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# Science and Technology Ontology: A Taxonomy of Emerging Topics

Mahender Kumar  Secure Cyber Systems Research Group, WMG, University of
Warwick, Coventry, United Kingdom, CV4 7AL  Ruby Rani  Mirko Bottarelli
Gregory Epiphaniou  Carsten Maple

(2023)

###### Abstract

Ontologies play a critical role in Semantic Web technologies by providing a
structured and standardized way to represent knowledge and enabling machines
to understand the meaning of data. Several taxonomies and ontologies have been
generated, but individuals target one domain, and only some of those have been
found expensive in time and manual effort. Also, they need more coverage of
unconventional topics representing a more holistic and comprehensive view of
the knowledge landscape and interdisciplinary collaborations. Thus, there
needs to be an ontology covering Science and Technology and facilitate
multidisciplinary research by connecting topics from different fields and
domains that may be related or have commonalities. To address these issues, we
present an automatic Science and Technology Ontology (S&TO) that covers
unconventional topics in different science and technology domains. The
proposed S&TO can promote the discovery of new research areas and
collaborations across disciplines. The ontology is constructed by applying
BERTopic to a dataset of 393,991 scientific articles collected from Semantic
Scholar from October 2021 to August 2022, covering four fields of science.
Currently, S&TO includes 5,153 topics and 13,155 semantic relations. S&TO
model can be updated by running BERTopic on more recent datasets.

###### keywords:

Science and Technology Ontology Unconventional Topics BERTopic Scientific
Knowledge Graph

##  1 Introduction

Ontologies are a valuable tool for representing and organising knowledge about
a specific topic or set of topics, using a set of concepts, relationships, and
rules within the domain [1, 2]. They have many applications, including data
annotation and visualisation [3], forecasting new research areas [4], and
scholarly data discovery [5]. Some topic ontologies created in different
domains include ACM Computing Classification System 111The ACM Computing
Classification System: http://www.acm.org/publications/class-2012., Physics
and Astronomy Classification Scheme (PACS) 222Physics and Astronomy
Classification Scheme: https://publishing.aip.org/publishing/pacs, replaced in
2016 by the Physics Subject Headings (PhySH) 333PhySH - Physics Subject
Headings: https://physh.aps.org/about., Mathematics Subject Classification
(MSC) 4442010 Mathematics Subject Classification:
https://mathscinet.ams.org/msc/msc2010.html., the taxonomy used in the field
of Mathematics, and Medical Subject Heading (MeSH) 555MeSH - Medical Subject
Headings: https://www.nlm.nih.gov/mesh.. Creating these large-scale taxonomies
is a complex and costly process that often requires the expertise of multiple
domain experts, making it a time-consuming and resource-intensive endeavour.
Consequently, these taxonomies are often difficult to update and maintain,
quickly becoming outdated as new information and discoveries emerge. As a
result, the practicality and usefulness of these taxonomies are significantly
limited. One of the most notable advancements in ontology generation is the
development of a large-scale automated ontology known as Computer Science
Ontology (CSO) [6]. CSO ontology defines a significant breakthrough in the
representation of research topics in the computer science domain, providing a
structured and comprehensive framework for organising and integrating
knowledge but limited to computer science concepts only.

Research Challenge: Understanding the dynamics associated with unconventional
topics, which present a more comprehensive and holistic perspective of the
knowledge landscape and interdisciplinary collaborations, poses a considerable
challenge. Constructing an ontology for such unconventional topics
necessitates recognising and collecting essential concepts and relationships
from multiple domains. Furthermore, unconventional topics may necessitate
multidisciplinary study, necessitating the integration of information from
many fields. By overcoming these challenges, there is an opportunity for
researchers and academicians to study new and developing areas of science and
technology, as well as facilitate interdisciplinary collaboration across
varied fields.

Contribution. This paper presents preliminary work to construct a S&TO
ontology that automatically generates a taxonomy of unconventional S&T topics.
S&TO ontology is built by applying BERTopic to a dataset of 393,991 scientific
articles collected from Semantic Scholar from October 2021 to August 2022,
covering four fields of science: computer science, physics, chemistry and
Engineering. Currently, S&TO includes 5,153 topics and 13,155 semantic
relations. Unlike existing ontology, S&TO ontology can provide many benefits
for knowledge representation and discovery, facilitating interdisciplinary
research and enabling dynamic updates.

Organisation. The rest of the paper is organized as follows. Section 2
discusses the dataset and methods for constructing the proposed S&TO. Section
3 gives the proposed S&TO. The experimental results are discussed in section
4. Section 5 presents the applications and Usecases of S&TO, and the
limitations of the current version are discussed in Section 6. Finally, the
conclusion is given in section 7.

##  2 Data and Methods

###  2.1 Semantic Scholar

Semantic Scholar has many academic publications from various fields, including
medical sciences, agriculture, geoscience, biomedical literature, and computer
science. We used the RESTful Semantic Scholar Academic Graph (S2AG) API to
retrieve a sample of these articles [7]. This API offers users on-demand
knowledge of authors, papers, titles, citations, venues, and more. We obtained
393,991 Science and Technology articles from Semantic Scholar using the S2AG
API. The API provides a dependable data source that allows users to link
directly to the related page on semanticscholar.org, making it a convenient
and accessible way to retrieve information about academic papers.

###  2.2 Methodologies

Figure 1: Data Flow

After downloading the dataset, we used the BERTopic method to obtain topics
from the articles—some articles representing the multi-discipline need to be
included as an outlier. To have these unconventional articles and reduce the
outlier percentage, we adjusted parameters with BERTopic during topic
clustering [8]. Table 3 lists the critical BERTopic parameters used in the
taxonomy generation.

As shown in Figure 1, our suggested topic modelling workflow consists of five
important steps: sentence embedding, dimension reduction, clustering, topic
quality, and topic representation. The Sentence Embedding stage, in
particular, involve turning textual input into numerical vectors that capture
the underlying semantics of the text. Dimension Reduction is then applied to
the vectors to lower their dimensionality and improve the effectiveness of the
clustering procedure. Clustering is the process of combining similar vectors
to generate coherent clusters of related material. The vectorisation ensures
that the extracted topics are high quality, whilst topic Representation
creates an interpretable summary of each topic. Overall, we provide a robust
and effective approach to extracting meaningful unconventional topics from
vast and heterogeneous datasets, utilising the power of BERTopic and careful
parameter optimisation to assure optimal outcomes.

Table 1: BERTopic Parameters Parameters | Values
---|---
Embedding parameter |
embedding_model | sentence_model
UMAP parameters |
n_neighbors | 2
n_components | 5
low_memory | true
HDBSCAN parameters |
5min_cluster_size | 10
metric | euclidean
cluster_selection_method | eom
prediction_data | true
min_samples | 1
CountVectoriser parameters |
Vocabulary | Vocabulary
Min_df | 10
c-TF-IDF parameters |
top_n_words | 30
Verbose | True
min_topic_size | 20
vectorizer_model | CountVectorizer
low_memory | True
calculate_probabilities | False
Diversity | 0.4
Table 2: BERTopic clustering model selection for outlier reduction Characteristics | HDBSCAN | K-means | HDBSCAN with
---|---|---|---
|  |  | Probability
Topic Quality | Good | Less | Good
Outlier | High | No | Low
Risk of missing | Less | Little | Least
unconventional topics |  |  |

  1. 1.

Sentence Embedding: We first transformed the input articles into numerical
representations before analysing them. For this purpose, we utilised sentence
transformers, the default embedding model used by BERTopic. This model can
determine the semantic similarity of different documents. As default, BERTopic
provides many pre-trained models among them we tried the following two: “`all-
MiniLM-L6-v2`“ and “`paraphrase-MiniLM-L12-v2`“. While various sentence
embedding models are available, we opted for the “`paraphrase-MiniLM-L12-v2`“
model in this work. This model effectively balances performance and speed,
making it a good fit for our requirements. Thus, we can effectively translate
textual data into numerical form and obtain relevant insights from large and
diverse datasets using sentence transformers in conjunction with BERTopic.

  2. 2.

Dimension Reduction: Clustering can be complex since embeddings are often
high-dimensional. To solve this problem, the dimensionality of the embeddings
is frequently reduced to a more practical level. We used the UMAP (Uniform
Manifold Approximation and Projection) technique, representing local and
global high-dimensional features in a lower-dimensional domain [9].
“`n_neighbors`“ and “`n_components`“ are two important parameters in the UMAP
method. These parameters have a considerable impact on the size of the
generated clusters. Larger values for these factors, in particular, result in
the formation of more important clusters. We got optimal clustering results
and extracted important topics from the input data by carefully tweaking these
parameters.

  3. 3.

Clustering: BERTopic splits the input data into clusters of similar embeddings
after the dimensionality reduction process. The clustering techniques’
accuracy directly impacts the quality of the generated topics. K-means [10],
Hierarchical Density-Based Spatial Clustering of Applications with Noise
(HDBSCAN) [11], and Agglomerative Clustering [12] are among the clustering
techniques provided by BERTopic. The advantages and drawbacks of these
clustering algorithms are summarised in Table 2, emphasising their capacity to
generate high-quality topics, manage outlier percentages, and limit the danger
of missing unconventional topics. HDBSCAN is a density-based clustering
algorithm used to find clusters of varying densities in a dataset. It works by
constructing a hierarchical tree of clusters based on the density of the data
points. It starts by identifying the points with the highest density and
forming a cluster around them. Then, it gradually adds lower-density points to
the cluster until a natural cutoff is reached, indicating the end of the
cluster. According to our findings, the HDBSCAN with the prediction_data
parameter set to “`True`“ was the best option. Our method efficiently balances
the above elements, allowing us to obtain meaningful and valuable insights
from large and complex datasets.

  4. 4.

Vectorisation: The CountVectorizer technology turns text documents into
vectors of phrase frequencies. However, it has significant drawbacks, such as
failing to consider a specific topic’s relative relevance in an article. To
fix this issue, we adopted C-TF-IDF (Class-Based TF-IDF), a variant of the
classic TF-IDF (Term Frequency-Inverse Document Frequency) method that
allocates weights to terms depending on their relevance to a specific class of
documents. We could fine-tune the model’s performance in BERTopic by adjusting
its parameters to optimise the clustering process using CountVectorizer with
C-TF-IDF. This method made it possible to create higher-quality topics that
are more fascinating and pertinent to the input data.

  5. 5.

Topic Representation: BERTopic can adjust TF-IDF to work at the cluster level
instead of the document level to obtain a concrete representation of topics
from the bag-of-words matrix. This modified TF-IDF is called c-TF-IDF. For
word x in class c, the c-TF-IDF value is:

$w_{x,c}=|tf_{x,c}|\times log(1+A/f_{x})$ (1)

Where $tf_{x,c}$ denotes the frequency of word x in class c, $f_{x}$ denotes
word x across all classes, and A denotes the average number of words per
class.

Figure 2: Schema of Topic Extraction and Topic Network Creation

##  3 Science and Technology Ontology Generation

To create a topic network, also known as a knowledge graph, the metadata
provided by the semantic scholar is utilised. The construction of topic
ontologies involves the definitions of the following components:

  * •

Topics: concepts of the topic ontology (e.g. Sports, Arts, Politics).

  * •

Predicates: kinds of relationships that define the semantic link established
between the ontology concepts. Many predicates can be defined in topic
ontologies: hierarchical (e.g. superTopicOf) and non-hierarchical (e.g., part
of, contribute to).

  * •

Relationships: according to predicates and the set of elements they link,
relationships are distinguished. They can be used to characterise the paths in
the graphs and denoted as a triplet (T1, P, T2), where T1 and T2 denote the
topics, and P denotes the predicate that links T1 and T2.

###  3.1 Topics

The KeyBERT tool is used on the associated publications to extract keywords
representing the essential concepts and topics within each document to produce
the vocabulary for BERTopic. These keywords are then sent into BERTopic, which
generates a complete collection of topics that capture the overarching topic
found in the dataset. The extracted topics are saved in a database’s
"topic_nets_topics" and "topic_nets_topics keywords" (see Figure 2). Each
topic’s weight denotes the number of papers for which it serves as the main
association, showing its relevance within the dataset.

There are two techniques to establish the relationship between papers and
topics:

  1. 1.

Probability: First, each paper’s BERTopic/HDBSCAN probabilities are saved as
entries in the "papers topics" table. These probabilities indicate how closely
each document relates to each extracted topic.

  2. 2.

Embedding similarity: Second, using the "main topic id" field, the major topic
associated with each paper, as identified by BERTopic, is directly linked to
the paper using a SQL trigger. This allows for efficient querying and analysis
of the topics and papers related to them within the corpus.

Figure 3: An instance of topic hierarchy

###  3.2 Relationship

The next step is to create topic networks using the relationship among topics.
Currently, the S&TO ontology is built on 393,991 scientific papers collected
from Semantic Scholar from October 2021 to August 2022. It covers four science
fields: computer science, physics, chemistry and Engineering. S&TO ontology
follows the data model SKOS666SKOS Simple Knowledge Organization System -
http://www.w3.org/2004/02/skos. and includes the following semantic
relationships:

  * •

“`relatedIdentical`“, It is a sub-metric of skos related, denotes that two
topics can be viewed as identical for assessing research topics. The
similarity between topics is calculated as cosine-similarity in the SQL stored
procedure create_topic_nets. The relationship between topics is established if
the similarity threshold is above 0.9.

