Copyright for this paper by its authors. Use permitted under Creative Commons
License Attribution 4.0 International (CC BY 4.0).

ARISDE2024: 1st International Workshop on Artificial Intelligence for
Sustainable Development, 1-3 July 2024

[orcid=0000-0002-3309-132X<EMAIL_ADDRESS>url=https://ioannismavromatis.com/, ]

[orcid=0000-0001-5372-7201<EMAIL_ADDRESS>url=https://www.linkedin.com/in/lecost/, ]

[orcid=0000-0002-3573-6240<EMAIL_ADDRESS>url=https://www.linkedin.com/in/aftabkhan00/, ]

[1]Corresponding author: Ioannis Mavromatis,
<EMAIL_ADDRESS>
# Computing Within Limits: An Empirical Study of Energy Consumption in ML
Training and Inference

Ioannis Mavromatis  Kostas Katsaros  Aftab Khan  Digital Catapult, NW1 2RA,
London, U.K.  Bristol Research & Innovation Laboratory, Toshiba Europe Ltd.,
Avon, BS1 4ND, Bristol, UK

(2024)

###### Abstract

Machine learning (ML) has seen tremendous advancements, but its environmental
footprint remains a concern. Acknowledging the growing environmental impact of
ML this paper investigates Green ML, examining various model architectures and
hyperparameters in both training and inference phases to identify energy-
efficient practices. Our study leverages software-based power measurements for
ease of replication across diverse configurations, models and datasets. In
this paper, we examine multiple models and hardware configurations to identify
correlations across the various measurements and metrics and key contributors
to energy reduction. Our analysis offers practical guidelines for constructing
sustainable ML operations, emphasising energy consumption and carbon footprint
reductions while maintaining performance. As identified, short-lived profiling
can quantify the long-term expected energy consumption. Moreover, model
parameters can also be used to accurately estimate the expected total energy
without the need for extensive experimentation.

###### keywords:

Machine Learning, Power Profiling, Energy Consumption, Sustainable AI, Green
AI

##  1 Introduction

In recent years, Machine Learning (ML) has seen remarkable advancements,
significantly impacting various sectors. However, concerns regarding its
environmental footprint have surfaced alongside its growth, showing that 2% of
global carbon emissions will occur from ML pipelines by 2030 [1]. The
intensive computation of training and deploying ML and Deep Learning (DL)
models contribute to substantial energy consumption and, thus, carbon
emissions. However, this presents a crucial challenge: how can the ML field
continue progressing while aligning with global sustainability goals?

ML is becoming increasingly prominent in current and future use cases and
human activities. For example, we see ML playing a pivotal role in 3D-media
content supporting Virtual Reality (VR) and Augmented Reality (AR) gaming,
interactive art installations, education, etc. [2]. We also see AI-native
future networks [3] enabling the synergy of AI and data exchange, or we even
see the use of Large Language Models (LLMs) and chatbots for everyday
activities from millions of people [4]. The flourishing of ML in all domains
makes the need for sustainable ML practices not merely a response to
environmental concerns but also a strategic imperative for companies and
organisations [5]. What is also clear from the above is that ML becomes part
of multiple steps in the delivery and operation pipelines of each use case
(e.g., the ICT infrastructure, the algorithms, the data processing and
analysis, etc.), necessitating ways to reduce energy consumption holistically
across the entire system [6].

Driven by the above, this paper presents an empirical study of the energy
consumption of an immersive media task, i.e., an image classification. Such a
task can be common in applications like hand gesture detection, interactive
educational games [7, 8], etc. Our investigation aims to offer practical
guidelines and best practices that researchers and practitioners can adopt in
their applications across the entire ML Operations (MLOps) lifecycle. Even
though this is a domain-specific task, our work can be leveraged by ML
practitioners aiming for energy-aware optimisations in their ML pipelines
across different fields and use cases.

There is recently increased interest in Green and Sustainable ML [5, 9].
Sustainable ML practices [5] encompass efficient use of computational
resources and holistic optimisation of ML pipelines that collectively lead to
reduced energy consumption, minimised carbon footprints, and economic
benefits. Our paper aims to provide insights into how the ML lifecycle can be
optimised for lower energy consumption without compromising performance. We
analyse various model architectures and hyperparameters, both for training and
inference, to identify areas where energy consumption can be reduced. Based on
our findings, we will critically comment on the key contributors to energy
reduction and provide ways for estimating the expected energy consumption
based on various model parameters.

The remainder of this paper is structured as follows: Sec. 2 presents recent
activities around sustainable ML and discusses their limitations. Green MLOps
is described in 3, outlining the energy consumed within an MLOps pipeline. The
methodology used for our extensive investigation is illustrated in Sec. 4.
Secs 5 and 6 present our results and lessons learned. Finally, the paper is
concluded in Sec. 7

##  2 Background and Sustainability Goals

The United Nations (UN) has recently presented its 2030 Agenda for Sustainable
Development, targeting 17 UN Sustainable Development Goals (SDGs)111UN
Sustainable Development Goals: https://sdgs.un.org/goals. All 17 SDGs are
increasingly important and should be considered by future systems and use
cases. Our work is especially linked with the goals below:

  * •

Goal 9: Industry, Innovation and Infrastructure \- Build resilient
infrastructure, promote inclusive and sustainable industrialisation and foster
innovation \- Our work is looking to propose a roadmap for future MLOps
frameworks development, paving the way for innovation and good practices
across the entire technology stack.

  * •

Goal 10: Reduced Innequalities \- Reduce inequality within and among countries
\- Reducing energy consumption, ML can become economically viable and
sustainable for everyone, fulfilling the 4Cs (Coverage, Capacity, Cost,
Consumption) requirements.

  * •

Goal 12: Responsible Consumption and Production \- Ensure sustainable
consumption and production patterns \- Green ML can significantly reduce the
demand for fossil fuel energy production and the total energy consumption.

  * •

Goal 13: Climate Action \- Take urgent action to combat climate change and its
impacts \- Reducing energy consumption across an entire MLOps pipeline can
lead to reduced carbon emissions.

Overall, this work aims to provide ways to tackle the energy-hungry tasks of
ML training and inference and potential avenues for sustainable MLOps
pipelines across the entire computing continuum of any given use case and
scenario.

###  2.1 Related Work

Many works present concepts and solutions around Green and Sustainable ML.
Some notable examples are [9, 5, 10], where the authors provide statistics on
how ML’s energy consumption will increase over time. Authors in [5] also
compare transformer models running in Google’s data centres. All papers
discuss the potential benefits of energy reduction from good practices (e.g.,
early existing, knowledge transfer, etc.) but do not systematically assess
those. Our work contributes towards that by conducting an empirical study on
real-world hardware.

Researchers have explored various energy reduction algorithms, e.g., pruning
[11] or quantisation [12]. These works are smaller-scale investigations and
focus on methods that affect the accuracy of a given model. On the contrary,
in our large-scale study, we explore ways for energy reduction without changes
in accuracy. Similarly to [13], our work will analyse tradeoffs across
different configurations and parameters that can be considered when designing
ML-enabled systems.

A notable work presenting various measurement campaigns is outlined in [14].
The authors focus on how various modifications in an ML pipeline can reduce
the environmental impact, targeting system-level holistic optimisations.
However, the individual measurements or the models used are not detailed. Our
work studies a set of well-known models and datasets to enable readers to
understand the differences between distinct hyperparameters and models.
Moreover, open-sourcing our code will enable other researchers to replicate
our study with different models, datasets or hardware.

The recent literature includes two relevant studies to our evaluation based on
real-world measurements [15, 16]. The authors of the first study [15] focused
primarily on shallow single-layer models. The authors of the second study [16]
investigated larger transformer-based models. However, neither includes a deep
exploration of how different model characteristics or hyperparameters affect
energy consumption. This is a key contribution of our paper.

##  3 Green MLOps: A Strategic Imperative

DevOps merges software development with IT operations to speed up development
time using automation and integration tools. MLOps, an extension of DevOps for
ML pipelines, focuses on managing the ML model lifecycle efficiently, tackling
issues like data management and reproducibility. All “production systems”
supporting ML-enabled applications usually integrate an MLOps framework [17].
Green MLOps extends the idea of MLOps, providing a framework that streamlines
ML operations in an energy-aware and cost-effective fashion [9].

###  3.1 Energy Consumption in MLOps

Figure 1: ML model development and deployment phase and the associated MLOps
life cycle.

MLOps (Fig. 1) typically involves a Data Processing phase for collecting,
curating, and labelling data, and assigning weights to features. In our
example image classification task, this might involve creating a dataset of
hand gestures. The Experimentation phase follows, where algorithms, model
architectures, and training methods are tested. For instance, this could
involve testing various models like VGG or YOLO and exploring different
hyperparameters, often using grid-search to ensure robustness.

Upon identifying viable solutions, the Training phase involves training the
selected models on larger, feature-rich datasets, refining the hyperparameters
as needed. After training, models enter the Inference phase. They are deployed
to make real-time decisions, e.g., running a hand gesture detection model on a
Virtual Reality headset (or a nearby computer). This phase includes continuous
performance monitoring and possible re-training.

Training, experimenting, and inferring consume significant energy [16].
Facebook’s AI research team [14] indicates that inference requires more
compute cycles than training, having a split of $10:20:70$ (in %) between
Experimentation, Training and Inference, respectively. Moreover, the energy
distribution across the entire MLOps pipeline is roughly $31:29:40$ (in %) for
the Data, Experimentation/Training, and Inference phases. As described in
[16], inadequate hyperparameter tuning and poor neural network management can
vastly increase energy consumption by up to $\times 2000$ times for Natural
Language Model (NLP) models and up to $\times 3000$ for a transformer-based
NLP.

###  3.2 A Comparison between Computation and Data Exchange

In an MLOps pipeline, energy consumption is the aggregated result of
computation and data exchange. Unlike other ICT systems, where computation and
data transfer energy use are roughly equal [18], ML pipelines are expected to
demand more energy for computation. This is because while activities like
Experimentation and Training may occur once (e.g., no retraining of the model
is required), the Inference and Data phases will always need continuous
computation. Additionally, trends like Federated Learning (FL), which
distributes training or inference to edge nodes, show even higher energy
consumption compared to centralised learning approaches (particularly
considering complex ML models) despite the decreased data exchange [19]. This
is due to the difficulty of parallelising computation, where training and
inference are executed across many clients on smaller datasets.

ML training and inference computation are tightly linked to the model
characteristics and hyperparameters, which motivated our investigation. Our
paper identifies different model characteristics and hyperparameters that
impact the energy consumed within an ML pipeline. We focus our investigation
on the Experimentation, Training and Inference phases of an MLOps pipeline and
compare parameters such as the model size, the batch size, the time required
for training and inference, the Multiply–Accumulate (MAC) operations, the
hardware utilisation and the model parameters as a function of the energy
consumed. Currently, no study compares the energy consumed by the computation
against the data exchange in large-scale ML deployments (e.g., as the one
presented from Facebook in [14]), highlighting a research gap.

##  4 Methodology

To investigate the above, measuring the absolute power at frequent intervals
and the time required for each experiment is essential. Hardware statistics
like the utilisation of resources and the model characteristics should also be
captured as part of our experimentation and correlated with the model
characteristics and hyperparameters. We implemented a framework that captured
all the above and produced the results found in the paper. Our codebase can be
found at github.com/ioannismavromatis/sustainable-ai.

###  4.1 Gathering Software-Based Energy Consumption Data

Monitoring energy consumption can be achieved through hardware or software
tools. Hardware methods offer precision [20] but face challenges in
synchronisation and control [21], particularly for brief measurements like
testing a shallow NN. These methods often require external clocks and costly
equipment, limiting accessibility for all ML practitioners. Our investigation
employs a software-based approach to measure energy consumption to overcome
these issues. This choice not only reduces cost and complexity but also
enhances consistency and scalability. Moreover, it allows for parallel
evaluations of multiple devices and facilitates testing in complex scenarios,
such as FL deployments.

