# Predictive Intent Maintenance with Intent Drift Detection in Next Generation
Networks

Chukwuemeka Muonagor, Mounir Bensalem and Admela Jukan  Technical University
of Braunschweig, Germany {c.muonagor, mounir.bensalem<EMAIL_ADDRESS>
###### Abstract

Intent-Based Networking (IBN) is a known concept for enabling the autonomous
configuration and self-adaptation of networks. One of the major issues in IBN
is maintaining the applied intent due the effects of drifts over time, which
is the gradual degradation in the fulfillment of the intents, before they
fail. Despite its critical role to intent assurance and maintenance, intent
drift detection was largely overlooked in the literature. To fill this gap, we
propose an intent drift detection algorithm for predictive maintenance of
intents which can use various unsupervised learning techniques (Affinity
Propagation, DBSCAN, Gaussian Mixture Models, Hierarchical clustering, K-Means
clustering, OPTICS, One-Class SVM), here applied and comparatively analyzed
due to their simplicity, yet efficiency in detecting drifts. The results show
that DBSCAN is the best model for detecting the intent drifts. The worst
performance is exhibited by the Affinity Propagation model, reflected in its
poorest accuracy and latency values.

##  I Introduction

Intent-Based Networking (IBN) has emerged as one of the key technology
solutions for autonomous networks [1]. IBN is a network management system that
makes use of natural language processing to compile humanly understandable
commands to the network management system, while using machine learning and
artificial intelligence in general, as well as orchestration of network
resources to automate the activation of these commands [2]. An example of
intent could be expressed as _permit all traffic from nodes A, F, and K to go
through port 80 and not traffic from other nodes in the system_ , which
expressed in this simple form can assure that the policy in the system is
enforced and maintained. According to Cisco [3], intent assurance is a core
part of any IBN system. It is also referred to as intent validation or
verification, which is a network management process that continuously monitors
the network to verify that the desired intent which has been applied is
functioning properly.

A typical intent-based network life cycle considers three phases: Phase 1 -
when the network operates without an active intent; Phase 2 - the period
during which the intent is active and the network behaviour complies with the
intent; Phase 3 - the intent failure. Let take an example of average latency
as network parameter to be monitored (Figure 1).When an intent is entered into
the network (Phase 2) after an hour of network operation when the latency was
4t ms (Phase 1) the latency value is brought down value of 1t ms instead. In
Phase 2, which is intent fulfillment period, the latency remains on the
average at this value of t ms for about 75 mins before it starts fluctuating
gradually in a generally increasing trend. The failure of the intent is
confirmed after one and half hours of these fluctuations in Phase 3 - the
intent failure period when the network finally goes back to its original state
where the latency was 4t ms before the intent was entered. It is precisely in
the time period where the latency fluctuating gradually in a generally
increasing trend that illustrates the period of the so-called _Intent Drift_.
Although a major part of intent maintenance should consider whether intent
drift occurs over time [4], only a few studies addressed intent drift
detection.

Figure 1: Intent life cycle and the appearance of drifts.

This paper proposes a novel function for the predictive maintenance of intents
in the IBN architecture, i.e., the intent drift detection. We develop and
apply an intent drift detection algorithm that can selectively test seven
machine learning (ML) models to detect intent drifts, which we analyze on
their ability to predictably maintain intents based on their resulting
accuracy, response time and complexity. Specifically, unsupervised learning
techniques (Affinity Propagation, DBSCAN, Gaussian Mixture Models (GMM),
Hierarchical clustering, K-Means clustering, OPTICS algorithm, and One-Class
SVM) are adapted to this end, while a greedy algorithm is also used for
comparison purposes. The results show that DBSCAN has the best accuracy, and
the lowest average response time (latency) in detecting intent drifts, though
its space complexity is high. The worst detection performance is exhibited by
Affinity Propagation algorithm reflected in its poorest accuracy value and
maximum latency. To the best of our knowledge, this is the first work to
address the problem of intent drift detection, and predictive intent
maintenance.

The rest of the paper is organized as follows. We present and discuss related
work in Section II. Section III proposes the reference IBN architecture with
intent drift detection module. Section IV focuses on intent drift detection
algorithms. Section V analyzes numerical results. Section VI concludes the
paper.