  * •

“`superTopicOf`“: It is a sub-metric of skos:narrower, which means that a
topic is a super-area of another topic in the graph. For example,
"streaminmg_rsi_retrieval_streaming_regression" is the super-topic of Topics
with topic_ids 78 and 101, as shown in Figure 3.

  * •

“`CommonArticles`“: It extracts common articles that appear in the two topics.
The link between two topics is evaluated as the sum of the probability
distribution by common articles assigned to the topics.

  * •

“`nSimilarTopics`“: It returns the top x number of similar topics for an input
keyword. For instance, the top 5 similar topics related to the keyword "motor"
are shown in Table 3.

Table 3: Top 5 similar topics to keyword "motor" Topic Id | Name
---|---
33826 | 33826_motorized_spindle_aerostatic_restrictors
50435 | 50435_multimotor_geartrain_rocker_truck
62709 | 62709_liner_motorbike_motored_honing
49677 | 49677_motorized_spindle_nanocatalysts_dragging
46619 | 46619_powertrains_powertrain_earthmoving_2025

.

##  4 Experimental Results and Discussion

In the literature [13], ontology has been evaluated by four methods: gold-
standard based [14], corpus-based [15], application-based [16], and structure-
based methods [17]. The gold-standard-based method compares the developed
ontology with the referenced ontology developed earlier. The corpus-based
method compares the significantly developed ontology with the contents of a
text corpus that covers a given domain. The application-based approach
considers applications and evaluations according to their performance across
use cases. The structure-based approach quantifies structure-based properties
such as size and ontology complexity.

Selecting the best evaluation approach and defining the rationale behind
evaluating a developed ontology is necessary. In the proposed study, the
science and technology ontology is in the early stages of development and will
grow in future work. Thus, the application-based approach should not be a good
evaluation approach because the proposed ontology is not proper for
application purposes as it is currently in development. The proposed ontology
is developed on a Semantic Scholar data set subset. Thus, the best reference
ontology should be a semantic scholar. However, using Semantic Scholar as the
gold standard ontology is impractical due to its unavailability.

Figure 4: An instance of knowledge graph in Neo4j Browser [18]

Structure-based evaluation is performed on several measures, including
knowledge coverage and popularity measures (i.e., number of properties and
classes) and structural measures (i.e., maximum depth, minimum depth, and
depth variance). These measures are adopted based on the belief that densely
populated ontologies with high depth and breadth variance are more likely to
result in meaningful semantic content. Structural metrics are related to the
semantic accuracy of adaptively modelled knowledge in the ontology [19].

In the context of the proposed S&TO ontology, we quantified some structural
measures by considering their taxonomic structure. S&TO ontology gives 5,153
topics and 13,155 semantic relations (of which 8052 topics are based on cosine
similarity and 5103 topics are based on probability distribution). Figure 4
shows an example of a knowledge graph covering topics and semantic
relationships. We used neo4j as the graph database to host the final ontology
[18]. Links in green indicate semantic relationships assessed using cosine
similarity, while links in yellow indicate relationships based on the
probability distribution of papers assigned to topics. S&TO covers the maximum
amount of articles for topic clustering and gives only 15.47% of articles as
an outlier, enabling the extraction of topics belonging to unconventional
articles.

###  4.1 Topics Details

This section discusses the structure of topic networks, topics in topic
networks and keywords related to topics in the network.

Table 4: Structure of Table ’topic_nets’ Field | Description
---|---
topic_net_id | PK777Primary Key, network unique identifier
created_on | Indicates when the network was created
status | Indicates the status of the network (NEW, DONE, etc)
year_month | Indicate the month from which this network was created

The "topic_nets" table (Table 4) gives information about the development and
status of topic modelling networks. Each network is assigned a unique
identifier known as a "topic_net_id," its primary key. The "created_on"
parameter specifies the date and time the network was established. The
"status" field offers information on the network’s current status, which might
take various values. This field assists in tracking the process’s progress and
ensuring that all networks are correctly generated and assessed. In addition,
the "year_month" parameter provides the month the network was founded. This
feature is beneficial for tracking the temporal evolution of themes within the
corpus since it allows researchers to understand how topics and their
associations change over time.

Table 5: Structure of Table ’topic_nets_topics’ Field | Description
---|---
topic_id | PK, unique identifier of the topic
topic_net_id | FK888Foreign Key to the topic_nets table number Integer Topic number
label | Topic label
topic_weight | Number of papers associated with this topic
embedding | Topic embedding used for cosine similarity
similar_topics | Array of topic ids related to similar topics

The "topic_nets_topics" table (Table 5) provides essential information about
the topics associated with the Topic networks. Each topic has a unique
identifier known as a "topic_id", the primary key. Each topic links to the
corresponding network the "topic_nets" using a unique identifier known as a
"topic_net_id", its foreign key. A descriptive label is assigned to the topic
based on the most common terms in the associated papers. The topic has
"topic_weight" which indicates the number of papers related to it, indicating
its importance and relevance within the corpus. It also stores the
"embedding", which will be used for cosine similarity calculations, and
"similar_topics", an array of topic ids related to searched topics.

Table 6: Structure of Table ’topic_nets_topics_keywords’ Field | Description
---|---
topic_id | PK, unique identifier of the topic
number | Topic number
row | Auto-incremented number, ordered by increasing score
keyword | The keyword
score | The score associated with the keyword for this topic

The "topic_nets_topics_keyword" table (Table 6) provides essential information
about the keywords associated with the topic, represented by a unique
identifier known as a "topic_id", which is the primary key. It stores the
fields such as: "number" representing the topic number, "row" is an auto-
incremented number, and "keyword" representing the name of the keywords. In
addition, "score" represents a weight associated with the keyword for that
topic.

The "papers_topics" table (table 7) illustrates the relationship between
academic papers and the topics they cover. The "corpus_id" denotes the unique
identifier of the corpus the papers that were extracted from. The "topic_id"
represents the unique identifier of the topic that the paper covers, making it
easier to track papers within that topic. The "probability" represents the
weight of the paper assigned to the topic. This weight indicates the degree to
which the paper covers the topic. The probability value is typically
normalized, which is scaled to a range between 0 and 1.

Table 7: Structure of Table ’papers_topics’ Field | Description
---|---
corpus_id | Part of PK, and FK to the papers table
topic_id | Part of PK, and FK to the topic_nets_topics table
row | Auto-incremented number, ordered by increasing probability
probability | Probability of the paper to be assigned to this topic

###  4.2 Relation Details

This section discusses the structure of topic relations, such as information
on edges and similarities among topics.

The "topic_nets_topics_edges" stores (Table 8) the information of edges among
the topics "topic_nets_topics" in the network "topic_nets". The edge is
established between the two topics represented by their unique identifier
(i.e., "topic_id1" and "topic_id2"). "edge_Weight" is the sum of possibilities
("possibility" field of Table 7) of papers sharing between two topics. The
strength of collaboration "str_of_col" represents the weight computed as
harmonic mean and normalised based on topics weights, shown in Eq (2).

$str\\_of\\_col=HarmonicMean\bigg{(}\frac{edge\\_weight(id1,id2)}{topic\\_weight(id1)},\frac{edge\\_weight(id1,id2)}{topic\\_weight(id2)}\bigg{)}$
(2)

The "topic_nets_topics_similarity" stores (Table 9) the information of edges
among the topics based on the similarity.

Table 8: Structure of Table ’topic_nets_topics_edges’ Field | Description
---|---
topic_id1 | Part of PK, and FK to the topic_nets_topics table
topic_id2 | Part of PK, and FK to the topic_nets_topics table
edge_weight | Sum of probabilities of papers sharing the two topics
str_of_col | Field weight computed as harmonic mean and normalised weight
Table 9: Structure of Table ’topic_nets_topics_similarities’ Field | Description
---|---
topic_id1 | Part of PK, and FK to the topic_nets_topics table
topic_id1 | Part of PK, and FK to the topic_nets_topics table
similarity | Cosine similarity between the two topics’ embeddings

##  5  Advantages and Use-cases

The proposed S&TO with unconventional topics could have the following
advantages and Use cases.

###  5.1 Knowledge expansion

S&TO can broaden the scope of knowledge representation beyond existing
ontologies by incorporating previously unconsidered topics. Offering a more
holistic and comprehensive view of the knowledge landscape can lead to new
insights and discoveries. In medical research, unconventional topics like
holistic therapies or mindfulness practice might be incorporated into the
ontology to provide a more comprehensive view of the more extensive health and
wellness landscape [20].

###  5.2 Interdisciplinary Collaboration

Second, by connecting topics from diversified fields that may have
commonalities or be connected, an unconventional topics ontology encourage
interdisciplinary collaboration. This can encourage the discovery of new
research areas and cross-disciplinary cooperation, leading to novel solutions
to complicated issues. For example, an ontology incorporating computer science
and psychology issues could make it easier for academics in both domains to
collaborate on human-computer interaction or affective computing [21].

###  5.3 Scalability and Adaptability

An unconventional topics ontology has the benefit of being easily updatable
and adaptable to reflect the most contemporary developments and topics,
resulting in a dynamic and flexible knowledge representation system. This
capability is significant in fast-paced sectors like technology and
healthcare, where new topics and concepts develop regularly. For example, an
ontology that includes topics relating to emerging technologies such as
artificial intelligence [22] or blockchain [23] can be easily updated to
include new concepts and trends.

##  6 Limitations

The current version of the proposed S&TO ontology has the following
limitations.

###  6.1 Limited dataset

The current version of S&TO ontology is built on the Semantic scholar dataset
covering 393,991 S&T articles from October 2021 to August 2022. However, it
could be built on more datasets.

###  6.2 Topic labelling

Since S&TO ontology utilises BERTopic, an unsupervised topic computation
library, ontology suffers from the consequences of unlabeled topics. Due to a
lack of labelled data, it may be challenging to determine the significance and
relevance of unconventional topics and distinguish them from noise or
irrelevant topics. This is incredibly challenging when working with massive,
complicated datasets containing various topics.

###  6.3 Domain coverage

S&TO can capture various research topics and domains by covering these four
domains: computer science, physics, chemistry and Engineering. However, many
other disciplines and subfields within science and technology are not yet
included in S&TO. For example, biology, environmental science, and
neuroscience are all essential areas of research that could be integrated into
an ontology to create a more comprehensive and multidisciplinary framework for
understanding scientific research. Expanding the coverage of S&TO to include
additional domains would have several potential benefits.

###  6.4 Topic quality

While S&TO depicts a significant effort towards organising and categorising
S&T topics, there is still room for improvement regarding the quality of the
topics included in the ontology.

##  7 Conclusion and Future Work

S&T Ontology, an automated ontology of science and technology that includes
all scientific study topics, was introduced in this paper. We constructed an
ontology encompassing four different science domains by utilising BERTopic on
a collection of 393,991 scientific articles acquired from Semantic Scholar
from October 2021 to August 2022. S&TO can be updated using BERTopic on recent
datasets, offering a dynamic and flexible foundation for knowledge
representation. S&TO ontology has the potential to broaden the scope of
knowledge representation and stimulate interdisciplinary collaboration, making
it a valuable resource for scientists and technologists.

The S&TO ontology constantly evolves and requires ongoing enhancements to meet
the expanding knowledge landscape’s demands. Currently, S&TO is growing, and
we are employing topic labelling techniques to improve the organisation and
comprehension of different topics by giving them meaningful "tags". This makes
it easy for users to browse the ontology and derive valuable insights.
Furthermore, we intend to improve the topic quality by investigating
additional methodologies and algorithms for topic modelling and clustering.
This will strengthen the ontology’s accuracy and efficacy in describing the
knowledge landscape. In addition, we intend to expand the ontology to a more
extensive dataset, allowing for the inclusion of more unconventional
categories and topics, which will improve and diversify the knowledge base.

