alpha^{\prime})^{2}}\
\mathrm{d}\alpha^{\prime}\\\
+\int_{\alpha+1}^{\infty}\frac{\alpha-\alpha^{\prime}}{\xi(\alpha)-\xi(\alpha^{\prime})}\frac{\xi(\alpha^{\prime})-\alpha^{\prime}}{(\alpha-\alpha^{\prime})^{2}}\
\mathrm{d}\alpha^{\prime}.$

That the first integral on the right-hand side converges follows from the
chord-arc condition (11) and from the fact that
$(\alpha-\alpha^{\prime})^{-2}$ is integrable on the relevant domain. That the
second integral on the right-hand side converges follows from these facts plus
the regularity of $\xi(\alpha^{\prime})-\alpha^{\prime}.$ The other integral
comprising $F$ may be treated in just the same way, and the conclusion for the
case $\xi(\alpha)-\alpha$ is decaying now follows.

For the case that $\xi(\alpha)-\alpha$ is periodic, the principal value at
positive and negative infinity is implicit in the cotangent summation formula
discussed above. For the principal value at $\alpha^{\prime}=\alpha,$ this is
essentially the same as in the real-line case, and we do not provide further
details (although the interested reader could consult Ambrose (2003)).

We now begin to prove (ii). We write the point $z$ as $z=x+iy,$ and we write
the integral as

$\mathrm{PV}_{\infty}\int_{-\infty}^{\infty}\frac{1}{z-\xi(\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime}=\mathrm{PV}_{\infty}\left(\int_{-\infty}^{x}\frac{1}{z-\xi(\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime}+\int_{x}^{\infty}\frac{1}{z-\xi(\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime}\right).$

We add and subtract $\frac{1}{z-\alpha^{\prime}}$ in each integral, arriving
at

(27)
$\mathrm{PV}_{\infty}\int_{-\infty}^{\infty}\frac{1}{z-\xi(\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime}=\mathrm{PV}_{\infty}\left(\int_{-\infty}^{x}\frac{1}{z-\alpha^{\prime}}\
\mathrm{d}\alpha^{\prime}+\int_{x}^{\infty}\frac{1}{z-\alpha^{\prime}}\
\mathrm{d}\alpha^{\prime}\right)\\\
+\int_{-\infty}^{\infty}\frac{1}{z-\xi(\alpha^{\prime})}-\frac{1}{z-\alpha^{\prime}}\
\mathrm{d}\alpha^{\prime}.$

Notice that the final integral on the right-hand side is no longer a principal
value (we will justify this shortly). Also, we can compute that the first
integral on the right-hand side is zero:

$\mathrm{PV}_{\infty}\left(\int_{-\infty}^{x}\frac{1}{z-\alpha^{\prime}}\
\mathrm{d}\alpha^{\prime}+\int_{x}^{\infty}\frac{1}{z-\alpha^{\prime}}\
\mathrm{d}\alpha^{\prime}\right)=0.$

For the remaining integral, we add the fractions, finding

(28)
$\mathrm{PV}_{\infty}\int_{-\infty}^{\infty}\frac{1}{z-\xi(\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime}=\int_{-\infty}^{\infty}\frac{\xi(\alpha^{\prime})-\alpha^{\prime}}{(z-\xi(\alpha^{\prime}))(z-\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime}.$

In this final integral, $\frac{1}{z-\xi(\alpha^{\prime})}$ is bounded away
from zero since $z$ is a positive distance from $\mathcal{C}.$ If
$\xi(\alpha^{\prime})-\alpha^{\prime}$ is in $L^{2}(\mathbb{R}),$ then since
the function $\frac{1}{z-\alpha^{\prime}}$ is also in $L^{2}(\mathbb{R}),$
this integral converges. This completes the proof in the case that
$\xi(\alpha)-\alpha$ is in $L^{2}(\mathbb{R}).$

In the periodic case, we just need to add and subtract one more time. We have

(29)
$\mathrm{PV}_{\infty}\int_{-\infty}^{\infty}\frac{1}{z-\xi(\alpha^{\prime})}\
\mathrm{d}\alpha^{\prime}=\int_{-\infty}^{\infty}\frac{\xi(\alpha^{\prime})-\alpha^{\prime}}{(z-\alpha^{\prime})^{2}}\
\mathrm{d}\alpha^{\prime}\\\
+\int_{-\infty}^{\infty}\frac{\xi(\alpha^{\prime})-\alpha^{\prime}}{z-\alpha^{\prime}}\left[\frac{1}{z-\xi(\alpha^{\prime})}-\frac{1}{z-\alpha^{\prime}}\right]\
\mathrm{d}\alpha^{\prime}.\\\
=\int_{-\infty}^{\infty}\frac{\xi(\alpha^{\prime})-\alpha^{\prime}}{(z-\alpha^{\prime})^{2}}\
\mathrm{d}\alpha^{\prime}+\int_{-\infty}^{\infty}\frac{(\xi(\alpha^{\prime})-\alpha^{\prime})^{2}}{(z-\xi(\alpha^{\prime}))(z-\alpha^{\prime})^{2}}\
\mathrm{d}\alpha^{\prime}.$

Again, the factor $\frac{1}{z-\xi(\alpha^{\prime})}$ is bounded above since
$z$ is a positive distance from $\mathcal{C}.$ Finally, the factor
$\frac{1}{(z-\alpha^{\prime})^{2}}$ is integrable on $\mathbb{R}.$ Since we
have assumed that $\xi(\alpha)-\alpha$ is bounded, this completes the proof in
the periodic case. ∎

Acknowledgements: The author wishes to thank Mark Hoefer for suggesting a bore
as an example. The author thanks Jon Wilkening for helpful discussions. The
author thanks Tom Beale for comments on an early version of this manuscript.
The author thanks Michael Siegel for extensive comments on an early version of
this manuscript.

Funding: The author is grateful to the National Science Foundation for support
through grant DMS-2307638.

Declaration of interests: The author reports no conflict of interest.

Author ORCIDs: D.M. Ambrose, https://orcid.org/0000-0003-4753-0319.

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