In software-based measurements, power consumption is typically assessed in two
ways. The first method estimates power based on a hardware component’s Thermal
Design Power (TDP) and its utilisation (in a linear relationship). TDP,
measured in Watts ($\text{\,}\mathrm{W}$), indicates the maximum power
consumption under theoretical full load. However, this method oversimplifies
the relationship between power consumption and utilisation [22], as modern
hardware can dynamically adjust the frequency and deactivate entire cores to
save energy. A more nuanced approach is based on the hardware’s capacitance
$C$, voltage $V$, and frequency $f$, as $P=\nicefrac{{1}}{{2}}\ CV^{2}f$, but
obtaining these values for all components is rather challenging.

As a workaround, manufacturers offer a solution by providing access to energy
data through Model Specific Registers (MSRs), like Nvidia’s Management Library
(NVML) for GPUs and Intel’s Running Average Power Limit (RAPL) for CPU and
DRAM usage. These methods are reliable with a reported variance of about
$\pm$5\text{\,}\mathrm{W}$$ in absolute values while following consistent
trends in relative measurements [23, 24]. For consumer CPUs that MSRs do not
provide DRAM measurements, DRAM’s energy consumption is approximated using
$P_{\mathrm{DRAM}}=\sum N_{\mathrm{DIMM}}\times P_{\mathrm{DIMM}}$, where
$N_{\mathrm{DIMM}}$ is the number of DIMMs and
$P_{\mathrm{DIMM}}=\nicefrac{{1}}{{2}}\ CV^{2}f$. The operational $V$ and $f$
are accessible from the OS, and $C$ is fixed for all our experiments. This
equation is a good approximation as voltage variations during DRAM operations
are almost negligible, and operational frequency does not change over time
[25].

###  4.2 Calculating Energy Usage in Machine Learning Processes

As discussed, our investigation will focus on the Experimentation, Training
and Inference phases. Training and Experimentation phases are very similar (a
model is trained using a set of preconfigured hyperparameters) and thus can be
approached similarly in our investigation. To measure the energy consumption
we define two metrics, i.e., $E_{\mathrm{tr}}$, which is the total energy
consumed during one training session (i.e., for a given model and dataset,
with a pre-defined set of hyperparameters and a fixed number of epochs), and
$E_{\mathrm{in}}$, which is the total energy during inference (i.e., for a
given model and dataset, inferring across all samples with a given batch
size). They are as follows:

$E_{\mathrm{tr}}=\int^{T_{\mathrm{tr}}}_{t=0}P_{\mathrm{tr}}(t)\,dt-\int^{T_{\mathrm{idl}}}_{t=0}P_{\mathrm{idle}}(t)\,dt$
(1)
$E_{\mathrm{in}}=\int^{T_{\mathrm{in}}}_{t=0}P_{\mathrm{in}}(t)\,dt-\int^{T_{\mathrm{id}}}_{t=0}P_{\mathrm{idle}}(t)\,dt$
(2)

where $T_{\mathrm{tr}}$ and $T_{\mathrm{in}}$ are the training and inference
times, $T_{\mathrm{id}}$ is a hardcoded time interval used for the idle
experiment, and $P_{\mathrm{tr}}$, $P_{\mathrm{in}}$ and $P_{\mathrm{id}}$ are
the power measurements during training, testing and when the system is idle.
Our framework captures the power consumption at frequent intervals $\Delta t$.
Denoting $t_{i}$ as the $i$-th time interval, the power $P(t_{i})$ (this could
be either for training or inference) is:

$P(t_{i})=P_{\mathrm{CPU}}(t_{i})+P_{\mathrm{GPU}}(t_{i})+P_{\mathrm{DRAM}}(t_{i})$
(3)

where $P_{\mathrm{CPU}}$, $P_{\mathrm{GPU}}$ and $P_{\mathrm{DRAM}}$ are the
power consumption, taken in real-time for the CPU, GPU and DRAM, respectively.
The energy within $i$-th interval can be calculated as the
$E(t_{i})=P(t_{i})\,\Delta t$. Based on that, the Eqs. (1) and (2) can be
approximated with the cumulative sum of all intervals, i.e.:

$E_{\mathrm{tr}}=\sum^{N_{\mathrm{tr}}}_{i=0}P_{\mathrm{tr}}(t_{i})\,\Delta
t-\sum^{N_{\mathrm{id}}}_{t=0}P_{\mathrm{id}}(t_{i})\,\Delta t$ (4)
$E_{\mathrm{in}}=\sum^{N_{\mathrm{in}}}_{t=0}P_{\mathrm{in}}(t_{i})\,\Delta
t-\sum^{N_{\mathrm{id}}}_{t=0}P_{\mathrm{id}}(t_{i})\,\Delta t$ (5)

where $N_{\mathrm{tr}}$, $N_{\mathrm{in}}$ and $N_{\mathrm{id}}$ are the total
number of intervals during training, inference, or idle, respectively. As
discussed, data exchange and processing, even though they play a significant
role in the energy consumed, will not be considered.

Table 1: Hardware Configurations (HCs). In brackets is the TDP for each
hardware component.

| HC-1 | HC-2 | HC-3
---|---|---|---
CPU∗ | i7-8700K ($95\text{\,}\mathrm{W}$) | i9-11900KF ($125\text{\,}\mathrm{W}$) | i5-12500 ($65\text{\,}\mathrm{W}$)
DRAM |  $4\mathrm{x}$16\text{\,}\mathrm{GB}$$ DDR4 |  $4\mathrm{x}$32\text{\,}\mathrm{GB}$$ DDR4 |  $2\mathrm{x}$16\text{\,}\mathrm{GB}$$ DDR5
$3600\text{\,}\mathrm{MHz}$ | $3200\text{\,}\mathrm{MHz}$ | $3200\text{\,}\mathrm{MHz}$
GPU+ | RTX 3080 ($320\text{\,}\mathrm{W}$) | RTX 3090 ($350\text{\,}\mathrm{W}$) | RTX A2000 ($70\text{\,}\mathrm{W}$)
$10\text{\,}\mathrm{GB}$ | $24\text{\,}\mathrm{GB}$ | $12\text{\,}\mathrm{GB}$

∗Intel Core, +Nvidia driver v530.30.02, CUDA v12.1

(a) Duration for HC-3.

(b) Duration for HC-2.

Figure 2: Training and inference duration (for $50$k samples).

###  4.3 Hardware Stats and Model Characteristics

Our framework collects utilisation statistics for all resources and model
characteristics. The NVML library provides the GPU (and its VRAM) utilisation.
For the CPU, the utilisation metrics were directly collected from the OS as a
function of each CPU core. The CPU utilisation is calculated as the average
utilisation at a given time between all cores. Similarly, DRAM’s utilisation
was also provided by the OS.

Concerning the model features, several key characteristics emerge as critical
indicators when considering the operational efficiency of the model. These
include the model size, the number of total and trainable parameters, buffer
size, and MACs. The model size, measured in bytes ($\text{\,}\mathrm{B}$), is
calculated when the model is decompressed and loaded in the VRAM. It includes
both the parameters and buffers and represents the overall footprint of the
model in memory.

The total number and the trainable parameters are key indicators of a model’s
complexity. These parameters are different when layers in the model are frozen
(i.e., are not updated). A larger number of parameters typically implies a
more complex model, which can potentially achieve higher accuracy but at the
cost of increased computational resources and memory usage. This complexity
can lead to longer training times and may require more powerful hardware.

The buffer size indicates the additional data structures often used for
storing intermediate outputs and constants that do not change during training,
such as batch normalisation parameters. While they do not directly contribute
to the model’s learning capacity, they impact the overall memory footprint. A
large buffer size can lead to inefficiencies in systems with limited memory.

Finally, the MAC is the fundamental operation in NNs, especially in
convolutional layers. The number of MACs provides an estimate of the
computational complexity of the model. Higher MACs generally indicate
increased computation for both training and inference, leading to longer
processing times and increased energy consumption. All the above-mentioned
model characteristics are calculated when the model is loaded in the GPU
before the execution of each experiment. For our investigation, either
independently or as a combination, these parameters will be explored towards
total energy consumption.

##  5 Results

For our investigation, we consider a simple image classification task. This
task was chosen due to the ample models and datasets available in the
literature. We conducted a thorough investigation to observe the behaviours of
different ML models. In brackets, we present the model variant chosen for our
experimentation. We picked: SimpleDLA, DPN (26), DenseNet (121), EfficientNet
(B0), GoogLeNet, LeNet, MobileNet, MobileNetV2, PNASNet, PreActResNet (18),
RegNet (X_200MF), ResNet (18), ResNeXt (29_2x64d), SENet (18), ShuffleNetV2,
and VGG (16), to capture a diverse range of architectures and sizes. All
experiments were conducted with the same hyperparameters (batch size of $128$,
learning rate $0.001$, SGD optimiser, categorical cross-entropy loss and
weight decay $5\times 10^{-4}$). We also fixed the seed to ensure consistency
across different runs.

We used three different Hardware Configurations (HCs) summarised in Tab. 1.
These three HCs provide diverse playgrounds to explore and identify their
differences or similarities and the correlations (Pearson $r$ and Spearman
$\rho$) of the different values. As the space in the paper is limited, we will
present a subset of the results and discuss the rest in the text. All results
can be found in our GitHub repository for further analysis.

Our investigation was based on the CIFAR-10 dataset [26]. The dataset consists
of $60000$ $32\times 32$ colour images in 10 classes, with $6000$ images per
class. The split between the training and testing set is $50000:10000$. For
our evaluation, we replicated the testing set 5 times (i.e., to $50$k
samples), so there are consistent samples between training and testing.

###  5.1 Initial Statistics

Starting with the maximum accuracy reached, most models achieved around
$87\%-91\%$ after 100 epochs. The shallower LeNet underperformed as expected,
reaching only around $68\%$, whereas MobileNet and EfficientNet reached $81\%$
and $83\%$, respectively. Comparing the time required for training and
inference (one epoch of training and $50$k samples of inference), we see the
results in Fig. 2. For most models, training takes three times longer than
inference due to backpropagation and parameter updating ($r\approx 0.9$ across
all models and all HCs). However, as seen, models like DPN, RegNet, etc. do
not adhere to this rule of thumb.

(a) Power usage by model (training).

(b) Power usage by model (inference).

Figure 3: Average power usage with HC-2.

Across different HCs, we observe large differences in the duration of the same
models. For example, PreActResNet at HC-2 (Fig. 2(b)) requires about
$5\mathrm{x}$ more time to train or infer compared to LeNet, but at HC-3 (Fig.
2(a)), that difference goes up to $26\mathrm{x}$. Moreover, even though there
is no observable difference during the training phase in terms of time
required (relatively – between models), for inference, we observe that a more
powerful GPU (HC-2), when fed relatively small models, parses the same number
of samples in about the same time, regardless of the model size. With the
inference dictating the energy consumption (as discussed in Sec. 3.1), as a
rule of thumb, models achieving similar accuracy but inferring quicker can
introduce significant energy benefits in the long term, even if they require
more time to train. For example, VGG and ResNet perform equally as well as
DenseNet or DPN, but only with a fraction of the energy required, making them
better candidates for long-term usage.

###  5.2 Power Consumption Measurements

Fig. 3 shows the average power consumed for HC-2 for training and inference.
All the larger models force the GPU to operate close to its TDP (Fig. 3(a)).
As expected, CPU and DRAM, being underutilised, show roughly equal and not
significantly high average power consumption across all models. However, this
is not the case for the inference (Fig. 3(b)). As shown, many models operate
$\geq 30\%$ lower than the GPU’s TDP (e.g., VGG), with the CPU and DRAM
showing similar results with the training. This is the case for the other two
HCs, with the difference being more prominent for HC-1 and less prominent for
HC-3.