## References

  * Saif et al. [2012] H. Saif, Y. He, H. Alani,  Semantic sentiment analysis of twitter,  in: The Semantic Web–ISWC 2012: 11th International Semantic Web Conference, Boston, MA, USA, November 11-15, 2012, Proceedings, Part I 11, Springer, 2012, pp. 508–524.
  * Osborne et al. [2016] F. Osborne, A. Salatino, A. Birukou, E. Motta,  Automatic classification of springer nature proceedings with smart topic miner,  in: The Semantic Web–ISWC 2016: 15th International Semantic Web Conference, Kobe, Japan, October 17–21, 2016, Proceedings, Part II 15, Springer, 2016, pp. 383–399.
  * Dudáš et al. [2018] M. Dudáš, S. Lohmann, V. Svátek, D. Pavlov,  Ontology visualization methods and tools: a survey of the state of the art,  The Knowledge Engineering Review 33 (2018) e10.
  * Salatino et al. [2018] A. A. Salatino, F. Osborne, E. Motta,  Augur: forecasting the emergence of new research topics,  in: Proceedings of the 18th ACM/IEEE on joint conference on digital libraries, 2018, pp. 303–312.
  * Fathalla et al. [2018] S. Fathalla, S. Vahdati, S. Auer, C. Lange,  Semsur: a core ontology for the semantic representation of research findings,  Procedia Computer Science 137 (2018) 151–162.
  * Salatino et al. [2018] A. A. Salatino, T. Thanapalasingam, A. Mannocci, F. Osborne, E. Motta,  The computer science ontology: a large-scale taxonomy of research areas,  in: The Semantic Web–ISWC 2018: 17th International Semantic Web Conference, Monterey, CA, USA, October 8–12, 2018, Proceedings, Part II 17, Springer, 2018, pp. 187–205.
  * Wade [2022] A. D. Wade,  The semantic scholar academic graph (s2ag),  in: Companion Proceedings of the Web Conference 2022, 2022, pp. 739–739.
  * Grootendorst [2022] M. Grootendorst,  Bertopic: Neural topic modeling with a class-based tf-idf procedure,  arXiv preprint arXiv:2203.05794 (2022).
  * McInnes et al. [2018] L. McInnes, J. Healy, J. Melville,  Umap: Uniform manifold approximation and projection for dimension reduction,  arXiv preprint arXiv:1802.03426 (2018).
  * Hartigan and Wong [1979] J. A. Hartigan, M. A. Wong,  Algorithm as 136: A k-means clustering algorithm,  Journal of the royal statistical society. series c (applied statistics) 28 (1979) 100–108.
  * McInnes et al. [2017] L. McInnes, J. Healy, S. Astels,  hdbscan: Hierarchical density based clustering.,  J. Open Source Softw. 2 (2017) 205.
  * Müllner [2011] D. Müllner,  Modern hierarchical, agglomerative clustering algorithms,  arXiv preprint arXiv:1109.2378 (2011).
  * Fernández et al. [2009] M. Fernández, C. Overbeeke, M. Sabou, E. Motta,  What makes a good ontology? a case-study in fine-grained knowledge reuse,  in: The Semantic Web: Fourth Asian Conference, ASWC 2009, Shanghai, China, December 6-9, 2009. Proceedings 4, Springer, 2009, pp. 61–75.
  * Maedche and Staab [2002] A. Maedche, S. Staab,  Measuring similarity between ontologies,  in: Knowledge Engineering and Knowledge Management: Ontologies and the Semantic Web: 13th International Conference, EKAW 2002 Sigüenza, Spain, October 1–4, 2002 Proceedings 13, Springer, 2002, pp. 251–263.
  * Brewster et al. [2004] C. Brewster, H. Alani, S. Dasmahapatra, Y. Wilks,  Data driven ontology evaluation (2004).
  * Sabou et al. [2007] M. Sabou, J. Gracia, S. Angeletou, M. d’Aquin, E. Motta,  Evaluating the semantic web: A task-based approach,  in: The Semantic Web: 6th International Semantic Web Conference, 2nd Asian Semantic Web Conference, ISWC 2007+ ASWC 2007, Busan, Korea, November 11-15, 2007. Proceedings, Springer, 2007, pp. 423–437.
  * Buitelaar et al. [2004] P. Buitelaar, T. Eigner, T. D. OntoSelect,  A dynamic ontology library with support for ontology selection in: Proc. of the demo session at the international semantic web conference,  Hiroshima, Japan, Nov (2004).
  * Fernandes and Bernardino [2018] D. Fernandes, J. Bernardino,  Graph databases comparison: Allegrograph, arangodb, infinitegraph, neo4j, and orientdb.,  in: Data, 2018, pp. 373–380.
  * Sánchez et al. [2015] D. Sánchez, M. Batet, S. Martínez, J. Domingo-Ferrer,  Semantic variance: an intuitive measure for ontology accuracy evaluation,  Engineering Applications of Artificial Intelligence 39 (2015) 89–99.
  * Howard [2019] J. Howard,  Artificial intelligence: Implications for the future of work,  American journal of industrial medicine 62 (2019) 917–926.
  * Xu [2019] W. Xu,  Toward human-centered ai: a perspective from human-computer interaction,  interactions 26 (2019) 42–46.
  * Zhang and Lu [2021] C. Zhang, Y. Lu,  Study on artificial intelligence: The state of the art and future prospects,  Journal of Industrial Information Integration 23 (2021) 100224.
  * Monrat et al. [2019] A. A. Monrat, O. Schelén, K. Andersson,  A survey of blockchain from the perspectives of applications, challenges, and opportunities,  IEEE Access 7 (2019) 117134–117151.

##  Appendix A BERTopic Parameters

Here, we summarised the parameters we set throughout S&TO development. The
following parameters have been set:

  * •

“`n_neighbors`“ refers to the number of neighbouring data points needed to
estimate the manifold. Large sample point embeddings produce a more global
perspective of the structure, while low values produce a narrower one. To get
a good strike, we set n=2 as the result of the estimation.

  * •

“`n_components`“ refers to the number of components after the reduction in
dominance. This value directly affects the clustering performance, so it is
necessary to set an optimal value. By default, it is set to 5 to reduce the
dimensionality as much as possible while maximizing the information in the
generated embeddings.

  * •

“`low_memory`“: It is set to TRUE because we use a huge dataset and need a lot
of memory.

  * •

“`min_cluster_size`“: The number of cluster generations highly relies on the
cluster size. It is necessary to adjust the minimum size. After several
experiments, a cluster size of 50 was found to be the optimal one. While high
value gives few clusters of considerable size, and low value gives
microclusters.

  * •

“`metric`“: metric, like HDBSCAN, calculates the distances. Here, we went with
Euclidean as, after reducing the dimensionality, we have low dimensional data,
and not much optimisation is necessary. However, if you increase
“`n_components`“ in UMAP, it would be advised to investigate metrics that work
with high dimensional data.

  * •

“`prediction_data`“: Make sure you always set this value to True, as it is
needed to predict new points later. You can set this to False if you do not
wish to predict any unseen data points.

  * •

“`min_samples`“: It is automatically set to “`min_cluster_size`“ and controls
the number of outliers generated. Setting this value significantly lower than
“`min_cluster_size`“ might help you reduce the amount of noise you will get.
Do note that outliers are typical to be expected, and forcing the output to
have no outliers may not properly represent the data.

  * •

“`top_n_words`“ refers to the number of words extracted per topic. In
practice, we keep this value below 30, preferably between 10 and 20. The
reasoning is that the more words representing a topic, the less relevant it
may be. In this case, the top words are most representative of the topic and
are the focus.

  * •

“`min_topic_size`“ specifies the minimum size of a topic. The lower the size
value, the more topics are created. If the value is set too high, no topics
may be created. We set this value too low, and we get many micro-clusters.

  * •

“`calculate_probabilities`“ give probabilities of all topics per document.
This could slow down the extraction of topics for a large number of many
documents.

  * •

“`low_memory`“ set to true ensures that less memory is used in the
calculations. This slows computation but allows UMAP to run on machines with
little memory.

  * •

“`diversity`“ reports a range of topic diversity from 0 to 1, where 0
indicates no diversity and 1 indicates a lot of diversity. Higher diverse
topics mean less coherent topics in smaller cluster sizes. In our case, the
diversity is assumed to be 0.4 or above.


#  On the well-posedness of a nonlinear diffusive
SIR epidemic model

Curtis Holliman and Harry Prieto

(Date: September 18, 2021)

###### Abstract.

This work considers an extension of the SIR equations from epidemiology that
includes a spatial variable. This model, referred to as the Kermack-McKendrick
equations (KM), is a pair of diffusive partial differential equations, and
methods developed for the Navier-Stokes equations and models of fluid dynamics
are adapted to prove that KM is well-posed in the homogenous Sobolev spaces
with exponent $0\leq s<2$.

###### Key words and phrases:

SIR equations, diffusion equations, Kermack-McKendrick equations, well-
posedness, COVID-19, time-weighted spaces, epidemic model, initial value
problem, bilinear estimates, well-posedness, Sobolev spaces, population
dynamics.

##  1\. Introduction

The outbreak of the COVID-19 pandemic galvanized the efforts to improve the
predictive power of the mathematics modeling the spread of disease. The most
well-known of these models is given by the SIR equations. The present paper
considers an extension of these equations that includes a spatial variable.
These equations, which we call the Kermack-McKendrick equations (KM), change
the SIR model from a set of ordinary differential equations (ODEs) into a
coupled pair of diffusive partial differential equations (PDEs). We
investigate the well-posedness of KM using methods that were originally
developed for fluid dynamics, in particular for the Navier-Stokes equations
(NS).

The SIR model was pioneered by W. Kermack and A. McKendrick in [20] and is an
example of a compartmental model. In their original formulation, the
population is partitioned into the disjoint groups, or compartments,
consisting of the susceptible $(S)$, infectious $(I)$, and recovered $(R)$
individuals. These quantities are strictly functions of time $t$, and the ODEs
they give rise to, called the SIR-equations, are given by

$\displaystyle S^{\prime}(t)$ $\displaystyle=-\beta SI,$ $\displaystyle
I^{\prime}(t)$ $\displaystyle=\beta SI-\mu I,$ (1.1) $\displaystyle
R^{\prime}(t)$ $\displaystyle=\mu I.$

Here $\beta$ and $\mu$ are constants representing the transmission and
recovery rates, respectively. For a detailed description of the basic
assumptions of the model and the technical underpinnings that lead to its
equations we refer to [5], [18], [21], [27]. For a review of the history of
the SIR equations, the interested reader may consult [2]

Since its inception, most of the attention has concentrated on using the SIR
model to understand disease transmission, and over the years important
applications to public health have been found [29]. A prime example of this is
vaccination, where the transition rate between compartments is accelerated,
since vaccinated individuals can be immediately placed in the $R$, (recovered)
compartment. Kermack and McKendrick applied their model to the 1906 bubonic
outbreak in Bombay [2], but the model has also been employed in a wide variety
of circumstances such as the evolution of the dengue outbreaks in Cuba (1997)
and Venezuela (2000) [16], the classical swine flu in the Netherlands
(1997-1998) [26], and many others.

Considerable work has been devoted to improving the SIR model itself. Notably,
the original model has been expanded by the addition of more compartments. For
instance, along with the traditional three compartments, some models also
include incubation and latency periods ($E$) [21]. More recently, some models
have incorporated compartments to account for immunization and vaccination in
populations [11], [30].

The descriptive and predictive power of the model has been applied to the
COVID-19 pandemic. The great interest generated by the topic and its timely
nature are evidenced by the explosion of the literature on the subject. For a
few applications of the SIR model in this context, we refer the readers to
[1], [7], [14].

One fundamental issue in using the SIR equations to model a pandemic, however,
is that it completely ignores spatial information. As compared to a localized
disease outbreak, the location and concentration of affected individuals in a
global setting would most certainly contribute to the time evolution of the
model. With this issue in mind, a generalized compartmental SIR model is
constructed by allowing individuals to move via random walks. For an
investigation of random walks in this context, we refer the reader to [25],
and for the foundational work on Brownian motion, which lies at the core of
these diffusion processes, we refer the reader to [8]. According to these
models, an individual moves randomly in a direction with the amplitude of the
Brownian motion equalling $D_{S}$ and $D_{I}$ for the Susceptible and Infected
individuals, respectively. After taking an expected value, these constants
become the coefficients in the diffusion linear symbol, and we obtain the
equations

$\begin{split}&S_{t}=D_{S}\Delta S-\beta SI,\\\ &I_{t}=D_{I}\Delta I+\beta
SI-\mu I.\end{split}$ (1.2)

It is worth noting that if there is no displacement of the individuals, then
$D_{S}=D_{I}=0$, and we obtain the original SIR equations. A detailed
description of how to obtain (1.2) appears in [6], and is reviewed in [9],
[10]. Additionally, the traveling wave solutions of (1.2) have been
investigated in [3], [17], [23], [28].

In this work, we will assume the spatial dimension in (1.2) to be one.
Additionally, we make the change of variables
$u(x,t)=S(x\sqrt{D_{S}/\beta},t/\beta)$ and
$v(x,t)=I(x\sqrt{D_{I}/\beta},t/\beta)$ to obtain a slightly simpler non-
dimensional version of (1.2). In this new version, the coefficient for the
linear $v$ term will be $-\mu\beta$ which we will relabel as $-\mu$. We refer
to this new system of equations as the Kermack-McKendrick equations (KM), and
they are given by

$\displaystyle u_{t}+uv-u_{xx}=0,$ $\displaystyle v_{t}-uv-v_{xx}-\mu v=0,$
(1.3) $\displaystyle u(x,0)=\varphi(x),\quad v(x,0)=\psi(x),\quad
t\in[0,T),x\in\mathbb{R}.$

We will approach the initial value problem (ivp) posed by (1.3) using theory
developed in the study of diffusion equations in fluid dynamics. To that end,
we adapt the methods developed and applied in [4], [16], [19]. Specifically,
we investigate the well-posedness of KM in the sense of Hadamard. In order to
rigorously state what we mean by well-posedness, we must also state precisely
what spaces we are taking the initial data and solutions to be in. We will
take the initial data $\varphi,\psi$ to be in the homogeneous Sobolev spaces
$\dot{H}^{s}$, and the solution $u$ to be in the intersection of $\dot{H}^{s}$
and the time-weighted $L^{4}$ spaces, which we will call $X^{s}$. The precise
definitions of these spaces are provided for the reader in (2.1) and (2.2),
respectively.