##  II Related Work

Previous work majorly focused on intent assurance and measures to overcome
failures [1, 5]. Studies in [6, 7, 8]resorted to ML techniques to build a
self-assurance IBN system on top of programmable networks. Alternatively to
intent assurance methods, the use of threshold can be adopted, for instance in
[9] with rerouting when packet losses exceed the threshold. The problem with
setting of thresholds or establishment of specific reference points for the
network is, according to [10], that when network behaviour is dynamic, fixed
thresholds specifications may not reflect actual network behavior. A drift in
the intent may be detected, due to dynamic network state changes, whereby IBN
should automatically take actions to revert the network back to compliance
state [11]. However, detecting this gradual degradation of intents and at the
same time avoiding false positives is a challenge.

The problem of intent drift can be solved by analyzing the network behaviour
during intent fulfillment and normal network operation, which in turn is a
problem of statistical or analytical modeling of the network behavior, or
based on time series analysis and forecasting, e.g., as in [12]. While
statistical modelling may not be best suited to a large amount of data, time-
series analysis and rule-based heuristic algorithms also have difficulty
scaling [13]. We opt for unsupervised machine learning here, due to their
simplicity, and due to the fact that the network data used in the analyses
would come unlabeled. The unsupervised learning models used in our algorithms
are [14]: Affinity Propagation, DBSCAN, Gaussian Mixture Models (GMM),
Hierarchical clustering, K-Means clustering, OPTICS, and One Class SVM.

##  III Intent Drift Detection

###  III-A Reference Architecture

The newly introduced module of Intent Drift Detection into the typical IBN
architecture [4] is illustrated in Figure 2. Generally, an IBN system fulfills
3 main objectives; the translation where the business intents are mapped into
policies, the activation where the intents and policies are deployed and
enforced into the network, and finally the assurance where validity of the
intent is verified and changes are detected, predicted and prevented, using AI
and ML techniques. We also illustrate how IBN interacts with the network
management systems, with examples of the selected functions [15], including
Security Management, QoS provisioning, etc.

Figure 2: Introducing Intent Drift Detection into the IBN Architecture.

In the IBN architecture, the Intent Entry Interface receives high-level
intents from the user and sends same to the Intent Manager, which parses them
into low-level intents and forwards them to the Policy Configurator. The
Intent Manager also sends the intents to the Intent Store. The Policy
Configurator matches the intents received from Intent Manager with the
policies in the Policy Store to generate the appropriate workflows to be sent
to the Network Manager. The Knowledge Base is the storage component of the IBN
system. It contains the Policy, Intent and Data Stores. The Policy Store
contains the policies that the Policy Configurator matches the intents with so
as to generate the appropriate workflows corresponding to the desired intents.
The Intent Store stores the received intents from the Intent Manager. When an
intent is deleted from the system, it is also deleted from the Intent Store.
The Data Store stores network data received from the Monitoring function block
of the Network Manager.

In this architecture, we propose to add a new component, i.e., the Intent
Drift Detector. This component retrieves data from the Data Store and analyses
it to be able to detect any drift in the fulfillment of the intents. It
enables predictive maintenance of intents, and prevents its failure. It should
be noted that while the existing IBN architectures consider an intent
assurance engine or module, it does not implement any explicit intent drift
detection. This is a novel architectural component introduced in this work
which replaces the conventional intent assurance engine. The latter is often
hard coded to check periodically the validity of an intent, which is not
efficient.

###  III-B Intent Drifts Exemplified

Figure 3: Examples of Intents for Security, QoS provisioning and Reachability

We now illustrate a few intents and then exemplify their drifts, see Figure 3.
Intents A and B are examples of security intents. Intent A entails that the
bandwidth allocation to the attack victim is to be increased by an extra 3
Gb/s in the event of the attack. Intent B requires that the entire incoming
traffic to the attack victim would be minimised, i.e., many packets would be
dropped while very few packets are allowed. Intents C and D are examples of
QoS provisioning intents. Intent C implies that there should not be any
restriction to the amount of bandwidth allocated to the host. Intent D
requires that the bandwidth allocation to all types of traffic to and from
each host across the entire network be minimised. Intent E is a load balancing
intent. Finally, Intent F is a reachability assurance intent that ensures that
the host or server is reachable for a period of 7200 secs. This implies that
all traffic directed to this host must be able to reach it.