Given that CPU and DRAM usage do not change drastically across different
models, in Fig. 4, we present an example of the power consumption as a
function of the utilisation and the usage of the GPU’s VRAM. For training, a
larger GPU VRAM use reflects, most of the time, higher utilisation and
increased power consumption. This is even more prominent during inference.
Moreover, the results indicate a high correlation between utilisation and
power consumption, but up to a certain point (e.g., $\rho\approx 0.81$ for
HC-3, $\rho\approx 0.55$ for HC-2). Beyond a power draw of
~$300\text{\,}\mathrm{W}$, any further increase did not translate to a
dramatic increase in the GPU utilisation. At this point, utilisation was
almost $100\%$, so performance was pretty much at its maximum. This is more
clear in Fig. 4(a), where, as said earlier, most models push the GPU to
operate close to its TDP. Our results in this investigation are consistent
with our previous work [27].

(a) Power usage by model (training).

(b) Power usage by model (inference).

Figure 4: Utilisation and power consumption (considering the GPU RAM usage) -
HC-1. Figure 5: Loss, energy and accuracy per epoch, averaged across all
models - the shaded areas show the range of values - HC-3.

We observe a linear relationship by investigating the time and energy
consumption, with $r=0.99$ (e.g., per epoch – for training, or per X number of
samples – for inference). Our results for that can be found in our repository.
However, comparing the model loss, accuracy, and total energy accumulated as
the number of epochs increases (average across all models while training –
Fig. 5), even though there is no correlation between accuracy and total energy
consumed, as the number of epochs increases, the range of values observed for
the energy, is greater (relatively) to the accuracy, thus replacing a model
can significantly benefit the energy consumption with no significant cost in
the accuracy.

(a) During the training phase.

(b) During the inference phase.

Figure 6: Total energy consumption as a function of the MACs per parameter -
HC-3.

MAC is usually a standard metric that one can use to calculate the complexity
of a model and, thus, the expected energy consumption. Comparing the MACs of
each model as a function of the total energy, we see a high correlation
between them, with $\rho\approx 0.8$ across all HCs (results in our
repository). However, our investigation showed that combining MACs with the
model (i.e., MACs per model parameter - Fig. 6) parameters provides a better
metric for that. For both training (Fig. 6(a)) and inference (Fig. 6(b)), we
see a strong correlation across them ($\rho\approx 0.9$ across all HCs).

Finally, comparing different batch sizes for training and inference (Fig. 7),
we see that smaller batch sizes increase the power consumption. There is a
direct correlation with the GPU utilisation for each model. For all setups,
there is a batch size that minimises the power consumption, with no further
improvements shown if the batch size is increased. Considering that smaller
batch sizes achieve higher accuracy [28], there is a tradeoff between the
accuracy and the energy consumption that can be further investigated.

Figure 7: Total energy consumption as a function of the batch size - HC-2.

##  6 Discussion

Starting with our initial observations (Sec. 5.1), it is clear that each
model’s unique architecture does not allow room for cross-model observations
(i.e., if a model’s energy consumption is low, there is no obvious way to say
that another model will have an equally low energy consumption). Further
investigation of the specific architectures and model layers and how they
affect energy consumption could identify more similarities. This could be
considered a future activity.

Broadly speaking (Fig. 5), benefits in energy reduction can outperform the
gains in accuracy on many occasions. Moreover, training and inference
durations are not correlated; thus, cross-phase or cross-hardware estimations
are not promising. Even though a rule of thumb could say that training will
require three times more time for the same number of samples, this is not
always the case.

Considering that time and total energy are linear, short-living profilers
(e.g., training for one epoch or inferring for a small number of samples) can
be used to extrapolate the total energy for larger scenarios. In addition, for
accuracy and duration, it was evident that models achieving comparable
accuracy but “running faster” can introduce huge energy benefits in the long
term. From what was shown in Fig. 3 and taking into account Facebook’s energy
split presented in Sec. 3.1, a less power-hungry model during inference should
be prioritised for a pipeline over a less energy-intensive model during
training. This observation can be combined with the total energy and time to
get even more accurate estimations for both training and inference across
different models. Moreover, strategies that analyse the learning curves (from
the initial epochs) in conjunction with the power could estimate the expected
total energy consumption, identifying models that better fit a given scenario.

As shown in Fig. 4, hardware devices’ power profiles are not exactly linear.
Usually, manufacturers push their devices to the limit to squeeze a narrow
increase in performance. Smart strategies like the one introduced in [27]
(power capping optimisations) can exploit that and significantly reduce the
total energy consumed. Finally, if an estimation of the model’s expected
energy is required, contrary to the literature that proposes using the model’s
MACs, we identified the MACs per model parameter as a more suitable candidate.
Similar correlations are observed across all setups, proving it is a uniform
solution for accurately estimating the expected energy consumption.

##  7 Conclusions

This work presented an extensive analysis across multiple ML models and
hardware setups to uncover techniques for improving sustainability without
sacrificing effectiveness – the investigation methodology combined software-
based power measurements with tracking of hardware utilisation and model
characteristics. The experiments demonstrated that for many models, reductions
in energy consumption can outpace marginal accuracy improvements, highlighting
the need to balance performance and efficiency. Additionally, assumptions
about energy use cannot be reliably made across training and inference or
hardware due to a lack of consistent correlations. However, normalising model
MACs by the number of parameters provides an excellent indicator of energy
consumption in most cases. The insights from this study can guide decisions
when constructing ML pipelines, whether choosing architectures and
hyperparameters or provisioning hardware resources. There remains ample
opportunity for future work to improve sustainability through novel
architectures optimised for efficiency and adopting best practices around
selective retraining, power capping, and inference-focused model selection.
Overall, the evidence clearly shows that with careful planning, ML can
continue advancing while aligning with environmental responsibility.

## Acknowledgment

This work is a contribution by Project REASON, a UK Government funded project
under the Future Open Networks Research Challenge (FONRC) sponsored by the
Department of Science Innovation and Technology (DSIT). This work was also
funded in part by Toshiba Europe Ltd. and Bristol Research and Innovation
Laboratory (BRIL).

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# Generic regularity of conservative solutions to the Novikov equation

Zhen $\mbox{He}^{1}$<EMAIL_ADDRESS>Wei
<EMAIL_ADDRESS>and Zhaoyang
<EMAIL_ADDRESS>
${}^{1}\mbox{Department}$ of Mathematics, Sun Yat-sen University, Guangzhou
510275, China
${}^{2}\mbox{School}$ of Science,
Shenzhen Campus of Sun Yat-sen University, Shenzhen 518107, China

###### Abstract

In this paper, we concentrate on the Novikov equation. For an open dense set
of $C^{3}$ initial data, we prove that the solution is piecewise smooth in the
t-x plane, while the gradient $uu_{x}$ can blow up along finitely many
characteristic curves. And we provide a description of the solution in a
neighborhood of each singular point.
2010 Mathematics Subject Classification: 35B65, 35D30, 35F25, 35L03, 35L60.

Keywords: Novikov equation; generic regularity; conservative weak solution;
singularity

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3  Families of perturbed solutions.
  4. 4 Generic property
    1. 4.1 Generic solutions of the semilinear system
    2. 4.2 Proof of Theorem 1.1
  5. 5 Generic singularity behavior
    1. 5.1 Proof of Theorem 1.2

##  1 Introduction

Consideration here is the initial-value problem for the Novikov equation in
the form

(1.1)
$\left\\{\begin{array}[]{ll}u_{t}-u_{xxt}+4u^{2}u_{x}=3uu_{x}u_{x}x+u^{2}u_{xxx},\quad
x\in\mathbb{R},~{}t>0,\\\\[4.30554pt] u|_{t=0}=u_{0},\quad
x\in\mathbb{R}.\\\\[4.30554pt] \end{array}\right.$

The equation was proposed by Novikov in [16]. In [8], Chen, Hu and Liu showed
that the Novikov equation can be viewed as a shallow water model.

The Novikov equation (1.1) can be rewritten as the following compact form

(1.2) $m_{t}+u^{2}m_{x}+\frac{3}{2}(u^{3})_{x}m=0,~{}~{}~{}m=u-u_{xx}.$

There is a huge literature devoted to the equations (1.1), local well-
posedness can be referred to [11, 12, 14, 20, 21, 22, 23]. Yan, Li and Zhang
[22, 23] present two blow-up results under some suitable conditions. Wu and
Yin [19] proved the global existence of weak solutions of the equation (1.1),
provided the initial data satisfy certain sign conditions. In [20], they also
obtained global strong solutions under some conditions. Chen, Zhang and Liu
[7] proved the existence and uniqueness of conservative solutions for the
Novikov equation in $H^{1}\cap W^{1,4}$.

Another related shallow water model is the Camassa-Holm (CH) equation

(1.3) $m_{t}+um_{x}+2u_{x}m=0,~{}~{}~{}m=u-u_{xx}.$

Recently, Li and Zhang [15] proved the generic property and the singular
behavior of the Camassa-Holm equation and the two-component Camassa-Holm
equation. The generic property is a property which is satisfied by almost all
elements of the whole set. Considering our PDE problem here, generic
regularity is the regularity of the solutions solved from an open and dense
subset in the space of initial data. Generic property is an interesting
problem in hyperbolic conservation laws. The original result was studied by
Schaeffer in [17], which showed that for the one space dimensional
conservation law, the generic solutions are piecewise smooth, with finitely
shocks in a bounded domain in the $(t,x)$ plane. The proof relies on the Hopf-
lax representation formula. For the $2\times 2$ temple class systems, Dafermos
and Geng proved a similar result in [9] by analyzing the solution along the
characteristics. For the general $n\times n$ system with $n\geq 3$, Caravenna
and Spinolo [6] proved that the generic property does not hold true.

Recently, Bressan and his collaborators [2, 3, 4] studied generic property and
singularity behavior for the variational wave equation. In [2], the authors
showed that for an open dense set of $C^{3}$ initial data, the solution is
piecewise smooth in the $(t,x)$ plane, while the gradient $u_{x}$ will blow up
along finite many characteristic curves. For an open dense set of initial
data, the authors in [3] provided a detailed asymptotic description of the
solution in a neighborhood of each singular point, where
$|u_{x}|\rightarrow\infty$ and analysed the different structure of
conservative and dissipative solutions. Yang [24] studied generic regularity
of energy conservative solutions to the rotation Camassa-Holm equation. Cai,
Chen, Shen and Tan [5] studied the generic property of conservative solutions
to the Hunter-Saxton type equations and give a new way to construct a Finsler
type metric which renders the flow uniformly Lipschitz continuous on bounded
subsets of $H^{1}(\mathbb{R^{+}})$.

To the best of our knowledge, the generic properties of the Novikov equation
has not been studied yet. In [7], the authors pointed out that the singularity
for the conservative solutions of the Novikov equation is different from the
Camassa-Holm equation. The wave speed $c(u)$ for the Camassa-Holm equation is
$u$ and for the Novikov equation is $u^{2}$. Because of the difference
nonlinearity of wave speed, the singularity behaviour is various. In this
paper, we give an exactly proof to show that why the singularity is various
between the Camassa-Holm equation and the Novikov equation.

The content of this paper is the following. In Section 2, we recall some basic
definitions and the related results about the conservative solutions the
Noivkov equation. Section 3 presents a perturbation lemma. In Section 4, we
study about the generic property of the Novikov equation. Section 5 devotes to
showing the asymptotic behavior for generic singularity.

We state our main results as follows

######  Theorem 1.1.

(Generic property) For any $T\textgreater 0$ fixed,there exists an open dense
set of initial data

$\mathscr{D}\subset(C^{3}(\mathbb{R})\cap H^{1}(\mathbb{R})\cap
W^{1,4}(\mathbb{R}))$

such that for $u_{0}\in\mathscr{D}$, the conserved solution u=u(t,x) of the
Novikov equation (2.4) is differentiable in the complement of finitely many
characteristic curves $\gamma_{i}$, within the domain $[0,T]\times\mathbb{R}$.