The idea of well-posedness was introduced in [15], and we say that the KM
equations are well-posed with initial data in
$(\varphi,\psi)\in\dot{H}^{s}\times\dot{H}^{s}$ and solution in $(u,v)\doteq
W(\varphi,\psi)\in X^{s}\times X^{s}$, if the following three conditions hold:

  * I)

Existence. For any initial data
$(\varphi,\psi)\in\dot{H}^{s}\times\dot{H}^{s}$, there exists a solution
$(u,v)\in X^{s}\times X^{s}$ to KM.

  * II)

Uniqueness. The solution $(u,v)\doteq W(\varphi,\psi)$ is unique in the space
$X^{s}\times X^{s}$.

  * III)

Continuity/Stability. The solution map $W:\dot{H}^{s}\times\dot{H}^{s}\to
X^{s}\times X^{s}$ is continuous.

With this definition in mind, we now state the primary result of this work.

######  Theorem 1.

Let $0\leq s<2$, $0<T<\frac{1}{6\mu}$ and
$(\varphi,\psi)\in\dot{H}^{s}\times\dot{H}^{s}$ satisfying the smallness
condition

$\displaystyle\|(\varphi,\psi)\|_{\dot{H}^{s}\times\dot{H}^{s}}\leq\frac{1}{18C_{\ell}C_{b}},$
(1.4)

where the constants $C_{\ell}$ and $C_{b}$ are given in Propositions 1 and 2,
respectively. Then the KM ivp (1.3) has a unique solution $(u,v)\in
X^{s}\times X^{s}$. Moreover, the solution map,
$W:\dot{H}^{s}\times\dot{H}^{s}\to X^{s}\times X^{s}$, which takes
$(\varphi,\psi)\mapsto(u,v)$, is Lipschitz continuous.

The proof of Theorem 1 revolves around the techniques developed in [19] to
prove the well-posedness of the Navier-Stokes (NS) equations

$\begin{split}&u_{t}+(u\cdot\nabla)u+\mu\Delta u+\nabla p=0,\\\
&\text{div}\;u=0,\end{split}$ (1.5)

where $p$ is the pressure of the fluid, and $\mu$ its viscosity. Here, the
Brownian motion amplitudes in (1.2), $D_{S}$ and $D_{I}$, act in a similar
manner as the viscosity coefficient $\mu$ in NS. The strategy that was
implemented for NS was built on the foundations developed in [12] and
consisted in showing that the associated integral operator had a fixed point
in a suitable space. These ideas have also been used in other hydrodynamic
equations as such the viscous Burgers (vB) equation

$\displaystyle u_{t}+uu_{x}+\mu u_{xx}=0,$ (1.6)

which was examined in [4], and the $k$-Burgers equation

$\displaystyle u_{t}+u^{k}u_{x}+\mu u_{xx}=0,\quad\quad k={1,2,3,\cdots}.$
(1.7)

which was investigated in [16].

Outline of the paper. The present paper is organized as follows. In section 2,
we provide a number of preliminaries, including the definitions of our
function spaces as well as linear estimates for the diffusion operator. In
section 3, we first reformulate KM as a fixed point problem and then prove
that this associated integral operator is a contraction mapping. In section 4,
we provide a proof of the bilinear estimate that was needed in order to
establish the contraction in section 3.

##  2\. Preliminaries and Linear Estimates

In this section, we set up our notation and collect the basic estimates that
will be used in the course of proving the main result.

Notation. We say $A\lesssim B$ if there exists a constant $C>0$ such that
$A\leq CB$. If $A\lesssim B$ and $B\lesssim A$ we write $A\simeq B$.

Function Spaces. The spaces that we will use are the combination of
homogeneous Sobolev spaces and time-weighted $L^{p}$ spaces, and we briefly
provide a definition of these spaces and their norms.

We begin with the homogenous Sobolev space, $\dot{H}^{s}$, which is a subspace
of the Tempered Distributions where the following norm is finite. We take the
the Riesz potential $D_{x}=(-\partial^{2}_{x})^{1/2}$, or equivalently, the
Fourier multiplier given by $\widehat{D_{x}f}(\xi)=|\xi|\widehat{f}(\xi)$, and
then define the $\dot{H}^{s}$-norm as

$\displaystyle\|u\|_{\dot{H}^{s}}\doteq\|D_{x}^{s}u\|_{L^{2}}=\Big{(}\int_{\mathbb{R}}|\xi|^{2s}|\widehat{u}(\xi)|^{2}d\xi\Big{)}^{1/2}.$
(2.1)

Next, we define our time-weighted $L^{p}$ spaces. For any fixed $T$,
$\alpha\geq 0$, we define the subspace $C^{\alpha}((0,T)$;$L^{p})\subset
C((0,T)$;$L^{p})$ by

$\displaystyle C^{\alpha}_{0}((0,T);L^{p})=\Big{\\{}u\in
C((0,T);L^{p}):\sup_{t\in(0,T)}t^{\alpha}\|u\|_{L^{p}}<\infty\;\;\text{and}\;\;\lim_{t\to
0^{+}}t^{\alpha}\|u\|_{L^{p}}=0\Big{\\}}.$

For given $s,\alpha,p$ we can now define $X^{s,\alpha,p}=\dot{H}^{s}\cap
C^{\alpha}_{0}((0,T);L^{p})$; however, in our particular case, there is a
relationship between $s$ and $\alpha$ that arises in our proof of the Bilinear
Estimate needed for Theorem 1. We thus will restrict our attention to

$\displaystyle\alpha\doteq\frac{1}{2}-\frac{s}{4}.$

Additionally, the only $p$ that we will utilize is $p=4$ as it also arises in
the Bilinear Estimate after applying the generalized Hölder’s inequality in
$L^{2}$. In view of these choices, we define $X^{s}\doteq
X^{s,\frac{1}{2}-\frac{s}{4},4}$, and take the norm to be

$\displaystyle\|u\|_{X^{s}}\doteq\sup_{t\in(0,T)}\|u\|_{\dot{H}^{s}}+\sup_{t\in(0,T)}t^{\alpha}\|u\|_{L^{4}},\quad\text{where}\quad\alpha=\frac{1}{2}-\frac{s}{4}.$
(2.2)

Finally, since we are working with two simultaneous equations, we define our
norms on the product spaces in the usual fashion. For pairs, we have

$\displaystyle\|(\varphi,\psi)\|_{\dot{H}^{s}\times\dot{H}^{s}}$
$\displaystyle\doteq\|\varphi\|_{\dot{H}^{s}}+\|\psi\|_{\dot{H}^{s}},$
$\displaystyle\|(u,v)\|_{X^{s}\times X^{s}}$
$\displaystyle\doteq\|u\|_{X^{s}}+\|v\|_{X^{s}}.$

The Diffusion Operator and Linear Estimates. We begin by considering the
diffusion equation

$\begin{split}&w_{t}+w_{xx}=0,\\\ &w(0,x)=w_{0}(x).\end{split}$ (2.3)

We take the solution operator $S$ of (2.3) as the Fourier multiplier given by

$\displaystyle\widehat{S(t)\varphi}(\xi)=e^{-t\xi^{2}}\widehat{\varphi}(\xi).$

Before we proceed with our estimates regarding the operator $S$, we state the
well-known Hardy-Littlewood-Sobolev inequality for the convenience of the
reader. For a proof of this estimate, we refer the reader to [22], §4.3.

######  Lemma 1 (Hardy-Littlewood-Sobolev).

Suppose that $0<\alpha<1$, $1<\alpha<p<1/\alpha$ and $q$ satisfies

$\displaystyle\frac{1}{q}=\frac{1}{p}-\alpha.$

If $f\in L^{p}$, then $D^{-\alpha}_{x}f\in L^{q}$ (where
$D_{x}=(-\partial^{2}_{x})^{1/2}$ is the Riesz potential) and there exists a
constant $C=C(\alpha,p)$ such that

$\displaystyle\|D^{-\alpha}_{x}f\|_{L^{q}}\leq C\|f\|_{L^{p}}.$ (2.4)

We now consider three estimates for $S$ that will be used throughout this
work. The first is an $L^{p}$-$L^{q}$ estimate that can be found in [13].

######  Lemma 2.

Let $S(t)$ be the solution operator for the heat equation (2.3) with initial
data $\varphi$ and $1\leq q\leq p\leq\infty$. Then for $t>0$ we have the
estimate

$\displaystyle\|S(t)\varphi\|_{L^{p}}\leq(4\pi
t)^{-\frac{1}{2}(\frac{1}{q}-\frac{1}{p})}\|\varphi\|_{L^{q}},$ (2.5)

######  Lemma 3.

For $s\geq 0$, there exists a positive constant $c_{s}$ such that

$\displaystyle\big{\|}D^{s}_{x}S(t)f\big{\|}_{L^{2}}\leq
c_{s}t^{-s/2}\big{\|}f\big{\|}_{L^{2}}.$

###### Proof.

By Parseval’s identity and Hölder’s inequality, we have

$\displaystyle\big{\|}D^{s}_{x}S(t)f\big{\|}_{L^{2}}=\frac{1}{2\pi}\big{\|}|\xi|^{s}e^{-\xi^{2}t}\widehat{f}(\xi)\big{\|}_{L^{2}}\leq\frac{1}{2\pi}\big{\|}|\xi|^{s}e^{-\xi^{2}t}\big{\|}_{L^{\infty}}\big{\|}\widehat{f}\big{\|}_{L^{2}}.$
(2.6)

We see that the $L^{\infty}$-norm of $|\xi|^{r}e^{-\xi^{2}t}$ is now easily
bounded by

$\displaystyle\big{\|}|\xi|^{s}e^{-\xi^{2}t}\big{\|}_{L^{\infty}}\leq\Big{(}\frac{s}{2e^{t}}\Big{)}^{\frac{r}{2}}t^{-\frac{s}{2}}.$
(2.7)

We can then continue bounding (2.6) by using (2.7) to get

$\displaystyle\big{\|}D^{s}_{x}S(t)f\big{\|}_{L^{2}}\leq\frac{1}{2\pi}\Big{(}\frac{s}{2e^{t}}\Big{)}^{\frac{s}{2}}t^{-\frac{s}{2}}\big{\|}\widehat{f}\big{\|}_{L^{2}}=\Big{(}\frac{s}{2e^{t}}\Big{)}^{\frac{s}{2}}t^{-\frac{s}{2}}\big{\|}f\big{\|}_{L^{2}}.$

We see then that taking $c_{s}=(\frac{s}{2})^{\frac{s}{2}}$ we obtain our
desired estimate. ∎

######  Proposition 1 (Linear Estimate).

For $0\leq s<2$, the mapping $\phi\mapsto S(t)\phi$, where $S$ is the solution
operator to the diffusion equation, continuously maps $\dot{H}^{s}\to X^{s}$,
and we have the inequality

$\displaystyle\|S(t)\varphi\|_{X^{s}}\leq C_{\ell}\|\varphi\|_{\dot{H}^{s}}.$
(2.8)

where the constant $C_{\ell}$ depends on $s$ and $T$.

###### Proof.

From the definition of the $X^{s}$-norm, we have

$\displaystyle\|S(t)\varphi\|_{X^{s}}=\sup_{t\in(0,T)}\|S(t)\varphi\|_{\dot{H}^{s}}+\sup_{t\in(0,T)}t^{\alpha}\|S(t)\varphi\|_{L^{4}}.$

The first term is handled with the usual methods using Plancherel’s Theorem
and the definition of the diffusion solution operator $S$. Indeed, we have

$\displaystyle\|S(t)\varphi\|_{\dot{H}^{s}}^{2}$
$\displaystyle=\int_{\mathbb{R}}|\xi|^{2s}e^{-t\xi^{2}}|\widehat{\varphi}(\xi)|^{2}d\xi\leq\int_{\mathbb{R}}|\xi|^{2s}|\widehat{\varphi}(\xi)|^{2}d\xi=\|\varphi\|_{\dot{H}^{s}}^{2}.$

Thus, the first term has the upper bound

$\displaystyle\sup_{t\in(0,T)}\|S(t)\varphi\|_{\dot{H}^{s}}\leq\sup_{t\in(0,T)}\|\varphi\|_{\dot{H}^{s}}.$

For the second term, we begin by restricting our attention to the quantity
under the $L^{4}$-norm. Applyling the Riesz Derivative and its inverse with
exponent $1/4$ gives us

$\displaystyle\|S(t)\varphi\|_{L^{4}}=\|D^{-1/4}_{x}D^{1/4}_{x}S(t)\varphi\|_{L^{4}}.$

Now we apply the Hardy-Littlewood-Sobolev Lemma, or Lemma 1, with $p=2,q=4$
and $\alpha=1/4$ to get

$\displaystyle\|D^{-1/4}_{x}D^{1/4}_{x}S(t)\varphi\|_{L^{4}}\lesssim\|D^{1/4}_{x}S(t)\varphi\big{\|}_{L^{2}}.$

Next, we again apply the Riesz Derivative along with its inverse, both with
exponent $s/2$, after which we can apply Lemma 3 with the corresponding
$\alpha=1/4-s/2$. Note that this Lemma requires $\alpha\geq 0$, which
corresponds to our hypothesis $0\leq s\leq 1/2$. We thus get

$\displaystyle\|D^{1/4}_{x}S(t)\varphi\big{\|}_{L^{2}}=\|D^{1/4-s/2}_{x}D^{s/2}_{x}S(t)\varphi\big{\|}_{L^{2}}\lesssim
t^{-1/8+s/4}\|D^{s/2}_{x}\varphi\|_{L^{2}}=t^{-1/8+s/4}\|\varphi\|_{\dot{H}^{s}}.$

Using this upper bound for the $L^{4}$-norm of $S(t)\phi$, we get the time-
weighted norm to be

$\displaystyle\sup_{t\in(0,T)}t^{\alpha}\|S(t)\varphi\|_{L^{4}}\lesssim\sup_{t\in(0,T)}t^{\alpha}t^{-1/8+s/4}\|\varphi\|_{\dot{H}^{s}}.$

From the definition of the $X^{s}$ space, we have $\alpha=1/2-s/4$ which
implies that the exponent for $t$ is
$\alpha-\frac{1}{8}+\frac{s}{4}=\frac{3}{8}$. Thus, we have the upper bound

$\displaystyle\sup_{t\in(0,T)}t^{\alpha-1/8+s/4}\|\varphi\|_{\dot{H}^{s}}\leq
T^{3/8}\|\varphi\|_{\dot{H}^{s}}.$

Putting our results together for each term gives the desired inequality (2.8),
where the associated constant with the upper bound depends on $s$ and $T$. ∎

##  3\. Proof of Theorem 1

We begin by first reformulating the KM equations as a fixed point problem and
then proving that the associated integral operator is a contraction. The
Banach Contraction Mapping Theorem then implies the existence and uniqueness
of solutions to the KM ivp. After this task is accomplished, we prove that the
solution map $W$ is Lipschitz continuous.