An intent drift can be caused by a manual intervention of the network
administrator: when Intent B is enforced, the Security Management drops
several incoming packets to Host when a DDoS attack is detected. Assuming that
an operator manually makes an intervention or changes by allowing some traffic
to access the host, this change can cause a drift in the security intent,
which needs an additional verification. Another drift example can be caused by
an automated reconfiguration: when intent E is enforced, the load balancing
distributes the traffic approaching the host. Assuming that the monitoring
system can suggest updates to the traffic distribution due to CPU, memory, or
bandwidth telemetry, this might lead to unbalanced traffic distribution, which
causes intent E to drift. Furthermore, intent drift can be caused by a
conflict between intents. For instance, an existing Intent B, and a new intent
C can conflict, since due to maximisation of QoS assurance for a host which is
what Intent C is, i.e., ensuring the usage of maximum bandwidth by that host,
there is conflict with intent B. Similarly, there maybe a conflict between the
existing intent A, and new intent D, etc.

##  IV Algorithms for Intent Drift Detection

In this section, we propose an intent drift detection algorithm Algorithm 1,
which uses different ML models to detect intent drifts from monitoring time
series data. The algorithm receives stored time series data,  X_train, which
can be specified depending on the intent, a new incoming data to be evaluated,
such as throughput,  X_test, which can be a recently measured telemetry, and
finally an algorithm selection, Model_Type, where a list of ML algorithm and
heuristics is provided to support drift detection. In Step 1, we check if the
chosen model requires a training phase, then run and save the learnt model. In
Step 2, we evaluate our new data either using the trained model, or run
clustering phase for models like K-means. Using the output of the evaluation
phase, the algorithm decides whether there is an intent drift or not depending
on how each ML and heuristic works (Step 3), and then send the results to the
user (Step 4).

###  IV-A Detection algorithms based on various ML models

####  IV-A1 Affinity Propagation

Affinity propagation creates clusters by sending messages between pairs of
samples until convergence. A dataset is then described using a small number of
exemplars, which are representative points that have high similarity to other
points in the same cluster, and low similarity to points in other clusters.
Affinity Propagation has the preference parameter, which is the “availability”
a(p, q) sent from point q to point p reflecting how much point q sees itself
as an exemplar of point p against other potential exemplars. For the detection
of intent drift according to Algorithm 1, the multiplier value is initialised
at first. Then, the learning process in Step 1 starts with the computation of
the suitable preference parameter automatically as the minimum of the pairwise
distances between the network throughput values, after which the training is
done by generating the number of clusters automatically by assigning samples
to exemplars. The higher the variation in the throughput data the higher the
number of generated clusters. The same process is repeated for new
measurements of throughput values. In Step 3, a drift in the intent is
indicated by an increase in the number of generated clusters from the data.

Algorithm 1 Intent Drift Detection Algorithm

1:Input:  X_train (time series data, eg. throughput), X_test (new data, e.g,.
new throughput values), Model_Type (algorithm type)

2:Output:  boolean (drift detection)

3:Step 1:

4:if method requires learning then

5: Run learning algorithm of Model_Type with X_train

6: Save trained model

7:end if

8:Step 2:

9:if method requires inference then

10: if inference uses trained model then

11: Evaluate trained model with X_test

12: end if

13: if inference uses math function then

14: Evaluate math function with X_test and X_train

15: end if

16: Save inference results

17:end if

18:Step 3:  Decide intent drift possibility based on Model_Type

19:Step 4:  Send drift detection results to user.

####  IV-A2 DBSCAN

The intent drift detection process implemented with DBSCAN is based on [14]
and [16]. We first initialise MinPts which is the minimum number of samples in
a given neighborhood. The learning process starts in Step 1. After the
training is done, the neighborhoods are generated automatically based on data
variability. The extracted throughput values are then assigned to different
neighborhoods or discovered clusters using equation 3. Throughput values that
are within the radius of the neighborhood. The more similar the throughput
values are, the less the number of generated or discovered clusters or
neighborhoods. The same process is repeated for new measurements of throughput
values. In Step 3, a drift in the intent is indicated by an increase in the
number of generated or discovered clusters from the data.

####  IV-A3 Hierarchical clustering

Hierarchical clustering builds nested clusters by merging or splitting them
successively. The linkage criteria determines the metric used for the merge
strategy. For the detection of intent drift according to Algorithm 1, the
parameters affinity and linkage are initialised at first. The learning process
in Step 1 starts with the automatic computation of the ideal distance
threshold value which is based on the mean of the extracted throughput data.
After which the training is done by generating the number of clusters
automatically by assigning samples to clusters. The same process is repeated
for new measurements of throughput values. A drift in the intent according to
Step 3 can be decided when the number of generated clusters from the data
increases.