From the above generic regularity, we can obtain the asymptotic description of
the solution in a neighborhood of each singular point, where
$|u_{x}|\rightarrow\infty$.

######  Theorem 1.2.

Consider generic initial data $u_{0}\in\mathscr{D}$ as in Theorem 1.1 with
$u_{0}\in C^{\infty}(\mathbb{R})$. Call $(u,v,\xi,x,t)$ the corresponding
solution of the semilinear system (2.17) and let u=u(x,t) be the solution to
the original equation (2.4). Consider a singular point $P=(t_{0},Y_{0})$ where
v=$\pi$, and set $(x_{0},t_{0})=(x(t_{0},Y_{0}),t(t_{0},Y_{0}))$. Generically,
at the singular point, u has following parameteric expression.

  * \scriptsize{1}⃝

If P is a point of Type $\mathcal{I}$ , i.e.$v_{Y}=0$ and $v_{YY}\neq 0$
$v=\pi,~{}~{}v_{YY}\neq 0$ then

(1.4)
$u(t,x)=A(x-x_{0})^{\frac{3}{4}}+B(t-t_{0})+\mathcal{O}(1)(|t-t_{0}|^{2}+|x-x_{0}|^{\frac{7}{8}})$

for some constant $A,B$.

  * \scriptsize{2}⃝

If P is a point of Type $\mathcal{I}\mathcal{I}$ , i.e.$v_{Y}\neq 0$ and
$v_{YY}=0$ $v=\pi,~{}~{}v_{YY}\neq 0$ then

(1.5)
$u(t,x)=A(x-x_{0})^{\frac{4}{5}}+B(t-t_{0})+\mathcal{O}(1)(|t-t_{0}|^{2}+|x-x_{0}|)$

for some constant $A,B$.

##  2 Preliminaries

In this section, we give some basic definitions, celebrated Thom’s
transversality theorem and an useful lemma. Then we recall some basic
definitions and results for the Novikov equation. The content of this section
can be found in many books and monographs [1, 10, 18, 13].

######  Definition 2.1.

[1, 10, 18, 13] (Map transverse to a submanifold) Let F: X$\rightarrow$Y be a
smooth map from manifold X to manifold Y. W is a submanifold of Y. We say F is
transverse to W at a point x$\in$X, denoted by F $\pitchfork_{x}$W, if

$\bullet$ either $F(x)\notin W$

$\bullet$ or $F(x)\in W$and $T_{F(x)}Y=(dF)_{x}(T_{x}X)+T_{F(x)}W$ Here
$T_{x}X$ means the tangent space of X at point x.

If F $\pitchfork_{x}$W for every $x\in X$, we say F is transverse to W , and
denote as F $\pitchfork$W

######  Definition 2.2.

[1, 10, 18, 13] Let $F:X\rightarrow Y$ be a smooth map from manifold $X$ to
$Y$. A point $y\in Y$ is a regular value if for every $x\in X$ one has

$T_{y}Y=(dF)_{x}(T_{x}X).$

In the special case where $W=\\{y\\}$ consists of a single point, $F\pitchfork
W$ if and only if $y$ is a regular value of F

######  Theorem 2.3.

[1, 10, 18, 13] (Transversality lemma) Let $X$, $\Theta$ and $Y$ be smooth
manifolds and $W$ a submanifolds of $Y$. Let $\theta\rightarrow\phi^{\theta}$
be a smooth map that to each $\theta\in\Theta$ associates a function
$\phi^{\theta}\in C^{\infty}(X,Y)$, and define $\Phi:X\times\Theta\rightarrow
Y$ by setting $\Phi(x,\theta)=\phi^{\theta}(x)$. If $\Phi\pitchfork W$ then
the set ${\theta\in\Theta;\phi^{\theta}\pitchfork W}$ is dense in $\Theta$.

######  Lemma 2.4.

[15] Consider an ODE system

(2.1) $\frac{d}{dt}u^{\epsilon}=f(u^{\epsilon}),\quad
u^{\epsilon}(0)=u_{0}+\epsilon_{1}v_{1}+...+\epsilon_{m}v_{m},$

where $u^{\epsilon}(t):\mathbb{R}\rightarrow\mathbb{R}$, $f$ is a Lipschitz
function. The system is well posed in [0,T). Assume the matrix

(2.2) $\displaystyle
D_{\epsilon}u_{0}^{\epsilon}=(v_{1},v_{2},...,v_{m})\in\mathbb{R}^{n\times
m},$

and the rank of this matrix is

(2.3) $\displaystyle rank(D_{\epsilon}u_{0}^{\epsilon})=k.$

Then for any $t\in[0,T)$, $rank(D_{\epsilon}u^{\epsilon})=k$.

Next we will recall some basic results about the Novikov equation. The
equation (1.1) can be written into

(2.4)
$\left\\{\begin{array}[]{l}u_{t}+u^{2}u_{x}+\partial_{x}P_{1}+P_{2}=0,\\\
u(0,x)=u_{0}(x),\quad x\in\mathbb{R},\end{array}\right.$

where

$P_{1}\triangleq p\ast(\frac{3}{2}uu_{x}^{2}+u^{3})\quad\textit{and}\quad
P_{2}\triangleq\frac{1}{2}p\ast u_{x}^{3}$

with $p(x)=\frac{1}{2}e^{-|x|}$, and we define the weak solution as follows.

######  Definition 2.5.

[7] The energy conservative solution $u=u(t,x)$ of (2.4) satisfies
1\. For any fixed t$\geq 0$,u(t,x)$\in H^{1}(\mathbb{R})\cap
W^{1,4}(\mathbb{R})$. The map $t\rightarrow u(t,\cdot)$ is Lipschitz
continuous under the $L^{4}$ metric.
2\. The solution u=u(t,x) satisfies the initial condition (2.4) in
$L^{4}(\mathbb{R})$, and

$\int\int_{\Lambda}{-u_{x}(\phi_{t}+u^{2}\phi_{x})+(-\frac{3}{2}uu_{x}^{2}-u^{3}+P_{1}+\partial_{x}P_{2})\phi}dxdt+\int_{\mathbb{R}}u_{0,x}\phi(0,x)dx=0$

for every test function $\phi\in C_{c}^{1}(\Lambda)$ with
$\Lambda=\\{(t,x)|t\in[0,\infty),x\in\mathbb{R}\\}$.
3\. The solution $u=u(t,x)$ is conservative if the balance law is satisfied in
the following sense:
There exists a family of Radon measures $\\{\mu_{(t)},t\in\mathbb{R}\\}$,
depending continuously on time and w.r.t the topology of weak convergence of
measures. For every $t\in\mathbb{R}^{+}$, the absolutely continuous part of
$\mu(t)$ w.r.t. the Lebesgue measure has density $u_{x}^{4}(t,\cdot)$, which
provides a measure-valued solution to the balance law

$\int_{\mathbb{R^{+}}}\\{\int(\phi_{t}+u^{2}\phi_{x})d\mu(t)+\int\big{(}4u^{3}u_{x}^{3}-4u_{x}^{3}(P_{1}+\partial_{x}P_{2})\phi
dx\big{)}\\}dt-\int_{\mathbb{R}}u_{0,x}^{4}\phi(0,x)=0$

for every test function $\phi\in C_{c}^{1}(\Lambda)$.

######  Theorem 2.6.

[7] Let $u_{0}\in H^{1}(\mathbb{R})\bigcap W^{1,4}(\mathbb{R})$ be an absolute
continuous function on x. Then the initial value problem (2.4) admits a unique
energy conservative solution $u(t,x)$ defined for all
$(t,x)\in\mathbb{R^{+}}\times\mathbb{R}$. The solution also satisfies the
following properties.

  * (1)

$u(t,x)$ is Hölder continuous with exponent $\frac{3}{4}$ on both $t$ and $x$
.

  * (2)

The first energy density $u^{2}+u_{x}^{2}$ is conserved for any time $t\geq
0$, i.e.

$\mathscr{E}(t)=\|u(t)\|_{H^{1}}^{2}=\|u(t)\|_{H^{1}}^{2}.$

  * (3)

The second energy density $u^{4}+2u^{2}u_{x}^{2}-\frac{1}{3}u_{x}^{4}$ is
conserved in the following sense:

    * (i)

An energy inequality is satisfied in $(t,x)$ coordinates:

$\mathscr{F}(t)=\int_{\mathbb{R}}(u^{4}+2u^{2}u_{x}^{2}-\frac{1}{3}u_{x}^{4})dx\geq\mathscr{F}(0)$

for any $t\geq 0$.

    * (ii)

Denote a family of Radon measures ${\nu(t),t\in\mathbb{R^{+}}}$, such that

$\nu_{(t)}(\mathscr{A})=\int_{\mathscr{A}}(u^{4}+2u^{2}u_{x}^{2})(t,x)dx-\frac{1}{3}\mu_{(t)}(\mathscr{A})$

for any Lebesgue measurable set $\mathscr{A}$ in $\mathbb{R}$. Then for any
$t\in\mathbb{R^{+}}$,

$\nu_{(t)}=\mathscr{F}(0)=\int_{\mathbb{R}}(u^{4}+2u^{2}u_{x}^{2}-\frac{1}{3}u_{x}^{4})(0,x)dx$

For any $t\in\mathbb{R^{+}}$, the absolutely continuous part of $\nu_{(t)}$
w.r.t Lebeugue measure has density
$u^{4}+2u^{2}u_{x}^{2}-\frac{1}{3}u_{x}^{4}$. For almost every
$t\in\mathbb{R^{+}}$, the singular part of $\nu_{(t)}$ is concentrated on the
set where $u=0$.

  * (4)

A continuous dependence result holds. Consider a sequence of initial data
$u_{0,n}$ such that $\|u_{0n}-u_{0}\|_{H^{1}\bigcap W^{1,4}}$, as
n$\rightarrow\infty$.Then the corresponding solutions $u_{n}(t,x)$ converge to
$u(t,x)$ uniformly for $(t,x)$ in any bounded sets.

By virtue of (2.4), a direct computation shows that conservation law holds

(2.5)
$\displaystyle(\frac{u^{2}+u_{x}^{2}}{2})_{t}+(\frac{u^{2}u_{x}^{2}}{2}+uP_{1}+u\partial_{x}P_{2})_{x}=0.$

And it is not hard to check that

$\displaystyle(u^{4}$
$\displaystyle+2u^{2}u_{x}^{2}-\frac{1}{3}u_{x}^{4})_{t}+\big{[}2u^{4}u_{x}^{2}-\frac{1}{3}u^{2}u_{x}^{4}+\frac{4}{3}u^{3}(P_{1}+\partial_{x}P_{2})\big{]}_{x}$
(2.6)
$\displaystyle+\frac{4}{3}[(P_{1}+\partial_{x}P_{2})^{2}-(P_{2}+\partial_{x}P_{1})^{2}]_{x}=0.$

Thus two conserved quantities can be derived from (2.5) and (2)

(2.7)
$\displaystyle\mathscr{E}(t)\triangleq\int_{\mathbb{R}}(u^{2}+u_{x}^{2})(t,x)dx=\mathscr{E}(0),$
(2.8)
$\displaystyle\mathscr{F}(t)\triangleq\int_{\mathbb{R}}(u^{4}+2u^{2}u_{x}^{2}-\frac{1}{3}u_{x}^{4})(t,x)dx=\mathscr{F}(0).$

To bound $P_{i}(i=1,2)$, we may firstly observe that

(2.9) $\displaystyle\|u\|_{L^{\infty}}^{2}$
$\displaystyle\leq\|u\|_{H^{1}}^{2}=\mathscr{E}(0),$
$\displaystyle\|u_{x}\|_{L^{4}}^{4}$
$\displaystyle=3\int_{\mathbb{R}}(u^{4}+2u^{2}u_{x}^{2})dx-\mathscr{F}(t)$
$\displaystyle\leq
3\big{(}\|u\|_{L^{\infty}}^{2}\int_{\mathbb{R}}(u^{2}+2u_{x}^{2})-\mathscr{F}\big{)}$
(2.10) $\displaystyle\leq
3(2\mathscr{E}^{2}(t)-\mathscr{F}(t))=3(2\mathscr{E}^{2}(0)-\mathscr{F}(0)),$

and (2.9)-(2.10) imply that

(2.11)
$\displaystyle\|u_{x}\|_{L^{3}}^{3}\leq\sqrt{3\mathscr{E}(0)[2\mathscr{E}^{2}(0)-\mathscr{F}(0)])}\triangleq
K.$

Then we can bound $P_{i}$ and $\partial_{x}P_{i}$ for i=1,2 as follows.