Reformulating (1.3) as a fixed point problem. We begin by taking the Fourier
transform in the spatial variable of both sides of the $u$ and $v$ equations
in (1.3). Additionally, using the properties
$\widehat{\partial_{x}u}=i\xi\widehat{u}$ and
$\widehat{\partial_{t}u}=\partial_{t}\widehat{u}$, we get

$\displaystyle\partial_{t}\widehat{u}+\widehat{uv}+\xi^{2}\widehat{u}=0,$
$\displaystyle\partial_{t}\widehat{v}-\widehat{uv}+\xi^{2}\widehat{v}+\mu\widehat{v}=0.$

Next, we isolate the terms corresponding to the linear heat equation on the
left-hand side and multiply by the integrating factor $e^{t\xi^{2}}$. The
integrating factor allows us to then write the left-hand side of each equation
as an exact derivative as for instance
$\partial_{t}(e^{t\xi^{2}}\widehat{u})=e^{t\xi^{2}}(\partial_{t}\widehat{u}+\xi^{2}u)$.
We thus obtain

$\displaystyle\partial_{t}(e^{t\xi^{2}}\widehat{u})=-e^{t\xi^{2}}\widehat{uv},$
$\displaystyle\partial_{t}(e^{t\xi^{2}}\widehat{v})=e^{t\xi^{2}}\Big{(}\widehat{uv}-\mu\widehat{v}\Big{)}.$

We now integrate both equations from $0$ to $t$ using the Fundamental Theorem
of Calculus and the initial data specified in (1.3) to get

$\displaystyle
e^{t\xi^{2}}\widehat{u}-\widehat{\varphi}=-\int_{0}^{t}e^{t^{\prime}\xi^{2}}\widehat{uv}dt^{\prime},$
$\displaystyle
e^{t\xi^{2}}\widehat{v}-\widehat{\psi}=\int_{0}^{t}e^{t^{\prime}\xi^{2}}\Big{(}\widehat{uv}-\mu\widehat{v}\Big{)}dt^{\prime}.$

Next we move the initial data to the right-hand side of each equation and
multiply both sides of both equations by $e^{-t\xi^{2}}$. Finally, applying
the inverse Fourier Transform gives us

$\displaystyle
u(x,t)=\frac{1}{2\pi}\int_{\mathbb{R}}e^{ix\xi}e^{-t\xi^{2}}\widehat{\varphi}d\xi-\frac{1}{2\pi}\int_{\mathbb{R}}e^{ix\xi}e^{-t\xi^{2}}\Big{(}\int_{0}^{t}e^{t^{\prime}\xi^{2}}\widehat{uv}dt^{\prime}\Big{)}d\xi,$
$\displaystyle
v(x,t)=\frac{1}{2\pi}\int_{\mathbb{R}}e^{ix\xi}e^{-t\xi^{2}}\widehat{\psi}d\xi+\frac{1}{2\pi}\int_{\mathbb{R}}e^{ix\xi}e^{-t\xi^{2}}\Big{(}\int_{0}^{t}e^{t^{\prime}\xi^{2}}(\widehat{uv}-\mu\widehat{v})dt^{\prime}\Big{)}.$

We see further that using the solution operator to the heat equation $S$, we
can also rewrite these equations as

$\displaystyle u(x,t)=S(t)\varphi-\int_{0}^{t}S(t-t^{\prime})(uv)dt^{\prime},$
$\displaystyle v(x,t)=S(t)\psi+\int_{0}^{t}S(t-t^{\prime})(uv-\mu
v)dt^{\prime}.$

With the desired form of our equations obtained, we define the operators

$\displaystyle T_{1}(u,v)\doteq
S(t)\varphi-\int_{0}^{t}S(t-t^{\prime})(uv)dt^{\prime},$ (3.1) $\displaystyle
T_{2}(u,v)\doteq S(t)\psi+\int_{0}^{t}S(t-t^{\prime})(uv-\mu v)dt^{\prime}.$
(3.2)

We note here that the initial data $\varphi,\psi$ are assumed to be fixed in
this construction. Should the dependence on the initial data be needed, such
as in the proof of Lipschitz continuity, we will denote the operator with an
appropriate subscript, for instance $T_{1}=T_{1,\varphi}$. Finally, we take
$T$ to be the two component operator

$\displaystyle T(u,v)\doteq(T_{1}(u,v),T_{2}(u,v)).$ (3.3)

We now can rewrite our equation as

$\displaystyle(u,v)=T(u,v).$ (3.4)

Our next objective will be to demonstrate that $T$ has a fixed point using the
Banach Fixed-Point Theorem. One of the key ingredients we will need to
establish a contraction is the following Bilinear Estimate, which is proved in
the next section.

######  Proposition 2 (Bilinear Estimate).

Let $0\leq s<2$. Then there exists a constant $C_{b}>0$ such that for $f,g\in
X^{s}$ we have

$\displaystyle\Big{\|}\int_{0}^{t}S(t-t^{\prime})(fg)dt^{\prime}\Big{\|}_{X^{s}}\leq
C_{b}\|f\|_{X^{s}}\|g\|_{X^{s}}.$ (3.5)

We now show the contraction, which completes the proof of well-posedness under
a smallness asummption.

######  Proposition 3 (Contraction).

Let $0\leq s<2$, $T<\frac{1}{6\mu}$, $\rho=\frac{1}{3C_{b}}$,
$(\varphi,\psi)\in\dot{H}^{s}\times\dot{H}^{s}$ satisfying the smallness
condition

$\displaystyle\|(\varphi,\psi)\|_{\dot{H}^{s}\times\dot{H}^{s}}\leq\frac{1}{18C_{\ell}C_{b}},$
(3.6)

where the constants $C_{\ell}$ and $C_{b}$ are given in Propositions 1 and 2,
respectively. Then the operator $T$ defined in (3.3) maps $T:B(0,\rho)\to
B(0,\rho)$, where
$B(0,\rho)\dot{=}\\{(u,v)\in{X^{s}}\times{X^{s}}:\|(u,v)\|_{{X^{s}}\times{X^{s}}}\leq\rho\\}$
and is a contraction mapping.

The key to proving this proposition lies in the Bilinear Estimate, Proposition
2, whose proof can be found in the next section. We further note that the
hypothesis on $s$ comes from the Bilinear Estimate.

###### Proof.

We begin by first demonstrating that $T:B(0,\rho)\rightarrow B(0,\rho).$
Noting that

$\displaystyle\|T(u,v)\|_{X^{s}\times
X^{s}}=\|T_{1}(u,v)\|_{X^{s}}+\|T_{2}(u,v)\|_{X^{s}},$

we proceed by examining the terms on the right-hand side separately.

Estimating $\|T_{1}(u,v)\|_{X^{s}}$. We have

$\displaystyle\|T_{1}(u,v)\|_{X^{s}}$
$\displaystyle=\Big{\|}S(t)\varphi-\int_{0}^{t}S(t-t^{\prime})(uv)dt^{\prime}\Big{\|}_{X^{s}},$
$\displaystyle\leq\Big{\|}S(t)\varphi\Big{\|}_{X^{s}}+\Big{\|}\int_{0}^{t}S(t-t^{\prime})(uv)dt^{\prime}\Big{\|}_{X^{s}}.$

For the first term, we use Proposition 1, which gives us

$\displaystyle\|S(t)\varphi\|_{X^{s}}\leq C_{\ell}\|\varphi\|_{\dot{H}^{s}}.$

The second term is bounded using Proposition 2, giving us

$\displaystyle\Big{\|}\int_{0}^{t}S(t-t^{\prime})(uv)dt^{\prime}\Big{\|}_{X^{s}}\leq
C_{b}\|u\|_{X^{s}}\|v\|_{X^{s}}=C_{b}\rho^{2}.$ (3.7)

Therefore, we get

$\displaystyle\|T_{1}(u,v)\|_{X^{s}}\leq
C_{\ell}\|\varphi\|_{\dot{H}^{s}}+C_{b}\rho^{2}.$ (3.8)

Estimating $T_{2}(u,v)$. We have

$\displaystyle\|T_{2}(u,v)\|_{X^{s}}$
$\displaystyle=\Big{\|}S(t)\psi+\int_{0}^{t}S(t-t^{\prime})(uv-\mu
v)dt^{\prime}\Big{\|}_{X^{s}},$
$\displaystyle\leq\Big{\|}S(t)\psi\Big{\|}_{X^{s}}+\Big{\|}\int_{0}^{t}S(t-t^{\prime})(uv)dt^{\prime}\Big{\|}_{X^{s}}+\mu\Big{\|}\int_{0}^{t}S(t-t^{\prime})vdt^{\prime}\Big{\|}_{X^{s}}.$

Again, the linear term is estimated using Proposition 1, giving us

$\displaystyle\|S(t)\psi\|_{X^{s}}\leq C_{\ell}\|\psi\|_{\dot{H}^{s}}.$ (3.9)

The second term was already estimated in (3.7). Thus, we will restrict our
attention to the third term, leaving out the coefficient of $\mu$ for the
moment. We begin with the definition of the $X^{s}$-norm, noting that we will
reinsert the constant $\mu$ at the end of the computation. We have

$\displaystyle\Big{\|}\int_{0}^{t}S(t-t^{\prime})vdt^{\prime}\Big{\|}_{X^{s}}=\sup_{t\in(0,T)}\Big{\|}\int_{0}^{t}S(t-t^{\prime})vdt^{\prime}\Big{\|}_{\dot{H}^{s}}+\sup_{t\in(0,T)}t^{\alpha}\Big{\|}\int_{0}^{t}S(t-t^{\prime})vdt^{\prime}\Big{\|}_{L^{4}}.$

Looking at the expression under the supremum, we see that for the Sobolev term
we have

$\displaystyle\Big{\|}\int_{0}^{t}S(t-t^{\prime})vdt^{\prime}\Big{\|}_{\dot{H}^{s}}$
$\displaystyle\leq\int_{0}^{t}\Big{\|}S(t-t^{\prime})vdt^{\prime}\Big{\|}_{\dot{H}^{s}},$
$\displaystyle=\int_{0}^{t}\Big{(}\int_{\mathbb{R}}|\xi|^{2s}e^{-(t-t^{\prime})\xi^{2}}|\widehat{v}(t^{\prime})|^{2}d\xi\Big{)}^{1/2}dt^{\prime},$
$\displaystyle\leq\int_{0}^{t}\Big{(}\int_{\mathbb{R}}|\xi|^{2s}|\widehat{v}(t^{\prime})|^{2}d\xi\Big{)}^{1/2}dt^{\prime},$
$\displaystyle=\int_{0}^{t}\|v\|_{\dot{H}^{s}}dt^{\prime}.$ (3.10)

We can now apply the Mean-Value Theorem, which tells us that for some
$t^{\star}\in(0,T)$, we have

$\displaystyle\int_{0}^{t}\|v\|_{\dot{H}^{s}}dt^{\prime}=t^{\star}\|v(t^{\star})\|_{\dot{H}^{s}}.$

Taking the supremum over $t\in(0,T)$ thus gives us

$\displaystyle\sup_{t\in(0,T)}\Big{\|}\int_{0}^{t}S(t-t^{\prime})vdt^{\prime}\Big{\|}_{\dot{H}^{s}}$
$\displaystyle\leq\sup_{t\in(0,T)}t^{\star}\|v(t^{\star})\|_{\dot{H}^{s}}\leq
T\sup_{t\in(0,T)}\|v\|_{\dot{H}^{s}}.$

For the time-weighted $L^{4}$-term, we first obtain an upper bound by passing
the norm into the integral. We then apply the $L^{p}$–$L^{q}$ Linear Estimate,
Lemma 2, with $p=q=4$.