####  IV-A4 K-Means

At first, the value of a multiplier, later used in Step 3, is initialised. The
input data are the old and new throughput values referred to as X_train and
X_test, respectively. In Step 1, the learning process is done which involves
first determining the optimal number of clusters using the Silhouette analysis
since number of clusters determines how accurate the algorithm would be, after
which training is done on the extracted throughput data using the determined
optimal number of clusters. These throughput values are partitioned into the
various clusters and their new centroid values are computed. An array of this
new centroid values is created and sorted. The differences between
neighbouring centroid values and the maximum of the differences are estimated,
the lower this maximum value, the more similar the entire throughput values
from different clusters are. The same process is repeated whenever new
measurements of the throughput values are received by the model. In Step 3,
the new maximum is compared with the old maximum. If the new maximum exceeds
the product of the old maximum and the multiplier, then there is drift.

####  IV-A5 Gaussian Mixture Models (GMM)

Gaussian mixture model generalises K-Means clustering, also taking note of the
input data covariance structure as well as the Gaussian components centroids.
One of the important parameters in GMM is the number of desired components.
The intent drift detection process using GMM is very similar to that of
K-Means. The number of clusters in K-Means is similar to the number of
components in GMM and the same process is followed according to the steps in
Algorithm 1.

####  IV-A6 OPTICS algorithm

DBSCAN and OPTICS algorithms are similar. For OPTICS algorithm, at first the
minimum number of samples, the reachability distance and the minimum cluster
size are initialised. Points with similar reachability distances are likely to
be in the same cluster. The learning process in Step 1 starts with generating
the neighborhoods automatically, assigning the throughput data to different
neighborhoods based on their reachability distances. In Step 3, a drift in the
intent is indicated by an increase in the number of generated or discovered
clusters from the data, after which Step 4 is implemented which reports the
intent drift.

####  IV-A7 Support Vector Machine (SVM)

The SVM algorithm used here is One-class SVM (1-SVM). According to Algorithm
1, the first action is to initialise the kernel function to be used and the nu
parameter which is upper bounded by the fraction of outliers considered by the
model to be in the data and lower bounded by the fraction of support vectors.
This is followed by implementing Step 1. Step 1 involves learning the 1-SVM on
the data to group the old extracted throughput values into normal data and
outliers. In Step 2, the trained model is used to make inference on the newly
extracted data to be able to classify them into either normal data or
outliers. The presence of outliers in the newly extracted data indicates
possibility of drift, and the drift detection result is sent to the
administrator at the controller level.

####  IV-A8 Greedy algorithm

Greedy algorithm is used for comparison. It is based on the determination of
the maximum measured throughput data at any point in time. When the maximum of
the incoming traffic over a short period exceeds this network monitoring
maximum with a large margin, which is defined at the inception, the intent
drift is detected.

###  IV-B Complexity Analysis

In terms of data size, aside the GMM and the Affinity Propagation algorithms,
the algorithms used here are scalable as they are not affected by the size of
data. K-Means, DBSCAN and OPTICS are the most scalable of the models. However,
in terms of number of nodes and with a relatively fixed amount of data, the
entire algorithms are scalable as they can be deployed on each node and also
as many nodes as possible.

The Affinity Propagation algorithm has a time complexity of the order
$O(N\textsuperscript{2}T)$ where $N$ is the number of samples and $T$ is the
number of iterations until convergence. Further, the memory complexity is of
maximum $O(N\textsuperscript{2})$. The time complexity of k-means algorithm is
$O(N\textsuperscript{2}$). The overall complexity is $O(N\textsuperscript{2})$
for DBSCAN, $O(N\textsuperscript{3})$ for GMM, O(N) for OPTICS, and $O(N)$ for
Greedy algorithm. The complexity of One-Class SVM is at best
$O(N\textsuperscript{2})$. The time complexity of the agglomerative
hierarchical clustering is $O(N\textsuperscript{3})$ because in each of the
$N-1$ iterations the $N\times N$ matrix dist matrix is scanned exhaustively
for the lowest distance, and the memory complexity is
$O(N\textsuperscript{2})$.