(2.12)
$\displaystyle\begin{split}&\|P_{1}(t)\|_{L^{\infty}},\|\partial_{x}P_{1}(t)\|_{L^{\infty}}\leq\|p\|_{L^{\infty}}\|\frac{3}{2}uu_{x}^{2}+u^{3}\|_{L^{1}}\leq\frac{3}{4}\mathscr{E}^{\frac{3}{2}}(0),\\\
&\|P_{1}(t)\|_{L^{2}},\|\partial_{x}P_{1}(t)\|_{L^{2}}\leq\|p\|_{L^{2}}\|\frac{3}{2}uu_{x}^{2}+u^{3}\|_{L^{1}}\leq\frac{3}{2\sqrt{2}}\mathscr{E}^{\frac{3}{2}}(0),\\\
&\|P_{2}(t)\|_{L^{\infty}},\|\partial_{x}P_{2}(t)\|_{L^{\infty}}\leq\frac{1}{2}\|p\|_{L^{\infty}}\|u_{x}^{3}\|_{L^{1}}\leq\frac{1}{4}K,\\\
&\|P_{2}(t)\|_{L^{2}},\|\partial_{x}P_{2}(t)\|_{L^{2}}\leq\frac{1}{2}\|p\|_{L^{2}}\|u_{x}^{3}\|_{L^{1}}\leq\frac{1}{2\sqrt{2}}K.\end{split}$

Following the idea in [7], the characteristic equation is

(2.13) $\displaystyle\frac{dx(t)}{dt}=u^{2}(t,x(t)).$

If we consider $Y=Y(t,x)$ is a characteristic coordinate, and denote $T=t$.
Then we consider function $f(t,x)=f(T,x(t,Y))$ as a function of $(T,Y)$ also
denoted by $f(T,Y)$. It is easy to check that

(2.14) $\displaystyle Y_{t}+u^{2}Y_{x}=0,$ (2.15) $\displaystyle
f_{t}+u^{2}f_{x}=f_{Y}(Y_{t}+u^{2}Y_{x})+f_{t}(T_{t}+u^{2}T_{x})=f_{T}.$

We also denote

(2.16) $\displaystyle v=2\arctan
u_{x},~{}~{}~{}q=\frac{(1+u_{x}^{2})^{2}}{Y_{x}}.$

Then the conservative solution is constructed by the following semilinear
system

(2.17) $\left\\{\begin{array}[]{l}u_{T}=-\partial_{x}P_{1}-P_{2},\\\
v_{T}=-u\sin^{2}\frac{v}{2}+2u^{3}\cos^{2}\frac{v}{2}-2\cos^{2}\frac{v}{2}(P_{1}+\partial_{x}P_{2}),\\\
q_{T}=q[(2u^{3}+u)-2(P_{1}+\partial_{x}P_{2})]\sin v,\end{array}\right.$

with the initial condition

(2.18) $\left\\{\begin{array}[]{l}u(0,\beta)=u_{0}(x(0,\beta)),\\\
v(0,\beta)=2\arctan(u_{0}^{\prime}(x(0,\beta))),\\\
q(0,\beta)=1,\end{array}\right.$

for every $\beta\in\mathbb{R}$.

######  Lemma 2.7.

[7] Let $(u,v,q,Y)$ be the solution to (2.17) and (2.18), with $q>0$. Then the
set of points

(2.19)
$\displaystyle\\{(t,x(t,Y),u(t,Y));(t,Y)\in\mathbb{R}^{+}\times\mathbb{R}\\}$

is the graph of a conservative solution to the Novikov equation (2.4)

To study the singularities of the solution $u$ of (2.4), we should focus on
the level sets

$\\{v(t,Y)=\pi\\}.$

##  3  Families of perturbed solutions.

######  Lemma 3.1.

Let $(u,v,q)$ be a smooth solution of the semilinear system (2.17), and let a
point $(t_{0},T_{0})\in\mathbb{R^{+}}\times\mathbb{R}$ be given.
If $(v,v_{Y},v_{YY})(t_{0},Y_{0})=(\pi,0,0)$, then there exists a three
parameter family of smooth solutions $(u^{\theta},v^{\theta},\xi^{\theta})$
depending smoothly on $\theta\in\mathbb{R}^{3}$ such that

  * (i)

When $\theta=0\in\mathbb{R}^{3}$, one recovers the original solution namely
$(u^{0},v^{0})=(u,v)$;

  * (ii)

At the point $(t_{0},Y_{0})$ when $\theta=0$, one has

$rank~{}D_{\theta}(v^{\theta},v_{Y}^{\theta},v_{YY}^{\theta})=3$

.

###### Proof.

Let $(u,v,\xi)$ be the smooth solution of (2.17). And taking derivatives to
the equation of $v$, we obtain

$\displaystyle\frac{\partial}{\partial_{T}}v_{Y}$
$\displaystyle=-u_{Y}\sin^{2}(\frac{v}{2})-u\cos(\frac{v}{2})\sin(\frac{v}{2})v_{Y}+6u_{Y}u^{2}\cos^{2}(\frac{v}{2})-2u^{3}(\sin\frac{v}{2}\cos\frac{v}{2}v_{Y})$
$\displaystyle~{}~{}~{}+\cos(\frac{v}{2})\sin(\frac{v}{2})v_{Y}(P_{1}+\partial_{x}P_{2})-2\cos^{2}\frac{v}{2}(\partial_{Y}P_{1}+\partial_{Y}\partial_{x}P_{2})$
$\displaystyle=-u_{Y}\sin^{2}(\frac{v}{2})-\frac{1}{2}uv_{Y}\sin
v+6u_{Y}u^{2}\cos^{2}(\frac{v}{2})-u^{3}v_{Y}\sin v$ (3.1)
$\displaystyle~{}~{}~{}+\frac{1}{2}\cos vv_{Y}(P_{1}+\partial_{x}P_{2})-(\cos
v+1)(\partial_{Y}P_{1}+\partial_{Y}\partial_{x}P_{2}).$

Following the same line

(3.2)
$\displaystyle\begin{split}\frac{\partial}{\partial_{T}}v_{YY}=&u_{YY}\sin^{2}\frac{v}{2}+\frac{1}{2}u_{YY}v_{Y}\sin
v-\frac{1}{2}u_{Y}v_{Y}\sin v-uv_{YY}\sin v-\frac{1}{2}v_{Y}^{2}\cos
v-\frac{1}{2}\sin vv_{Y}^{2}(P_{1}+\partial_{x}P_{2})\\\
&~{}~{}~{}+6u_{YY}u^{2}\cos^{2}\frac{v}{2}+12u_{Y}^{2}u\cos^{2}\frac{v}{2}+3u_{y}v_{Y}u^{2}\sin
v-3u^{2}u_{Y}v_{Y}\sin v-u^{3}v_{YY}\sin v-u^{3}v_{Y}^{2}\cos v\\\
&~{}~{}~{}+\frac{1}{2}\cos
vv_{Y}(\partial_{Y}P_{1}+\partial_{Y}\partial_{x}P_{2})+\sin
vv_{Y}(\partial_{Y}P_{1}+\partial_{Y}\partial_{x}P_{2})-(\cos
v+1)(\partial_{Y}^{2}P_{1}+{\partial_{Y}}^{2}\partial_{x}P_{2})\\\
&=u_{YY}(\sin^{2}\frac{Y}{2}+\frac{1}{2}v_{Y}\sin
v+6u^{2}\cos^{2}\frac{v}{2}-u^{3}\sin v)+v_{Y}(-\frac{1}{2}u_{Y}\sin
v-\frac{1}{2}v_{Y}\cos v-u^{3}v_{Y}\cos v)\\\ &~{}~{}~{}-\frac{1}{2}\sin
vv_{Y}^{2}(P_{1}+\partial_{x}P_{2})+v_{Y}(\partial_{Y}P_{1}+\partial_{Y}\partial_{x}P_{2})(\frac{1}{2}\cos
v+\sin v)-(\cos
v+1)(\partial_{Y}\partial_{x}P_{1}+\partial_{Y}^{2}\partial_{x}P_{2}).\end{split}$
$\displaystyle\frac{\partial}{\partial_{t}}q_{Y}=$
$\displaystyle\xi_{Y}[(2u^{3}+u)-2(P_{1}+\partial_{x}P_{2})]\sin
v+q[(2u^{3}+u)-2(P_{1}+\partial_{x}P_{2})]\cos vv_{Y}$ (3.3)
$\displaystyle+q[(6u^{2}u_{Y}+u_{Y})-2(\partial_{Y}P_{1}+\partial_{Y}\partial_{x}P_{2})]\sin
v,$

with

$\displaystyle
P_{1}(Y)=\frac{1}{2}\int_{-\infty}^{+\infty}e^{-|\int_{\bar{Y}}^{Y}\xi\cos^{4}\frac{v}{2}(T,\tilde{Y})d\tilde{Y}|}(\frac{3}{8}u\sin^{2}v+u^{3}\cos^{4}\frac{v}{2})\xi(T,\bar{Y})d\bar{Y},$
$\displaystyle
P_{2}(Y)=\frac{1}{8}\int_{-\infty}^{+\infty}e^{-|\int_{Y}^{\bar{Y}}(\xi\cos^{4}\frac{v}{2})(T,\tilde{Y})d\tilde{Y}|}(\xi\sin
v\sin^{2}\frac{v}{2})(T,\bar{Y})d\bar{Y},$
$\displaystyle\partial_{x}P_{1}(Y)=\frac{1}{2}(\int_{Y}^{+\infty}-\int_{-\infty}^{Y})e^{-|\int_{\bar{Y}}^{Y}\xi\cos^{4}\frac{v}{2}(T,\tilde{Y})d\tilde{Y}|}(\frac{3}{8}u\sin^{2}v+u^{3}\cos^{4}\frac{v}{2})\xi(T,\bar{Y})d\bar{Y},$
(3.4)
$\displaystyle\partial_{x}P_{2}(Y)=\frac{1}{8}(\int_{Y}^{+\infty}-\int_{-\infty}^{Y})e^{-|\int_{Y}^{\bar{Y}}(\xi\cos^{4}\frac{v}{2})(T,\tilde{Y})d\tilde{Y}|}(\xi\sin
v\sin^{2}\frac{v}{2})(T,\bar{Y})d\bar{Y}.$

And we have identities about $Y$ derivatives as follows,

$\displaystyle\partial_{Y}P_{i}$
$\displaystyle=\xi\cos^{4}\frac{v}{2}\partial_{x}P_{i},~{}~{}~{}~{}~{}~{}i=1,2,$
$\displaystyle\partial_{Y}\partial_{x}P_{2}$ $\displaystyle=-\frac{1}{4}\sin
v\cos^{2}\frac{v}{2}\xi+\xi\cos^{4}\frac{v}{2}P_{2},$ (3.5)
$\displaystyle\partial_{Y}^{2}\partial_{x}P_{2}=\xi^{2}\cos^{8}\frac{v}{2}(-\frac{3}{2}$
$\displaystyle
u_{x}^{2}u_{xx}+\partial_{x}P_{2})=\xi^{2}\cos^{8}\frac{v}{2}(-\frac{3}{16}v_{Y}q\sin^{2}\frac{v}{2}+\partial_{x}P_{2}).$