$\displaystyle\sup_{t\in(0,T)}t^{\alpha}\Big{\|}\int_{0}^{t}S(t-t^{\prime})vdt^{\prime}\Big{\|}_{L^{4}}\leq\sup_{t\in(0,T)}t^{\alpha}\int_{0}^{t}\|v\|_{L^{4}}dt^{\prime}.$

Again, we will apply the Mean-Value Theorem, giving us for some
$t^{\star}\in(0,T)$,

$\displaystyle\sup_{t\in(0,T)}t^{\alpha}\int_{0}^{t}\|v\|_{L^{4}}dt^{\prime}=\sup_{t\in(0,T)}t^{\alpha}t^{\star}\|v(t^{\star})\|_{L^{4}}dt^{\prime}.$

To bound this above and remove the dependence on $t^{\star}$, we note that
$t^{\star}=(t^{\star})^{\alpha}(t^{\star})^{1-\alpha}$ and $0\leq\alpha\leq
1$. Then we use the fact that $t^{\alpha}\leq T^{\alpha}$ and
$(t^{\star})^{1-\alpha}\leq T^{1-\alpha}$. Thus we get

$\displaystyle\sup_{t\in(0,T)}t^{\alpha}t^{\star}\|v(t^{\star})\|_{L^{4}}$
$\displaystyle\leq
T\sup_{t\in(0,T)}(t^{\star})^{\alpha}\|v(t^{\star})\|_{L^{4}}\leq
T\sup_{t\in(0,T)}t^{\alpha}\|v\|_{L^{4}}.$

Thus we can conclude

$\displaystyle\mu\Big{\|}\int_{0}^{t}vdt^{\prime}\Big{\|}_{X^{s}}\leq\mu
T\|v\|_{X^{s}}\leq\mu T\rho.$ (3.11)

Putting these results together, (3.9) and (3.11), we get

$\displaystyle\|T_{2}(u,v)\|_{X^{s}}\leq
C_{\ell}\|\psi\|_{\dot{H}^{s}}+C_{b}\rho^{2}+\mu T\rho.$ (3.12)

Estimating $\|T(u,v)\|_{X^{s}\times X^{s}}$. Putting our results together,
(3.7) and (3.12), we get

$\displaystyle\|T(u,v)\|_{X^{s}\times X^{s}}$
$\displaystyle=\|T_{1}(u,v)\|_{X^{s}}+\|T_{2}(u,v)\|_{X^{s}},$
$\displaystyle\leq
C_{\ell}\|(\varphi,\psi)\|_{\dot{H}^{s}\times\dot{H}^{s}}+2C_{b}\rho^{2}+\mu
T\rho,$

where $C_{\ell}$ is the constant associated with the upper bound from Lemma 1
and $C_{b}$ is the constant associated with the Bilinear Estimate, Proposition
2. Now using the hypothesis, $\rho=\frac{1}{3C_{b}}$ and $T<\frac{1}{6\mu}$,
we see that

$\displaystyle
C_{\ell}\|(\varphi,\psi)\|_{\dot{H}^{s}\times\dot{H}^{s}}+2C_{b}\rho^{2}+\mu
T\rho$
$\displaystyle<C_{\ell}\|(\varphi,\psi)\|_{\dot{H}^{s}\times\dot{H}^{s}}+2C_{b}\Big{(}\frac{1}{3C_{b}}\Big{)}\rho+\mu\frac{1}{6\mu}\rho,$
$\displaystyle=C_{\ell}\|(\varphi,\psi)\|_{\dot{H}^{s}\times\dot{H}^{s}}+\frac{5}{6}\rho.$

Thus, in order for $T$ to map $B(0,\rho)$ into $B(0,\rho)$, we must have

$\displaystyle
C_{\ell}\|(\varphi,\psi)\|_{\dot{H}^{s}\times\dot{H}^{s}}\leq\frac{1}{6}\rho,$

which is equivalent to our hypothesis

$\displaystyle\|(\varphi,\psi)\|_{\dot{H}^{s}\times\dot{H}^{s}}\leq\frac{1}{18C_{\ell}C_{b}}.$

Contraction. Recalling that the initial data $(\varphi,\psi)$ are fixed, we
take two pairs of functions $(u_{1},v_{1}),(u_{2},v_{2})\in X^{s}\times X^{s}$
and compute the norm of $T(u_{1},v_{1})-T(u_{2},v_{2})$. We break this
computation down by first examining the component operators $T_{1}$ and
$T_{2}$.

Estimating the $T_{1}$ difference.  We add and subtract a mixed term
$u_{1}v_{2}$ and regroup terms to obtain

$\displaystyle T_{1}(u_{1},v_{1})-T_{1}(u_{2},v_{2})$
$\displaystyle=-\int_{0}^{t}S(t-t^{\prime})\Big{(}u_{1}v_{1}-u_{2}v_{2}\Big{)}dt^{\prime},$
$\displaystyle=-\int_{0}^{t}S(t-t^{\prime})\Big{(}u_{1}(v_{1}-v_{2})\Big{)}dt^{\prime}-\int_{0}^{t}S(t-t^{\prime})\Big{(}v_{2}(u_{1}-u_{2})\Big{)}dt^{\prime}.$

We now examine this difference in the $X^{s}$ norm. After applying the
triangle inequality, we use Bilinear Estimate, Proposition 2, to further bound
the nonlinearities. We get

$\displaystyle\|T_{1}(u_{1},v_{1})-T_{1}(u_{2},v_{2})\|_{X^{s}}$
$\displaystyle\quad\quad\quad\leq\Big{\|}\int_{0}^{t}S(t-t^{\prime})\Big{(}u_{1}(v_{1}-v_{2})\Big{)}dt^{\prime}\Big{\|}_{X^{s}}+\Big{\|}\int_{0}^{t}S(t-t^{\prime})\Big{(}v_{2}(u_{1}-u_{2})\Big{)}dt^{\prime}\Big{\|}_{X^{s}},$
$\displaystyle\quad\quad\quad\leq
C_{b}\|u_{1}\|_{X^{s}}\|v_{1}-v_{2}\|_{X^{s}}+C_{b}\|v_{2}\|_{X^{s}}\|u_{1}-u_{2}\|_{X^{s}}.$

Now using the hypothesis that both $\|u_{1}\|_{X^{s}}$ and $\|v_{2}\|_{X^{s}}$
are bounded by the constant $\rho$, and recalling that the norm on the product
space is the sum of the norms, we get

$\displaystyle\|T_{1}(u_{1},v_{1})-T_{1}(u_{2},v_{2})\|_{X^{s}}\leq\rho
C_{b}\|(u_{1}-u_{2},v_{1}-v_{2})\|_{X^{s}\times X^{s}}.$ (3.13)

Estimating the $T_{2}$ difference. We first break up the integral to separate
the linear term, giving us

$\displaystyle T_{2}(u_{1},v_{1})-T_{2}(u_{2},v_{2})$
$\displaystyle\quad\quad\quad\quad=\int_{0}^{t}S(t-t^{\prime})\Big{(}u_{1}v_{1}-u_{2}v_{2}\Big{)}dt-\int_{0}^{t}S(t-t^{\prime})\mu(v_{1}-v_{2})dt^{\prime}.$

The first term under the $X^{s}$ norm is precisely the same as that of
$T_{1}$. Thus, adding and subtracting a mixed term and then applying the
Bilinear estimate leads to the same result. Thus we get

$\displaystyle\|T_{2}(u_{1},v_{1})-T_{2}(u_{2},v_{2})\|_{X^{s}}$
$\displaystyle\quad\quad\quad\leq\rho
C_{b}\|(u_{1}-u_{2},v_{1}-v_{2})\|_{X^{s}\times
X^{s}}+\mu\Big{\|}\int_{0}^{t}S(t-t^{\prime})(v_{1}-v_{2})dt^{\prime}\Big{\|}_{X^{s}}.$
(3.14)

To estimate the linear term, we begin with the definition of the $X^{s}$-norm,
noting that we will reinsert the constant $\mu$ at the end of the computation.
We have

$\displaystyle\Big{\|}\int_{0}^{t}S(t-t^{\prime})(v_{1}-v_{2})dt^{\prime}\Big{\|}_{X^{s}}$
$\displaystyle\quad\quad\quad\
=\sup_{t\in(0,T)}\Big{\|}\int_{0}^{t}S(t-t^{\prime})(v_{1}-v_{2})dt^{\prime}\Big{\|}_{\dot{H}^{s}}+\sup_{t\in(0,T)}t^{\alpha}\Big{\|}\int_{0}^{t}S(t-t^{\prime})(v_{1}-v_{2})dt^{\prime}\Big{\|}_{L^{4}}.$

The expression under the supremum can be bounded in precisely the same fashion
as (3.10), giving us

$\displaystyle\Big{\|}\int_{0}^{t}S(t-t^{\prime})(v_{1}-v_{2})dt^{\prime}\Big{\|}_{\dot{H}^{s}}$
$\displaystyle\leq\int_{0}^{t}\|v_{1}-v_{2}\|_{\dot{H}^{s}}dt^{\prime}.$

We can now apply the Mean-Value Theorem, which tells us that for some
$t^{\star}\in(0,T)$, we have

$\displaystyle\int_{0}^{t}\|v_{1}-v_{2}\|_{\dot{H}^{s}}dt^{\prime}=t^{\star}\|v_{1}(t^{\star})-v_{2}(t^{\star})\|_{\dot{H}^{s}}^{2}.$

Taking the supremum over $t\in(0,T)$ thus gives us

$\displaystyle\sup_{t\in(0,T)}\Big{\|}\int_{0}^{t}S(t-t^{\prime})(v_{1}-v_{2})dt^{\prime}\Big{\|}_{\dot{H}^{s}}$
$\displaystyle\leq\sup_{t\in(0,T)}t^{\star}\|v_{1}(t^{\star})-v_{2}(t^{\star})\|_{\dot{H}^{s}},$
$\displaystyle\leq T\sup_{t\in(0,T)}\|v_{1}-v_{2}\|_{\dot{H}^{s}}.$ (3.15)

For the time-weighted $L^{4}$-term, we apply the $L^{p}$–$L^{q}$ Linear
Estimate, Lemma 2, with $p=q=4$.

$\displaystyle\sup_{t\in(0,T)}t^{\alpha}\Big{\|}\int_{0}^{t}S(t-t^{\prime})(v_{1}-v_{2})dt^{\prime}\Big{\|}_{L^{4}}$
$\displaystyle\leq\sup_{t\in(0,T)}t^{\alpha}\int_{0}^{t}\|S(t-t^{\prime})(v_{1}-v_{2})\|_{L^{4}}dt^{\prime},$
$\displaystyle\leq\sup_{t\in(0,T)}t^{\alpha}\int_{0}^{t}\|v_{1}-v_{2}\|_{L^{4}}dt^{\prime}.$

Again, we will apply the Mean-Value Theorem, giving us for some
$t^{\star}\in(0,T)$,

$\displaystyle\sup_{t\in(0,T)}t^{\alpha}\int_{0}^{t}\|v_{1}-v_{2}\|_{L^{4}}dt^{\prime}=\sup_{t\in(0,T)}t^{\alpha}t^{\star}\|v_{1}(t^{\star})-v_{2}(t^{\star})\|_{L^{4}}dt^{\prime}.$

To bound this above and remove the dependence on $t^{\star}$, we note that
$t^{\star}=(t^{\star})^{\alpha}(t^{\star})^{1-\alpha}$ and $0\leq\alpha\leq
1$. Then we use the fact that $t^{\alpha}\leq T^{\alpha}$ and
$(t^{\star})^{1-\alpha}\leq T^{1-\alpha}$. Thus we get

$\displaystyle\sup_{t\in(0,T)}t^{\alpha}t^{\star}\|v_{1}(t^{\star})-v_{2}(t^{\star})\|_{L^{4}}$
$\displaystyle\leq
T\sup_{t\in(0,T)}(t^{\star})^{\alpha}\|v_{1}(t^{\star})-v_{2}(t^{\star})\|_{L^{4}},$
$\displaystyle\leq T\sup_{t\in(0,T)}t^{\alpha}\|v_{1}-v_{2}\|_{L^{4}}.$ (3.16)

Putting everything together, (3) and (3), we get the estimate for the term of
$T_{2}$ as

$\displaystyle\mu\Big{\|}\int_{0}^{t}S(t-t^{\prime})(v_{1}-v_{2})dt^{\prime}\Big{\|}_{X^{s}}$
$\displaystyle\leq\mu
T\big{(}\sup_{t\in(0,T)}\|v_{1}-v_{2}\|_{\dot{H}^{s}}+\sup_{t\in(0,T)}t^{\alpha}\|v_{1}-v_{2}\|_{L^{4}}\big{)},$
$\displaystyle\leq 2\mu T\|v_{1}-v_{2}\|_{X^{s}}.$ (3.17)

Finally, we obtain the full estimate for $T_{2}$ as (3), where the second term
is bounded by (3.17). Hence

$\displaystyle\|T_{2}(u_{1},v_{1})-T_{2}(u_{2},v_{2})\|_{X^{s}}\leq(\rho
C_{b}+2\mu T)\|(u_{1}-u_{2},v_{1}-v_{2})\|_{X^{s}\times X^{s}}.$ (3.18)

Combined Estimate for the $T$ difference. Putting together our estimates for
$T_{1}$ and $T_{2}$, namely, (3.13) and (3.18), we have

$\displaystyle\|T(u_{1},v_{1})-T(u_{2},v_{2})\|_{X^{s}\times X^{s}}$
$\displaystyle\quad\quad\quad=\|T_{1}(u_{1},v_{1})-T_{1}(u_{2},v_{2})\|_{X^{s}}+\|T_{2}(u_{1},v_{1})-T_{2}(u_{2},v_{2})\|_{X^{s}},$
$\displaystyle\quad\quad\quad\leq(2\rho C_{b}+2\mu
T)\|(u_{1}-u_{2},v_{1}-v_{2})\|_{X^{s}\times X^{s}}.$ (3.19)

Our hypotheses that $\rho=\frac{1}{3C_{b}}$ and $T<\frac{1}{6\mu}$ now allow
us to conclude that

$\displaystyle 2\rho C_{b}+2\mu T<1,$

and thus $T$ is indeed a contraction mapping. ∎

With the proof of Proposition 3 complete, we now are ready to prove Theorem 1.