##  V Measurements and Results

###  V-A Network Emulation Setup and Evaluation Metrics

We used a network emulation setup which contains three hosts and one virtual
switch hosted as virtual machines provisioned with the virtualisation
platform, VirtualBox, together with an external host where the VirtualBox
platform is hosted. The number of hosts was decided arbitrarily. The external
host serves as the controller where the IBN software runs and controls how the
rest of the hosts which are the virtual machines operate, and also has the
network manager emulator that implements the workflows as commands to the
other hosts through ssh. The network data are the ingress throughput at a
host. Two iterations each were run on three virtual machines and the
performance of the models were measured. The intent drift detection algorithms
are implemented on the hosts while they send the results of their analyses to
the controller in the external host to be viewed at the intent entry
interface.

Since any intent implementation needs to be specific to the intent, we
consider two specific intents in the analyses which are applied one after the
other. These intents are Intents B and D respectively which have been
exemplified earlier. Intent B is an already-existing security and reliability
intent which is to deploy an action in the event of an attack so as to
mitigate the attack. In the event of the attack the execution of the security
intent installed at the controller machine which is to mitigate the attack is
triggered. The second intent, Intent D is a QoS intent that entails minimising
ingress bandwidth to a host due to load or congestion. This implies that only
part of the ingress traffic is selected and routed to the host while the
remaining traffic is dropped. For each of the two intents, during the period
of the intent fulfillment, the ingress throughput to the host is observed and
analysed with the aid of the models to determine when possible drift in the
intent starts occurring and/or when the intent fails entirely. The measurement
of the ingress throughput is conducted by the host with the aid of the ifstat
linux command every second, and the data can be extracted in batches
periodically or singularly. Here, an arbitrary choice of every nine seconds is
made.

Each measurement iteration contains normal operation, intent fulfillment,
intent drift, and intent failure phases. Our focus is on the intent drift
phase. The metrics used to judge the performances of the different models are
the accuracies of the models observed over the period of the entire
iterations, the average latencies of the models while detecting the intent
drift periods, and the space complexity of the models.

For the two intents, Intents B and D, two of the iterations run on different
machines were selected and the plots of the network behaviour capturing their
life cycles as illustrated in Figures 4 (Intent B - security intent) and 5
(Intent D - QoS intent). Figure 4 shows that the host experienced an attack at
about 140s in time, at which a traffic restriction action according to the
existing reliability intent is deployed to mitigate this attack. This precedes
the intent fulfillment phase were the attack is mitigated for over 120s due to
the action deployed by the intent. This is followed by an intent drift period
where the intent fulfillment starts experiencing degradation. This intent
drift period lasts for a while before the intent fails eventually when the
host starts to experience an attack again. The life cycle for the QoS intent
is shown in Figure 5. The intent fulfillment phase starts from almost 180s and
lasts for some time before the drift in the intent starts being noticed at
almost 400s.

Figure 4: Iteration 1 Figure 5: Iteration 2

###  V-B Performances of the Models

####  V-B1 Model Accuracy

From Figure 6, it can be seen that DBSCAN had the highest accuracy (0.846) of
all the models in detecting intent drift. This is followed by GMM and
Hierarchical clustering methods with a joint accuracy of 0.768, and K-Means
clustering, then SVM. The higher accuracy of the DBSCAN model can be
attributed to two reasons. One is the fact that in DBSCAN algorithms number of
clusters are generated automatically according to statistical dispersion of
the data, otherwise known as data variability as opposed to the K-Means, GMM
and some other methods that make use of fixed number of clusters which can’t
always reflect network behaviour or the Greedy algorithm that only computes
the maximum values of the throughput data. Another reason is that DBSCAN
algorithms are also robust to noise and outliers since they are not assigned
to any cluster. The low accuracy of SVM can be attributed to the fact that it
tends to suffer when there is no clear separation between the normal and
abnormal data, which is the case in this work.

Figure 6: Accuracies of the Models

####  V-B2 Average Latency

The resilience and reliability of networks is dependent on the amount of time
it takes the network to detect a problem or failure and implement a recovery
strategy to ensure resilience. From Figure 7, it is noticed that the SVM model
had the lowest average latency in reporting the drifts. In a case like this
where the amount of data is not very large, SVM tends to execute fast. Worthy
of note is that SVM reported drifts more frequently than any other model even
when actual drifts didn’t occur, this contributed to its low latency even
though it had poor accuracy. DBSCAN algorithm had the second lowest delay
while detecting the drift in the network which is followed by GMM. Other
algorithms like KMeans, Optics and the Greedy Algorithm showed very high
latencies with Affinity Propagation being the poorest due to its general
inability to quickly detect the drifts. This can be attributed to the fact
that DBSCAN works well with data that are not very well-separated, e.g.,
network data, and does its clustering as new data are observed unlike K-Means
and the others whose strength is with well-separated data, so the former would
require less time to draw conclusion that a drift in the intent exists based
on the data it observes.