We now construct families
($\bar{u}^{\theta},\bar{v}^{\theta},\bar{q}^{\theta}$) of perturbations of the
initial data along the charactristic as

(3.6)
$\displaystyle\bar{u}^{\theta}(Y)=\bar{u}(Y)+\sum_{i=1,2,3}\theta_{i}U_{i}(Y),$
(3.7)
$\displaystyle\bar{v}^{\theta}(Y)=\bar{v}(Y)+\sum_{i=1,2,3}\theta_{i}V_{i}(Y),$
(3.8)
$\displaystyle\bar{q}^{\theta}(Y)=\bar{q}(Y)+\sum_{i=1,2,3}\theta_{i}Q_{i}(Y).$

Therefore, an ODE system of dimension 5 can be constrcuted as follows:

(3.9) $\displaystyle\frac{\partial}{\partial t}\begin{pmatrix}u\\\ v\\\ q\\\
v_{Y}\\\ v_{YY}\\\ \end{pmatrix}=\begin{pmatrix}-\partial_{x}P_{1}-P_{2}\\\
-u\sin^{2}\frac{v}{2}+2u^{3}\cos^{2}\frac{v}{2}-2\cos^{2}\frac{v}{2}(P_{1}+\partial_{x}P_{2})\\\
q[(2u^{3}+u)-2(P_{1}+\partial_{x}P_{2})]\sin v\\\ A\\\ B\end{pmatrix},$

with $A\triangleq-u_{Y}\sin^{2}(\frac{v}{2})-\frac{1}{2}uv_{Y}\sin
v+6u_{Y}u^{2}\cos^{2}(\frac{v}{2})-u^{3}v_{Y}\sin v+\frac{1}{2}\cos
vv_{Y}(P_{1}+\partial_{x}P_{2})-(\cos
v+1)(\partial_{Y}P_{1}+\partial_{Y}\partial_{x}P_{2})$ and $B\triangleq
u_{YY}(\sin^{2}\frac{Y}{2}+\frac{1}{2}v_{Y}\sin
v+6u^{2}\cos^{2}\frac{v}{2}-u^{3}\sin v)+v_{Y}(-\frac{1}{2}u_{Y}\sin
v-\frac{1}{2}v_{Y}\cos v-u^{3}v_{Y}\cos v)-\frac{1}{2}\sin
vv_{Y}^{2}(P_{1}+\partial_{x}P_{2})+v_{Y}(\partial_{Y}P_{1}+\partial_{Y}\partial_{x}P_{2})(\frac{1}{2}\cos
v+\sin v)-(\cos
v+1)(\partial_{Y}\partial_{x}P_{1}+\partial_{Y}^{2}\partial_{x}P_{2})$.

We construct a family of solutions
$(\bar{u}^{\theta},\bar{v}^{\theta},\bar{\xi}^{\theta})$ to (3) of
perturbations of the initial data as in (3.6)-(3.8). Take derivative w.r.t.
$\theta$,

(3.10) $\displaystyle\frac{\partial}{\partial
t}\begin{pmatrix}D_{\theta}{u}^{\theta}\\\ D_{\theta}{v}^{\theta}\\\
D_{\theta}{q}^{\theta}\\\ D_{\theta}{v}_{Y}^{\theta}\\\
D_{\theta}{v}_{YY}^{\theta}\\\
\end{pmatrix}=\begin{pmatrix}D_{\theta}f_{1}^{\theta}\\\
D_{\theta}f_{2}^{\theta}\\\ D_{\theta}f_{3}^{\theta}\\\
D_{\theta}f_{4}^{\theta}\\\ D_{\theta}f_{5}^{\theta}\\\ \end{pmatrix}$

where $f_{i}^{\theta}$(i=1,2,3,4,5) are the perturbation of the right-hand
side of (3.10). Then we can get

(3.11) $\frac{\partial}{\partial t}\begin{pmatrix}D_{\theta}{u}^{\theta}\\\
D_{\theta}{v}^{\theta}\\\ D_{\theta}{q}^{\theta}\\\
D_{\theta}{v}_{Y}^{\theta}\\\ D_{\theta}{v}_{YY}^{\theta}\\\ \end{pmatrix}\\\
=\begin{pmatrix}D_{u}f_{1}^{\theta}&D_{v}f_{1}^{\theta}&D_{q}f_{1}^{\theta}&D_{v_{Y}}f_{1}^{\theta}&D_{v_{YY}}f_{1}^{\theta}\\\
D_{u}f_{2}^{\theta}&D_{v}f_{2}^{\theta}&D_{q}f_{2}^{\theta}&D_{v_{Y}}f_{2}^{\theta}&D_{v_{YY}}f_{2}^{\theta}\\\
D_{u}f_{3}^{\theta}&D_{v}f_{3}^{\theta}&D_{q}f_{3}^{\theta}&D_{v_{Y}}f_{3}^{\theta}&D_{v_{YY}}f_{3}^{\theta}\\\
D_{u}f_{4}^{\theta}&D_{v}f_{4}^{\theta}&D_{q}f_{4}^{\theta}&D_{v_{Y}}f_{4}^{\theta}&D_{v_{YY}}f_{4}^{\theta}\\\
D_{u}f_{5}^{\theta}&D_{v}f_{5}^{\theta}&D_{q}f_{5}^{\theta}&D_{v_{Y}}f_{5}^{\theta}&D_{v_{YY}}f_{5}^{\theta}\\\
\end{pmatrix}\begin{pmatrix}D_{{\theta}_{1}}\bar{u}^{\theta}&D_{{\theta}_{2}}\bar{u}^{\theta}&D_{{\theta}_{3}}\bar{u}^{\theta}\\\
D_{{\theta}_{1}}\bar{v}^{\theta}&D_{{\theta}_{2}}\bar{v}^{\theta}&D_{{\theta}_{3}}\bar{v}^{\theta}\\\
D_{{\theta}_{1}}\bar{q}^{\theta}&D_{{\theta}_{2}}\bar{q}^{\theta}&D_{{\theta}_{3}}\bar{q}^{\theta}\\\
D_{{\theta}_{1}}\bar{v}_{Y}^{\theta}&D_{{\theta}_{2}}\bar{v}_{Y}^{\theta}&D_{{\theta}_{3}}\bar{v}_{Y}^{\theta}\\\
D_{{\theta}_{1}}\bar{v}_{YY}^{\theta}&D_{{\theta}_{2}}\bar{v}_{YY}^{\theta}&D_{{\theta}_{3}}\bar{v}_{YY}^{\theta}\\\
\end{pmatrix}.$

Utilizing Lemma 2.4, we only need to prove the Lipschitz continuity of $f$.
Because of the smoothness of $(u,v,\xi)$, we only need to consider the
Lipschitz continuity of $P_{i}$ and $\partial_{x}P_{i}$ (i=1,2). And for the
sake of simplcity, here we only detail the analysis for $\partial_{u}(P_{1})$
and $\partial_{u}(\partial_{x}P_{1})$. All other derivatives can be estimated
by the same methods.

$\displaystyle|\partial_{u}P_{1}|$
$\displaystyle=|\frac{1}{2}\int_{-\infty}^{\infty}e^{-\int_{Y}^{\bar{Y}}(q\cos^{4}\frac{v}{2}(T,\tilde{Y})d\tilde{Y})}(\frac{3}{8}\sin^{2}v+3u^{2}\cos^{4}\frac{v}{2})\xi(T,\bar{Y})d\bar{Y}|$
$\displaystyle\leq
C\|\Gamma\|_{L^{\infty}}(\|v\|_{L^{2}}^{2}+\|u\|_{L^{2}}^{2})$ (3.12)
$\displaystyle\leq C\mathscr{E}^{2}(0).$

In the same way, we have

$\displaystyle|\partial_{u}\partial_{x}P_{1}|$
$\displaystyle=|\frac{1}{2}(\int_{Y}^{+\infty}-\int_{-\infty}^{Y})e^{-|\int_{\bar{Y}}^{Y}q\cos^{4}\frac{v}{2}(T,\tilde{Y})d\tilde{Y}|}(\frac{3}{8}\sin^{2}v+3u^{2}\cos^{4}\frac{v}{2})\xi(T,\bar{Y})d\bar{Y}|$
$\displaystyle\leq
C\|\Gamma\|_{L^{\infty}}(\|v\|_{L^{2}}^{2}+\|u\|_{L^{2}}^{2})$ (3.13)
$\displaystyle\leq C\mathscr{E}^{2}(0)$

By choosing suitable perturbation $V_{i}$(i=1,2,3), we can make

(3.14) $\displaystyle rank\begin{pmatrix}D_{\theta}\bar{v}^{\theta}\\\
D_{\theta}\bar{v}_{Y}^{\theta}\\\ D_{\theta}\bar{v}_{YY}^{\theta}\\\
\end{pmatrix}=3$

when $\theta=0$. ∎

##  4 Generic property

### 4.1. Generic solutions of the semilinear system

In this section we first study smooth solutions to the semilinear system
(2.17) , determining the generic structure of the level sets $\\{v=\pi\\}$.

######  Lemma 4.1.

Consider a compact domain of the form

(4.1) $\displaystyle\Gamma\triangleq\\{(t,Y);0\leq t\leq T,|Y|\leq M\\}.$

Call $\mathbb{S}$ the family of all $C^{2}$ solutions to the semilinear
system, with $q\textgreater 0$ for all
$(t,Y)\in\mathbb{R}_{+}\times\mathbb{R}$. Moreover, call
$\mathbb{S}^{\prime}\subset\mathbb{S}$ the subfamily of all solutions
$(u,v,q)$ such that for $(t,Y)\in\Gamma$ the following value is never
attained:

(4.2) $\displaystyle(v,v_{Y},v_{YY})=(\pi,0,0).$

Then $\mathbb{S}^{\prime}$ is a relatively open and dense subset of
$\mathbb{S}$, in the topology induced by $C^{2}(\Gamma)$.

###### Proof.

  * 1.

Denote $\mathbb{S}_{1}$ to be the subset of solutions for which
$(v,v_{Y},v_{YY})=(\pi,0,0)$ is never attained on $\Gamma$. Since $\Gamma$ is
a compact domain, each $\mathbb{S}_{1}$ is a relatively open subset of
$\mathbb{S}$, in the topology of $C^{2}(\Gamma)$.

  * 2.

Let $(u,v,q)$ be any $C^{2}$ solution of the semilinear system, with
$q\textgreater 0$. For any $(t_{0},Y_{0})\in\Gamma$, two cases can occur.

    * CASE 1

$(v,v_{Y},v_{YY})(t_{0},Y_{0})\neq(\pi,0,0)$. In this case, by continuity,
there exists a neighborhood $\mathscr{N}$ of $(t_{0},Y_{0})$ in the $t-Y$
plane where $(v,v_{Y},v_{YY})\neq(\pi,0,0)$.

    * CASE 2

$(v,v_{Y},v_{YY})=(\pi,0,0)$. By Lemma 3.4 we can find a three-parameter
family of solutions $(u^{\theta},v^{\theta},q^{\theta})$ such that the
3$\times$3 Jacobian matrix of the map

(4.3)
$\displaystyle(\theta_{1},\theta_{2},\theta_{3})\rightarrow(v^{\theta}(t,Y),v_{Y}^{\theta}(t,Y),v_{YY}^{\theta}(t,Y))$

has rank 3 at the point $(t_{0},Y_{0})$, when $\theta=0$. By continuity, this
matrix still has rank 3 on a neighborhood $\mathscr{N}$ of $(t_{0},Y_{0})$,
for $\theta$ small enough.