###### Proof of Theorem 1.

We see that the fixed point from Proposition 3 gives us the existence and
uniqueness of the solutions to KM ivp (1.3), thus we restrict our attention to
proving that the solution map $W$ is Lipschitz continuous.

Let
$(\varphi_{1},\psi_{1}),(\varphi_{2},\psi_{2})\in\dot{H}^{s}\times\dot{H}^{s}$,
and $(u_{1},v_{1}),(u_{2},v_{2})\in X^{s}\times X^{s}$ be the corresponding
solutions to the KM initial value problem with these initial data
respectively. Thus, for $W$ the solution operator to KM, our objective is to
estimate the difference

$\displaystyle\|W(\varphi_{1},\psi_{1})-W(\varphi_{2},\psi_{2})\|_{X^{s}\times
X^{s}}=\|(u_{1},v_{1})-(u_{2},v_{2})\|_{X^{s}\times X^{s}}.$ (3.20)

We see that we can reframe this question using the operator $T$ established
above in (3.3). To use $T$, however, we must have fixed initial data. Thus we
will assume that $T$ in this instance uses the initial data
$(\varphi,\psi)=(0,0)$. Thus, we can rewrite the difference inside of the norm
in (3.20) as

$\displaystyle
W(\varphi_{1},\psi_{1})-W(\varphi_{2},\psi_{2})=(S(t)(\varphi_{1}-\varphi_{2}),S(t)(\psi_{1}-\psi_{2}))+(T(u_{1},v_{1})-T(u_{2},v_{2})).$

We therefore can now rewrite (3.20) as

$\displaystyle\|W(\varphi_{1},\psi_{1})-W(\varphi_{2},\psi_{2})\|_{X^{s}\times
X^{s}}$
$\displaystyle\quad\quad\quad\leq\|(S(t)(\varphi_{1}-\varphi_{2}),S(t)(\psi_{1}-\psi_{2}))\|_{X^{s}\times
X^{s}}+\|(T(u_{1},v_{1})-T(u_{2},v_{2})\|_{X^{s}\times X^{s}}.$

For the first term on the right-hand side, we use our linear estimate from
Lemma 1, giving us

$\displaystyle\|(S(t)(\varphi_{1}-\varphi_{2}),S(t)(\psi_{1}-\psi_{2}))\|_{X^{s}\times
X^{s}}\leq
C_{\ell}\|(\varphi_{1}-\varphi_{2},\psi_{1}-\psi_{2})\|_{\dot{H}^{s}\times\dot{H}^{s}}.$

For the second term, we will use the work from proving that $T$ is a
contraction. Setting our contraction constant as $C=2\rho C_{b}+2\mu T<1$, we
use (3.19) to give us

$\displaystyle\|(T(u_{1},v_{1})-T(u_{2},v_{2})\|_{X^{s}\times X^{s}}$
$\displaystyle\leq C\|(u_{1}-u_{2},v_{1}-v_{2})\|_{X^{s}\times X^{s}},$
$\displaystyle=C\|W(\varphi_{1},\psi_{1})-W(\varphi_{2},\psi_{2})\|_{X^{s}\times
X^{s}}.$

Thus, we get

$\displaystyle\|W(\varphi_{1},\psi_{1})-W(\varphi_{2},\psi_{2})\|_{X^{s}\times
X^{s}}$ $\displaystyle\quad\quad\quad\leq
C_{\ell}\|(\varphi_{1}-\varphi_{2},\psi_{1}-\psi_{2})\|_{\dot{H}^{s}\times\dot{H}^{s}}+C\|W(\varphi_{1},\psi_{1})-W(\varphi_{2},\psi_{2})\|_{X^{s}\times
X^{s}}.$

Simplifying this inequality, gives us

$\displaystyle\|W(\varphi_{1},\psi_{1})-W(\varphi_{2},\psi_{2})\|_{X^{s}\times
X^{s}}\leq\frac{C_{\ell}}{1-C}\|(\varphi_{1}-\varphi_{2},\psi_{1}-\psi_{2})\|_{\dot{H}^{s}\times\dot{H}^{s}},$

which implies that $W$ is Lipschitz continuous. ∎

##  4\. Bilinear Estimates

We begin with a proof of Proposition 2. This proof in turn requires analogous
estimates in $\dot{H}^{s}$ and the time-weighted $L^{4}$-space, which were
proved in Lemma 5.

###### Proof of Proposition 2.

Starting with the definition of the $X^{s}$-norm, and passing the norms inside
the integrals to bound above, we get

$\displaystyle\Big{\|}\int_{0}^{t}S(t-t^{\prime})(fg)dt^{\prime}\Big{\|}_{X^{s}}$
$\displaystyle\quad\quad\quad\leq\sup_{t\in(0,T)}\int_{0}^{t}\Big{\|}S(t-t^{\prime})(fg)\Big{\|}_{\dot{H}^{s}}dt^{\prime}+\sup_{t\in(0,T)}t^{\alpha}\int_{0}^{t}\Big{\|}S(t-t^{\prime})(fg)dt^{\prime}\Big{\|}_{L^{4}}dt^{\prime}.$

We now apply Lemma 5 to each term on the right-hand side of this equation. For
the first term, we get

$\displaystyle\sup_{t\in(0,T)}\int_{0}^{t}\Big{\|}S(t-t^{\prime})(fg)(t^{\prime})\Big{\|}_{\dot{H}^{s}}dt^{\prime}\lesssim\sup_{t\in(0,T)}t^{2\alpha}\|f\|_{L^{4}}\|g\|_{L^{4}}.$
(4.1)

For the second term, we have

$\displaystyle\sup_{t\in(0,T)}t^{\alpha}\int_{0}^{t}\Big{\|}S(t-t^{\prime})(fg)\Big{\|}_{L^{4}}dt^{\prime}\lesssim\sup_{t\in(0,T)}t^{2\alpha}\|f\|_{L^{4}}\|g\|_{L^{4}}.$
(4.2)

Combining (4.1) and (4.2), we get

$\displaystyle\Big{\|}\int_{0}^{t}S(t-t^{\prime})(fg)dt^{\prime}\Big{\|}_{X^{s}}$
$\displaystyle\lesssim\sup_{t\in(0,T)}t^{2\alpha}\|f\|_{L^{4}}\|g\|_{L^{4}},$
$\displaystyle\leq\Big{(}\sup_{t\in(0,T)}t^{\alpha}\|f\|_{L^{4}}\Big{)}\Big{(}\sup_{t\in(0,T)}t^{\alpha}\|g\|_{L^{4}}\Big{)},$
$\displaystyle\leq\|f\|_{X^{s}}\|g\|_{X^{s}}.$

The constant associated with the upper bound comes from Lemma 5, and as its
particular value is used to establish the Contraction in Theorem 1, we label
it as $C_{b}$. ∎

To complete the argument for Proposition 2, we use the following beautiful
calculus estimate related to the Beta distribution. A proof is given in [16],
A.2.

######  Lemma 4.

Let $0<a$, $b<1$ and $r=a+b-1$ with $0\leq r\leq 1$, then we have the
following bound

$\displaystyle B(a,b,t)\dot{=}\int_{0}^{t}(t-y)^{-b}y^{-a}dy\leq
c_{a,b}t^{-r}.$ (4.3)

With this estimate in hand, we now proceed to compute the following estimates
on the components of the $X^{s}$-norm, which completes our arguments.

######  Lemma 5.

Let $f$ and $g\in X^{s}$, with $0\leq s<2$ and $0\leq t<T<\infty$. Then the
following inequalities hold

$\displaystyle\sup_{t\in(0,T)}\int_{0}^{t}\Big{\|}S(t-t^{\prime})(fg)\Big{\|}_{\dot{H}^{s}}dt^{\prime}$
$\displaystyle\lesssim\sup_{t\in(0,T)}t^{2\alpha}\|f\|_{L^{4}}\|g\|_{L^{4}},$
(4.4)
$\displaystyle\sup_{t\in(0,T)}t^{\alpha}\int_{0}^{t}\Big{\|}S(t-t^{\prime})(fg)\Big{\|}_{L^{4}}dt^{\prime}$
$\displaystyle\lesssim\sup_{t\in(0,T)}t^{2\alpha}\|f\|_{L^{4}}\|g\|_{L^{4}}.$
(4.5)

###### Proof.

The argument for each of these estimates follows similar lines, but we examine
each one separately.

Estimating (4.4). Using the definition of the Riesz Derivative, we begin with

$\displaystyle\int_{0}^{t}\Big{\|}S(t-t^{\prime})(fg)\Big{\|}_{\dot{H}^{s}}dt^{\prime}=\int_{0}^{t}\Big{\|}D_{x}^{s}S(t-t^{\prime})(fg)\Big{\|}_{L^{2}}dt^{\prime}$

We now apply Lemma 3, which requires $s\geq 0$, to obtain

$\displaystyle\int_{0}^{t}\Big{\|}D_{x}^{s}S(t-t^{\prime})(fg)\Big{\|}_{L^{2}}dt^{\prime}\leq\int_{0}^{t}(t-t^{\prime})^{-\frac{s}{2}}\|f(t^{\prime})g(t^{\prime})\|_{L^{2}}dt^{\prime}.$

Next, the generalized Hölder inequality allows us to break the $L^{2}$ norm in
the integrand into a product of $L^{4}$ norms, giving us

$\displaystyle\int_{0}^{t}(t-t^{\prime})^{-\frac{s}{2}}\|f(t^{\prime})g(t^{\prime})\|_{L^{2}}dt^{\prime}\leq\int_{0}^{t}(t-t^{\prime})^{-\frac{s}{2}}\|f(t^{\prime})\|_{L^{4}}\|g(t^{\prime})\|_{L^{4}}dt^{\prime}.$

We now multiply and divide by $(t^{\prime})^{2\alpha}$ and then pull out the
factor
$(t^{\prime})^{2\alpha}\|f(t^{\prime})\|_{L^{4}}\|g(t^{\prime})\|_{L^{4}}$ by
taking a supremum over time. This gives us the upper bound

$\displaystyle\int_{0}^{t}(t-t^{\prime})^{-\frac{s}{2}}\|f(t^{\prime})\|_{L^{4}}\|g(t^{\prime})\|_{L^{4}}dt^{\prime}\leq\sup_{t\in(0,T)}t^{2\alpha}\|f(t)\|_{L^{4}}\|g(t)\|_{L^{4}}\int_{0}^{t}(t-t^{\prime})^{-\frac{s}{2}}(t^{\prime})^{-2\alpha}dt^{\prime}.$
(4.6)

The remaining integral can now be handled by Lemma 4. The choice of
multiplying and dividing by $(t^{\prime})^{2\alpha}$ is now apparent as this
allows us to satisfy the hypothesis for $r=\frac{s}{2}+2\alpha-1$. Noting that
we require $0\leq r\leq 1$, we see that our definition of
$\alpha=\frac{1}{2}-\frac{s}{4}$ gives us

$\displaystyle
r=\frac{s}{2}+2\alpha-1=\frac{s}{2}+2\Big{(}\frac{1}{2}-\frac{s}{4}\Big{)}-1=0.$

We see here that this $r$ restriction is always satisfied regardless of the
value of $s$. The additional requirements of Lemma 4 are satisfied so long as
we take $0\leq s<2$ and $0<\alpha\leq 1/2$. Thus for a constant that only
depends on $s$, as we have $\alpha$ a function of $s$, we get

$\displaystyle\int_{0}^{t}(t-t^{\prime})^{-\frac{s}{2}}(t^{\prime})^{-2\alpha}dt^{\prime}<c_{s}.$

We therefore further bound (4.6) by

$\displaystyle\sup_{t\in(0,T)}t^{2\alpha}\|f(t)\|_{L^{4}}\|g(t)\|_{L^{4}}\int_{0}^{t}(t-t^{\prime})^{-\frac{s}{2}}(t^{\prime})^{-2\alpha}dt^{\prime}\leq
c_{s}\sup_{t\in(0,T)}t^{2\alpha}\|f(t)\|_{L^{4}}\|g(t)\|_{L^{4}}.$

This chain of inequalities thus establishes (4.4).

Estimating (4.5). Before estimating the full expression on the left-hand side
of (4.5) we first estimate its integrand. To accomplish this, we begin by
using Lemma 2 which is the $L^{p}$-$L^{q}$ Heat Kernel Estimate and the
generalized Hölder inequality to get

$\displaystyle\|S(t-t^{\prime})(fg)\|_{L^{4}}$
$\displaystyle\lesssim(t-t^{\prime})^{-\frac{1}{8}}\|f(t^{\prime})g(t^{\prime})\|_{L^{2}}\lesssim(t-t^{\prime})^{-\frac{1}{8}}\|f(t^{\prime})\|_{L^{4}}\|g(t^{\prime})\|_{L^{4}}.$

This upper bound for $\|S(t-t^{\prime})(fg)\|_{L^{4}}$ thus gives us

$\displaystyle\sup_{t\in(0,T)}t^{\alpha}\int_{0}^{t}\Big{\|}S(t-t^{\prime})(fg)\Big{\|}_{L^{4}}dt^{\prime}\lesssim\sup_{t\in(0,T)}t^{\alpha}\int_{0}^{t}(t-t^{\prime})^{-\frac{1}{8}}\|f(t^{\prime})\|_{L^{4}}\|g(t^{\prime})\|_{L^{4}}dt^{\prime}.$
(4.7)

While our construction would suggest examining $0<\alpha\leq\frac{1}{2}$, we
in fact are able to prove this estimate for $0<\alpha\leq\frac{7}{8}$; though
the values of $\alpha>\frac{1}{2}$ are unused. To continue estimating (4.7),
our argument splits based into the cases where $0<\alpha\leq\frac{7}{16}$ and
$\frac{7}{16}<\alpha\leq\frac{7}{8}$. The strategy in both cases is similar,
with the difference lying in the quantity we multiply and divide by in order
to utilize Lemma 4.

The case $0<\alpha\leq\frac{7}{16}$. In this case, we multiply and divide by
$(t^{\prime})^{\frac{7}{8}}$ inside the integrand and then pull out the factor
of
$(t^{\prime})^{\frac{7}{8}}\|f(t^{\prime})\|_{L^{4}}\|g(t^{\prime})\|_{L^{4}}$
by taking a supremum over time. We thus get

$\displaystyle\sup_{t\in(0,T)}t^{\alpha}\int_{0}^{t}(t-t^{\prime})^{-\frac{1}{8}}\|f(t^{\prime})\|_{L^{4}}\|g(t^{\prime})\|_{L^{4}}dt^{\prime}$
$\displaystyle\quad\quad\quad\quad\leq\sup_{t\in(0,T)}t^{\alpha}\Big{[}\sup_{t^{\prime}\in(0,t)}(t^{\prime})^{\frac{7}{8}}\|f(t^{\prime})\|_{L^{4}}\|g(t^{\prime})\|_{L^{4}}\Big{]}\int_{0}^{t}(t-t^{\prime})^{-\frac{1}{8}}(t^{\prime})^{-\frac{7}{8}}dt^{\prime}.$
(4.8)

The integral in (4.8), can now be handled with the Beta distribution estimate,
Lemma 4. We see that the corresponding $r$ will be $r=1/8+7/8-1=0$, and
therefore

$\displaystyle\int_{0}^{t}(t-t^{\prime})^{-\frac{1}{8}}(t^{\prime})^{-\frac{7}{8}}dt^{\prime}\lesssim
1.$

To handle the composition of suprema, we first note that as $\alpha>0$, we
have $t^{\alpha}\leq T^{\alpha}$. Then the interior supremum can be bounded
above by taking the supremum to be over the full interval $(0,T)$. Putting
these estimates together gives us

$\displaystyle\sup_{t\in(0,T)}t^{\alpha}\Big{[}\sup_{t^{\prime}\in(0,t)}(t^{\prime})^{\frac{7}{8}}\|f(t^{\prime})\|_{L^{4}}\|g(t^{\prime})\|_{L^{4}}\Big{]}\int_{0}^{t}(t-t^{\prime})^{-\frac{1}{8}}(t^{\prime})^{-\frac{7}{8}}dt^{\prime}$
$\displaystyle\quad\quad\quad\quad\quad\lesssim
T^{\alpha}\sup_{t\in(0,T)}t^{\frac{7}{8}}\|f(t)\|_{L^{4}}\|g(t^{\prime})\|_{L^{4}}.$

Thus, so long as $0<2\alpha\leq 7/8$ we see that $t^{\frac{7}{8}-2\alpha}$
will be positive, and we thus continue our estimation by multiplying and
dividing by $t^{2\alpha}$. After bounding above by pulling out the factor of
$t^{\frac{7}{8}-2\alpha}$, we get

$\displaystyle
T^{\alpha}\sup_{t\in(0,T)}t^{\frac{7}{8}}\|f(t)\|_{L^{4}}\|g(t^{\prime})\|_{L^{4}}\leq
T^{\frac{7}{8}-\alpha}\sup_{t\in(0,T)}t^{2\alpha}\|f(t)\|_{L^{4}}\|g(t^{\prime})\|_{L^{4}}.$

Chaining these inequalities gives us (4.5).

The case $\frac{7}{16}<\alpha\leq\frac{7}{8}$. To continue bounding (4.7), we
follow a similar approach to the above case but alternatively multiply and
divide by $(t^{\prime})^{2\alpha}$. We thus obtain

$\displaystyle\sup_{t\in(0,T)}t^{\alpha}\int_{0}^{t}(t-t^{\prime})^{-\frac{1}{8}}\|f(t^{\prime})\|_{L^{4}}\|g(t^{\prime})\|_{L^{4}}dt^{\prime}$
$\displaystyle\quad\quad\quad\quad\leq\sup_{t\in(0,T)}t^{\alpha}\Big{[}\sup_{t^{\prime}\in(0,t)}(t^{\prime})^{2\alpha}\|f(t^{\prime})\|_{L^{4}}\|g(t^{\prime})\|_{L^{4}}\Big{]}\int_{0}^{t}(t-t^{\prime})^{-\frac{1}{8}}(t^{\prime})^{-2\alpha}dt^{\prime}.$
(4.9)

Our hypothesis of $\frac{7}{16}<\alpha\leq\frac{7}{8}$ in this case satisfies
the requirement of Lemma 4. We see that for $r=\frac{1}{8}+2\alpha-1$ to
satisfy $0\leq r\leq 1$ is equivalent to
$\frac{7}{16}\leq\alpha\leq\frac{15}{16}$. Thus the Beta distribution estimate
implies that we have

$\displaystyle\int_{0}^{t}(t-t^{\prime})^{-\frac{1}{8}}(t^{\prime})^{-2\alpha}dt^{\prime}\lesssim
t^{-\frac{1}{8}-2\alpha+1}.$

Continuing our estimation on (4.9), and noting that
$\alpha-r=\frac{7}{8}-\alpha$, we get

$\displaystyle\sup_{t\in(0,T)}t^{\alpha}\Big{[}\sup_{t^{\prime}\in(0,t)}(t^{\prime})^{2\alpha}\|f(t^{\prime})\|_{L^{4}}\|g(t^{\prime})\|_{L^{4}}\Big{]}\int_{0}^{t}(t-t^{\prime})^{-\frac{1}{8}}(t^{\prime})^{-2\alpha}dt^{\prime}$
$\displaystyle\quad\quad\quad\quad\lesssim\sup_{t\in(0,T)}t^{\frac{7}{8}-\alpha}\Big{[}\sup_{t^{\prime}\in(0,t)}(t^{\prime})^{2\alpha}\|f(t^{\prime})\|_{L^{4}}\|g(t^{\prime})\|_{L^{4}}\Big{]}$
(4.10)

To bound the exterior supremum, we see that our hypothesis in this case
implies that $\frac{7}{8}-\alpha\geq 0$. We therefore get

$\displaystyle\sup_{t\in(0,T)}t^{\frac{7}{8}-{\alpha}}\Big{[}\sup_{t^{\prime}\in(0,t)}(t^{\prime})^{2\alpha}\|f(t^{\prime})\|_{L^{4}}\|g(t^{\prime})\|_{L^{4}}\Big{]}\leq
T^{\frac{7}{8}-{\alpha}}\sup_{t\in(0,T)}t^{2\alpha}\|f(t)\|_{L^{4}}\|g(t)\|_{L^{4}}.$

Chaining these inequalities thus gives us (4.5). ∎

## References

  * [1] S. Alanazi, M. Kamruzzaman, M. Alruwaili, N. Alshammari, S. Alqahtani, and A. Karime, Measuring and preventing COVID-19 using the SIR model and machine learning in smart health care, J. Healthc. Eng. 2020, 8857346, 2020.
  * [2] N. Bacaër, A history of mathematical population dynamics, Springer, New York, 2010.
  * [3] Z. Bai, and S. Zhang, Traveling waves of a diffusive SIR epidemic model with a class of nonlinear incidence rates and distributed delay, Commun. Nonlin. Sci. Numer. Simul. 22, 1370–1381, 2015.
  * [4] D. Bekiranov, The initial-value problem for the generalized Burgers equation, Differ. Integral Equ. 9, no. 6, 1253–1265, 1996.
  * [5] F. Brauer, P. van den Driessche, and J. Wu, eds., Mathematical epidemiology, Lecture notes in mathematics 1945, mathematical biosciences subseries, Springer, New York, 2008.
  * [6] E. Codling, M. Plank, and S. Benhamou, Random walk models in biology, J. R. Soc. Interface. 5, 813–834, 2008.
  * [7] I. Cooper, A. Mondal, and C. Antonopoulos, A SIR model assumption for the spread of COVID-19 in different communities, Chaos, Solit. Fractals. 139, 110057, 2020.
  * [8] A. Einstein, Investigations on the theory of Brownian movement, Dover, New York, 1956.
  * [9] R. Ducasse, Qualitative properties of spatial epidemiological models, arXiv:2005.06781, 2020.
  * [10] A. Fofana, and A. Hurford, Mechanistic movement models to understand epidemic spread, Phil. Trans. R. Soc. B 372, 20160086, 2017.
  * [11] M. Fudolig, and R. Howard, The local stability of a modified multi-strain SIR model for emerging viral strains, PLoS ONE. 15(12), 10.1371, 2020.
  * [12] Y. Giga, Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations. 62, no. 2, 186–212, 1986.
  * [13] M. Giga, Y. Giga, and J. Saal, Nonlinear partial differential equations: Asymptotic behavior of solutions and self-similar solutions, Birkhäuser, Springer, New York, 2010.
  * [14] S. Gounane, Y. Barkouch, A. Atlas, M. Bendahmane, F. Karami, and D. Meskine, An adaptive social distancing SIR model for COVID-19 disease spreading and forecasting, Epidemiol. Methods. 10(s1), 20200044, 2021.
  * [15] J. Hadamard, Lectures on Cauchy’s problem in linear partial differential equations, Dover, New York, 1953.
  * [16] J. Holmes, The Cauchy problem for several nonlinear evolution equations, PhD diss., University of Notre Dame, Notre Dame, IN, March, 2015.
  * [17] Y. Hosono, and B. Ilyas, Traveling waves for a simple diffusive epidemic model, Math. Models Methods Appl. Sci. 5, 935–966, 1995.
  * [18] D. Jones, M. Plank, and B. Sleeman, Differential equations and mathematical biology, 2nd ed., Mathematical and computational biology series, CRC P., Boca Raton, FL, 2009.
  * [19] T. Kato, and H. Fujita, On the nonstationary Navier-Stokes system, Rend. Sem. Mat. Univ. Padova. 32, 243–260, 1962.
  * [20] W. Kermack, and A. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. Series A, containing papers of a mathematical and physical character, 115 (772), 700–721, 1927.
  * [21] M. Li, An Introduction to Mathematical Modeling, Mathematics of Planet Earth 2, Springer, New York, 2018.
  * [22] E. Lieb, and M. Loss, Analysis, 2nd ed., Graduate Studies in Mathematics 14, AMS, Providence, RI, 2001.
  * [23] H. McCallum, N. Barlow, and J. Hone, How should pathogen transmission be modelled?, Trends Ecol. Evol. 16, 295–300, 2001.
  * [24] E. Rosti, S. Olivieri, M. Cavaiola, A. Seminara, and A. Mazzino, Fluid dynamics of COVID-19 airborne infection suggests urgent data for a scientific design of social distancing, Sci. Rep. 10, 22426, 2020.
  * [25] J. Skellam, Random dispersal in theoretical populations, Biometrika. 38(1/2), 196–218, 1951.
  * [26] A. Stegeman, A. Elbers, J. Smak, and M. de Jong, Quantification of the transmission of classical swine fever virus between herds during the 1997–1998 epidemic in the Netherlands, Prev. Vet. Med., 42(3):219–234, 1999.
  * [27] W. Wang,and X. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM J. Appl. Math. 71, 147–168, 2011.
  * [28] Z. Wang, and J. Wu, Travelling waves of a diffusive Kermack–Mckendrick epidemic model with non-local delayed transmission, Proc. R. Soc. A 466, 237–261, 2010.
  * [29] H. Weiss, The SIR Model and the foundations of public health, Mater. Mat., 2013 (3), 2013.
  * [30] Z. Xu, B. Wu, and U. Topcu, Control strategies for COVID-19 epidemic with vaccination, shield immunity and quarantine: A metric temporal logic approach, PLoS ONE 16(3), e0247660, 2021.

Curtis Holliman (Corresponding author)

Department of Mathematics

The Catholic University of America

Washington, DC 20064

E-mail<EMAIL_ADDRESS>
Harry Prieto
Department of Mathematics
The Catholic University of America
Washington, DC 20064
E-mail<EMAIL_ADDRESS>