Now we choose finitely many points $(t_{i},Y_{i}),i=1,....,n$, such that the
corresponding open neighborhoods $\mathscr{N}_{(t_{i},Y_{i})}$ cover the
compact set $\Gamma$. Call $n_{\mathcal{I}}$ the cardinality of the set of
indices

(4.4)
$\displaystyle\mathcal{I}\triangleq\\{i;(v,v_{Y},v_{YY})(t_{i},Y_{i})=(\pi,0,0)\\}$

for which CASE 2 applies. For each $i\in\mathcal{I}$, our previous
construction provided a 3-parameter family of perturbations. Together, all
these perturbed solutions depend on $N=3n_{\mathcal{I}}$ parameters.

  * 3.

Let $\Omega\supset\Gamma$ be an open set contained in the union of the
neighborhoods $\mathscr{N}_{(t_{i},Y_{i})}$ and call
$B_{\epsilon}\triangleq\\{\theta\in\mathbb{R}^{N};|\theta|\leq\epsilon\\}$ the
open ball of radius $\epsilon$ in $\mathbb{R}^{N}$. We shall construct a
family $(u^{\theta},v^{\theta},q^{\theta})$ of smooth solutions to the
semilinear system (2.17), such that the map

(4.5)
$\displaystyle(t,Y,\theta)\rightarrow(v^{\theta}(t,Y),v_{Y}^{\theta}(t,Y),v_{YY}^{\theta}(t,Y))$

from $\Omega\times B_{\epsilon}$ into $\mathbb{R}^{3}$ has ($\pi$,0,0) as a
regular value. Toward this goal, we need to combine perturbations based at
possibly different points $(t_{i},Y_{i})$ into a single N-parameter family of
perturbed solutions. Let $(u,x,q)(t,Y)$ be a solution to the semilinear
system. For each $k=1,...,N,$ let a point $(t_{k},Y_{k})$ be given, together
with a number $U_{k}\in\mathbb{R}$ and functions $V_{k},Q_{k}\in
C_{c}^{\infty}(\mathbb{R})$. By the previous analysis, if we construct a
family of initial data $(\bar{v},\bar{v}_{Y},\bar{v}_{YY})$ with rank 3, we
will obtain a one-parameter family of perturbed solutions to the semilinear
system with rank 3 in the neighborhood of 0 for the parameter. So the family
of the perturbed solution is determined as follows. For
$|\epsilon|\textless\epsilon_{k}$ sufficiently small, we can determine the
unique solution of the perturbed semilinear system as

(4.6)
$\displaystyle(u^{\epsilon},v^{\epsilon},q^{\epsilon})\triangleq\Phi_{k}^{\epsilon}(u,v,q).$

Given($\theta_{1},...,\theta_{N}$), we can define a perturbation of the
original solution $(u,v,q)$ as the composition of N-parameter perturbations:

(4.7)
$\displaystyle(u^{\theta},v^{\theta},q^{\theta})=\Phi_{N}^{\theta_{N}}\circ...\circ\Phi_{1}^{\theta_{1}}(u,v,q).$

  * 4.

At every point $(t_{i},Y_{i})$ where i$\in\mathcal{I}$, using Lemma 3.1, we
can obtain a three-parameter families of perturbed solutions so that Jacobian
matrix of (4.3) is of full rank on neighborhood $\mathscr{N}$ of point
$(t_{0},Y_{0})$, for $\theta$ sufficiently small. Now we can choose finitely
many points $(t_{i},Y_{i}),~{}i=1,...,n$ such that the corresponding open
neighborhood $\mathbb{N}_{(t_{i},Y_{i})}$ covers the compact set $\Gamma$. So
we obtain a $N-$parameter family of solutions such that the value $(\pi,0,0)$
is a regular value for the map (4.3) from $\Gamma\times B_{\epsilon}$ into
$\mathbb{R}^{3}$. By the transversality theorem, for a.e. $\theta$, the map:

(4.8) $\displaystyle
F:(t,Y)\rightarrow(v^{\theta}(t,Y),v_{Y}^{\theta}(t,Y),v_{YY}^{\theta}(t,Y))$

is transverse to $(\pi,0,0)$.

By the definition of transversality, either

$F(t,Y)\neq(\pi,0,0)$

or

$F(t,Y)=(\pi,0,0)$ and
$T_{(\pi,0,0)}\mathbb{R}^{3}=(dF)_{(t,Y)}(T_{(t,Y)}\Omega)$

Since $\Omega$ is only two-dimensional,
$T_{(\pi,0,0)}\mathbb{R}^{3}=(dF)_{(t,Y)}(T_{(t,Y)}\Omega)$ cannot happen. So
the only choice is $F(t,Y)\neq(\pi,0,0)$. This means that for a.e. $\theta$
suffiently small, the corresponding solution
$(u^{\theta},v^{\theta},\xi^{\theta})$ has property that
$(v^{\theta},v_{Y}^{\theta},v_{YY}^{\theta})\neq(\pi,0,0)$ for all
$(t,Y)\in\Gamma$. This proves that the set $\mathbb{S}_{1}$ is dense in
$\mathbb{S}$.

∎

From now on we are going to prove our main result.

### 4.2. Proof of Theorem 1.1

###### Proof.

Consider the space $\mathbb{S}=C^{3}(\mathbb{R})\cap H^{1}(\mathbb{R})\cap
W^{1,4}(\mathbb{R})$ , with norm

(4.9)
$\displaystyle\|u\|_{\mathbb{S}}\triangleq\|u_{0}\|_{C^{3}}+\|u_{0}\|_{H^{1}}+\|u_{0}\|_{W^{1,4}}.$

Given initial data $\tilde{u}_{0}\in\mathbb{S}$, consider the open ball

(4.10) $\displaystyle
B_{\delta}\triangleq\\{u_{0}\in\mathbb{S};\|u_{0}-\tilde{u}\|\textless\delta\\}.$

We will prove that, for any $\tilde{u}_{0}\in\mathbb{S}$ ,there exists a
radius $\delta$ >0 and an open dense subset $\mathscr{D}\subset B_{\delta}$
such that for every initial data $u_{0}\in\mathscr{D}$ the conservative
solution $u=u(t,x)$ is of class $C^{1}$ in the complement of finitely many
characteristic curves $\gamma_{i}$ within the domain $[0,T]\times\mathbb{R}$.

Step 1. (Construction of $\mathscr{D}$.) If
$u_{0}u_{0,x}=\mathcal{O}(\epsilon^{\frac{1}{2}})$, then the blow up time of
$uu_{x}$ along the chararcteristics is of order $-\frac{1}{\epsilon^{2}}$.
Since $u_{0}\in\mathbb{S}$, we know $u_{0}u_{0,x}\rightarrow 0$ as
x$\rightarrow\infty$. So if we can take r >0 such that when $|x|$>0, we will
have $|uu_{x}|$<$\frac{1}{T_{1}}$, which means the singularity of u in set
$[0,T_{1}]\times\mathbb{R}$ only appears in the compact set
$A\triangleq[0,T_{1}]\times[-r-\|u^{2}\|_{L^{\infty}}T_{1},r+\|u^{2}\|_{L^{\infty}}T_{1}]$,
where$\|u\|_{L^{\infty}}\triangleq\max\\{u(t,x),(t,x)\in[0,T_{1}]\times\mathbb{R}\\}$.
In the (T,Y) plane ,it is reasonable for us to take a domain $\Gamma$ such
that $A\subset\Lambda(\Gamma)$, where $\Lambda$ is the map from (t,Y) to
(t,x(t,Y))

The subset $\mathscr{D}\subset B_{\delta}$ is defined as follows.
$u_{0}\in\mathscr{D}$ if $u_{0}\in B_{\delta}$ and, for the corresponding
solution $(u,v,\xi)$ of (2.17) with initial data, the value (4.2) are never
attained for $(t,x)\in A$.

Step 2. ($\mathscr{D}$ is open). We begin by proving that $\mathscr{D}$ is
open, in the topology of $C^{3}$. Take a sequence of initial data
$u_{0}^{\nu}\notin\mathscr{D}$ that converges to $u_{0}$. By definition of
$\mathcal{D}$, there is point $(t^{\nu},Y^{\nu})$ satisfying

(4.11)
$\displaystyle(v^{\nu},v_{Y}^{\nu},v_{YY}^{\nu})(t^{\nu},Y^{\nu})=(\pi,0,0),~{}~{}~{}~{}~{}(t^{\nu},x^{\nu}(t^{\nu},Y^{\nu}))\in
A,$

for all $\nu\geq 1$ . Since we have the domain $A$ is compact, by taking a
subsequence, we can assume $(t^{\nu},Y^{\nu})$ converges to some point $(t,Y)$
and

(4.12)
$\displaystyle(v,v_{Y},v_{YY})(t,Y)=(\pi,0,0),~{}~{}~{}~{}~{}~{}(t,x(t,Y))\in
A,$

which implies $u_{0}\notin\mathscr{D}$. The other case can be proved in the
same way. So $\mathscr{D}$ is open.

Step 3. ($\mathscr{D}$ is dense). Given $u_{0}\in B_{\delta}$, by a small
perturbation, we can assume $u_{0}\in C^{\infty}$

By lemma 4.1, we can construct a sequence of solutions
$(u^{\nu},v^{\nu},\xi^{\nu})$ of (2.17) such that

  * (i)

For every bounded set $\Omega\subset\mathbb{R}^{2}$ and $k\geq 1$,

(4.13)
$\displaystyle\lim_{\nu\rightarrow+\infty}\|(u^{\nu}-u,v^{\nu}-v,\xi^{\nu}-\xi,x^{\mu}-x)\|_{C^{k}(\Omega)}=0.$

  * (ii)

For every $\nu\geq 1$ the values in (4.2) are never attained for any
$(t,Y)\in\Gamma$.

Consider the sequence of solution $u^{\nu}(t,x)$ with graph

(4.14)
$\displaystyle\biggl{\\{}(u^{\nu}(t,Y),t^{\nu}(t,Y),x^{\nu}(t,Y));(t,Y)\in\mathbb{R}^{2}\biggr{\\}}\subset\mathbb{R}^{3}$

The corresponding sequence of initial data satisfies

(4.15) $\displaystyle\|u^{\nu}(0,\cdot)-u_{0}\|_{C^{l}(I)}\rightarrow
0,~{}~{}~{}~{}~{}as~{}\nu\rightarrow\infty$

for every bounded set $I$.

In order to obtain the convergence for the far field, we modify the sequence.
Define a cutoff function $\phi\in C_{c}^{\infty}$:

$\displaystyle\phi(x)=1~{}~{}~{}~{}~{}\text{if}~{}~{}~{}|x|\leq\rho,$ (4.16)
$\displaystyle\phi(x)=0~{}~{}~{}~{}~{}\text{if}~{}~{}~{}|x|\geq\rho+1,$

with $\rho\gg r+\|u^{2}\|_{L^{\infty}}T$ sufficiently large. For each $\nu\geq
1$ consider the initial data

(4.17) $\displaystyle\tilde{u}_{0}^{\nu}\triangleq\phi
u_{0}^{\nu}+(1-\phi)u_{0}.$

Then, we have

(4.18)
$\displaystyle\lim_{\nu\rightarrow+\infty}\|\tilde{u}^{\nu}-u_{0}\|_{\mathbb{S}}=0.$

Moreover, if $\rho$>0 large enough, we have

(4.19)
$\displaystyle\tilde{u}^{\nu}(t,x)=u^{\nu}(t,x),~{}~{}~{}~{}\forall(t,x)\in
A,$

while $\tilde{u}^{\nu}(t,x)$ is $C^{2}$ on the outer domain. Then we can find
that $\tilde{u}^{\nu}(t,x)\in\mathscr{D}$ for all $\nu\geq 1$ sufficiently
large, which leads to $\mathscr{D}$ being dense on $B_{\delta}$.
Step 4. By previous argument, we know that $u$ is $C^{2}$ on the outer domain
$\\{(t,x)|0\leq\ t\leq T,|x|\textgreater\|u^{2}\|_{L^{\infty}}T|\\}$. So it
suffices to study the singularity of $u$ on the inner domain $A$. Consider a
point $(t_{0},Y_{0})\in\Gamma$, there could be two situations:

  * i)

$v(t,Y)\neq\pi$. Then we can have
$x_{Y}=\frac{\xi}{(1+u_{x}^{2})^{2}}=cos^{4}\frac{v}{2}q$,

$0\textless x_{Y}\textless\infty,$

so the map $(t,Y)\rightarrow(t,x)$ is locally invertible in a neighborhood of
$(t_{0},Y_{0})$. We then conclude that the function u is $C^{2}$ in a
neighborhood of the point$(t_{0},x(t_{0},Y_{0}))$.

  * ii)

$v(t_{0},Y_{0})=\pi$. As is pointed out in [7] the singularity happens on the
level set $\\{u=0\\}$. By continuity, there exists $\epsilon$>0 such that the
values (4.2) are never attained in the open neighborhood

(4.20) $\displaystyle\Gamma^{\prime}\triangleq\\{(t,Y);t\in[0,T],|Y|\leq
M+\epsilon\\}.$

For (4.2), there are two states that can occur.

    * (1)

$(v_{Y}=0,v_{YY}\neq 0)$ By the implicit function theorem and (4.2), we can
conclude that the set

(4.21)
$\displaystyle\mathcal{S}^{v_{Y}}\triangleq\\{(t,Y)\in\Gamma^{\prime};v_{Y}(t,Y)=0\\}$

is a one dimensional embedded manifold of class $C^{1}$.

To prove the number of connected components of $\mathcal{S}^{v_{Y}}$ that
intersect the compact set $\Gamma$ is finite, we shall assume the contrary
that $P_{1},P_{2},....$is a sequence of points in
$\mathcal{S}^{v_{Y}}\cap\Gamma$ belonging to distinct components. Taking a
subsequence we can assume the convergence $P_{i}\rightarrow\bar{P}$ for some
$\bar{P}\in\mathcal{S}^{v_{Y}}\cap\Gamma$. By assumption, $v_{YY}\neq 0$.
Hence, by the implicit function theorem, there is a neighborhood $\mathcal{N}$
of $\bar{P}$ such that $\gamma\triangleq\mathcal{S}^{v_{Y}}\cap\mathcal{N}$ is
a connected $C^{1}$ curve. Hence $P_{i}\in\Gamma$ for all i large enough,
providing a contradiction.

    * (2)

$(v_{Y}\neq 0)$ By the implicit function theorem and (4.2), we can conclude
that the set

(4.22)
$\displaystyle\mathcal{S}^{v}\triangleq\\{(t,Y)\in\Gamma^{\prime};v(t,Y)=\pi\\}$

is a one dimensional embedded manifold of class $C^{2}$. The proof of that the
number of connected components of $S^{v}$ is finite is very similar, thus it
will be omitted.

∎

######  Remark 4.2.

We notice that the argument in Step 4 is differ from the Camassa-Holm
equation. For the Camassa-Holm equation, if $v(t_{0},Y_{0})=0$ for some point
$(t_{0},Y_{0})\in\Gamma$ then one can see that $v_{t}\neq 0,v_{Y}\neq 0$.
However, for the Novikov equation, the case $v_{Y}=0$ will happen. We believe
that this difference is caused by the energy concentration. For the Camassa-
Holm equation when the characteristic meet tangentially, they will separate
immediately. However, for the Novikov equation, when the characteristics
tangentially touch each other, they will stay for a period of time.

##  5 Generic singularity behavior

For smooth data $u_{0}\in C^{\infty}(\mathbb{R})$, the solution
$(t,Y)\rightarrow(x,t,u,v,q)(t,Y)$ of the semilinear system (2.17), with
initial data as in (2.18), remains smooth on the entire t-Y plane. Yet the
smoothness of the solution u of (2.4) is still needed to study because the
coordinate change:(Y,t)$\rightarrow$(x,t) is not smoothly invertible. By
definitions, its Jacobian matrix is computed by

(5.1) $\displaystyle\begin{pmatrix}x_{Y}&x_{t}\\\
t_{Y}&t_{t}\end{pmatrix}=\begin{pmatrix}qcos^{4}\frac{v}{2}&u\\\
0&1\end{pmatrix}.$

And we will observe that the matrix is invertible when $v\neq\pi$. To study
the set of points in the $t-x$ plane where $u$ is singular, we thus need to
look at points where $v=\pi$.

Our main theorem provides us with a detailed description of the solution
u(t,x) in the neighborhood of each one of these singular points. For
simplicity, we shall assume that the initial data $u_{0}$ are smooth, so we
shall not need to count how many derivatives are actually used to derive the
Talyor approximations.

### 5.1. Proof of Theorem 1.2

###### Proof.

1\. Let P be the point of Type $\mathcal{I}$. Recalling (2.16) and (2.17) then
we will have

(5.2) $\displaystyle
u_{Y}=u_{x}x_{Y}=\frac{1}{2}q\cdot\sin~{}v\cdot\cos^{2}\frac{v}{2}.$

In a similar way, we obtain

$\displaystyle u_{YY}$ $\displaystyle=\frac{1}{2}q_{Y}\cdot\sin v\cdot\
cos^{2}\frac{v}{2}+\frac{1}{2}q\cdot v_{Y}\cdot\cos
v\cdot\cos^{2}\frac{v}{2}-\frac{v_{Y}}{4}\cdot q\sin^{2}v,$ $\displaystyle
u_{Yt}$
$\displaystyle=\frac{1}{2}qsin^{2}vcos^{2}\frac{v}{2}\big{(}(2u^{3}+u)sin^{2}\frac{v}{2}-2(P_{1}+\partial_{x}P_{2})+(2u^{3}+u)cos^{2}\frac{v}{2}\big{)}$
(5.3)
$\displaystyle+\frac{1}{2}(1-4sin^{2}\frac{v}{2})cos^{2}\frac{v}{2}\big{(}-usin^{2}\frac{v}{2}+2u^{3}cos^{2}\frac{v}{2}-2cos^{2}\frac{v}{2}(P_{1}+\partial_{x}P_{2})\big{)}.$

At the point P, it provides us with that $v_{Y}=0,~{}v_{t}=0$ and $v_{YY}\neq
0$. If we denote $u_{Y^{n}}$ that u is differentiated with Y by n times. It is
not hard to check

(5.4) $\displaystyle
u_{Y^{3}}=0,~{}~{}~{}u_{Y^{4}}=0,~{}~{}~{}u_{Y^{5}}=0,~{}~{}~{}u_{Y^{6}}=\frac{6}{8}qv_{YY}^{3}cosvsin^{2}\frac{v}{2}-3v_{YY}^{3}qcos^{2}v,$

which means $u_{Y^{6}}\neq 0$ Following the same method, we obtain that

(5.5) $\displaystyle
u_{Y^{2}t}=0,~{}~{}~{}u_{Y^{3}t}=0,~{}~{}~{}u_{Y^{4}t}=0,~{}~{}~{}u_{Y^{5}t}=\frac{6}{8}qv_{YY}^{2}v_{Yt}cosvsin^{2}\frac{v}{2}-3v_{YY}^{2}v_{Yt}qcos^{2}v.$

we will have $u_{Y^{5}t}=0$.

So we have Taylor approximations of u at the singular point $P=(t_{0},Y_{0})$

(5.6) $u(t,Y)=B_{1}(t-t_{0})+B_{3}(Y-Y_{0})^{6}+\mathcal{O}(1)({\lvert
t-t_{0}\rvert}^{2},{\lvert Y-Y_{0}\rvert}^{7})$

By (2.16) and (2.17), we can obtain

(5.7) $\displaystyle x_{Y}=\frac{\xi}{(1+u_{x}^{2})^{2}}=qcos^{4}\frac{v}{2},$

similarly we have

$\displaystyle x_{YY}$
$\displaystyle=q_{Y}cos^{4}\frac{v}{2}-2qv_{Y}cos^{3}\frac{v}{2}sin\frac{v}{2}$
(5.8) $\displaystyle x_{Yt}$
$\displaystyle=q_{t}cos^{4}\frac{v}{2}-2qv_{t}sin\frac{v}{2}cos^{3}\frac{v}{2}$

so it is easy to check that in the point P

$x_{Y^{i}}=0,~{}~{}~{}~{}(i=1,2,3,4,5,6,7)$

and

$\displaystyle x_{Y^{8}}$ $\displaystyle=36qv_{YY}^{4}sin^{4}\frac{v}{2},$
(5.9) $\displaystyle x_{Y^{7}t}$
$\displaystyle=36qv_{YY}^{3}v_{Yt}sin^{4}\frac{v}{2}.$

So we have the Taylor approximations of x at the singular point
$P=(t_{0},Y_{0})$

(5.10) $x(t,Y)=x(t_{0},Y)+A_{2}(Y-Y_{0})^{8}+\mathcal{O}(1)(\lvert
t-t_{0}\rvert^{2},\lvert Y-Y_{0}\rvert^{9}).$

We combine (5.6) and (5.10) to deduce (1.4).

2.If P is of type $\mathcal{I}\mathcal{I}$,we have

(5.11) $\displaystyle v=\pi,~{}~{}~{}v_{Y}\neq 0,~{}~{}~{}v_{YY}=0,$

which implies

(5.12) $\displaystyle
u_{Y}=0,~{}~{}~{}~{}~{}~{}~{}u_{YY}=0,~{}~{}~{}~{}~{}~{}~{}u_{Y^{3}}=0,$
(5.13) $\displaystyle
u_{Y^{4}}=\frac{1}{4}qv_{Y}^{3}cosv\sin^{2}\frac{v}{2}-\frac{{v_{Y}}^{3}}{2}q\cos^{2}v=\frac{1}{2}v_{Y}^{3}q\neq
0.$

And

(5.14) $\displaystyle
u_{Yt}=0,~{}~{}~{}~{}~{}u_{Y^{2}t}=0,~{}~{}~{}~{}~{}u_{Y^{3}t}=0.$

Then we obtain the Taylor approximation of u is

(5.15) $\displaystyle
u(t,Y)=B_{1}(t-t_{0})+B_{2}(Y-Y_{0})^{4}+\mathcal{O}(1)(|t-t_{0}|^{2}+|Y-Y_{0}|^{5}).$

At point P,

(5.16) $\displaystyle
x_{Y}=0,~{}~{}~{}x_{Y^{2}}=0,~{}~{}~{}x_{Y^{3}}=0,~{}~{}~{}x_{Y^{4}}=0,$
(5.17) $\displaystyle
x_{Yt}=0,~{}~{}~{}x_{Y^{2}t}=0,~{}~{}~{}x_{Y^{3}t}=0~{}~{}~{},~{}~{}~{}x_{Y^{4}t}=0,$
(5.18) $\displaystyle x_{Y^{5}}=\frac{3}{2}qv_{Y}^{3}cos^{3}v\neq 0.$

This leads

(5.19) $x(t,Y)=x(t_{0},Y_{0})+A_{2}(Y-Y_{0})^{5}+\mathcal{O}(1)(\lvert
t-t_{0}\rvert^{2},\lvert Y-Y_{0}\rvert^{6}).$

So (1.5) can be concluded from (5.15) and (5.19). ∎

Acknowledgments This work was partially supported by the National Natural
Science Foundation of China (No.12171493 and No.11701586), the National Key
R&D Program of China ( No. 2021YFA1002100), and the Natural Science Foundation
of Guangdong province (No. 2021A1515010296 and 2022A1515011798).

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