Figure 7: Average Latencies of Models

####  V-B3 Space Complexity

Space complexity can be estimated as the amount of allocated bytes, whereby
algorithms that generally require less spaces run faster. From Figure 8 which
illustrates the average memory usages of the models it can be noticed that
OPTICS, DBSCAN, Hierarchical clustering, and Affinity Propagation methods
required generally more amount of memory than the others followed by SVM. This
high complexity of DBSCAN and the others is due to the fact that they spend
time to generate the appropriate clusters themselves automatically before
assigning data points to these clusters. GMM and KMeans have the lowest
complexity. This is because they view the dataset as well-separated and try to
accommodate noise, unlike the previously mentioned algorithms that take strong
cognisance of the data variability and do extra work of noticing noise and
treating it as outlier.

Figure 8: Average Memory Usages of Models

##  VI Conclusions and Outlook

We conclude that the concept of intent drift detection is very important in
predictive intent maintenance. The DBSCAN model had the best accuracy in
detecting the drift due to its automatic cluster generation and tendency to
ignore noise and outliers. It also responded second fastest to classifying
data traffic as anomaly which is indicated by its lowest average latency of
0.8s, with SVM being the fastest to report a drift. The worst performance was
exhibited by the Affinity Propagation model. This is reflected in its poorest
accuracy value of 0.428 and maximum latency of 5.152s and also its inability
to detect the intent drift phase quick enough. The OPTICS algorithm had the
highest complexity while GMM and K-Means had the least complexities.

There are many open areas for further work in this subject. The problem of
intent drift is much more complex than illustrated here. Reducing the amount
of false positives to the barest minimum is part of the basis for further
development in this area of research. Further, the intent drift detection
technique is peculiar to the type of intent, so replicating this work for
other types of intents in different kinds of network is of very good interest.
Also, other methods like neural networks and reinforcement learning can’t
entirely be ruled out and need to be studied on efficiency.

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# Gradient Bounds and Coarsening Estimates for Some Molecular Beam Epitaxy
Models

Daniel Oliveira da Silva  Department of Mathematics
Nazarbayev University
Qabanbai Batyr Avenue 53
010000 Nur-Sultan
Republic of Kazakhstan<EMAIL_ADDRESS>and  Magzhan Biyar
Department of Mathematics
Nazarbayev University
Qabanbai Batyr Avenue 53
010000 Nur-Sultan
Republic of Kazakhstan<EMAIL_ADDRESS>
###### Abstract.

We prove global well-posedness for some molecular beam epitaxy models. This
requires proving some bounds on the gradient of solutions. As a consequence of
these estimates, we also obtain bounds on the coarseness of solutions.

###### Key words and phrases:

Molecular beam epitaxy; Cahn-Hilliard; well-posedness; roughness

###### 2010 Mathematics Subject Classification:

35K25; 35K58; 35Q81

##  1\. Introduction

Molecular beam epitaxy (MBE) is a method of depositing very thin layers of
crystals onto surfaces such that the orientation of the crystals relative to
the substrate is well-defined. Such manufacturing methods are used in the
fabrication of various types of objects, such as optical components and
semiconductor devices. In many models of epitaxial dynamics, the profile of
the thin film is described by a partial differential equation of the general
form

(1) $u_{t}+\Delta^{2}u+\nabla\cdot\mathbf{J}(\nabla u)=0.$

Here, the function $u:\Omega\times\mathbb{R}\longrightarrow\mathbb{R}$ is
called the _height function_ , and represents the height of the thin film
layer in a comoving reference frame. The set $\Omega\subset\mathbb{R}^{2}$
represents the surface being coated, and the vector function $\mathbf{J}$ is
called the _surface diffusion current_. This equation is similar to the Cahn-
Hilliard equation

$u_{t}+\Delta^{2}u+\Delta F(u)=0.$

However, it is known that there are differences in the behavior of solutions
to these two equations; see, for example, [12, 8, 9] for discussions.

In the modeling of epitaxial processes with equations such as (1), there are
two questions of fundamental importance:

  * •

Well-posedness: Do unique solutions to (1) exist for a given class of initial
data?

  * •

Coarsening: How much do solutions to equation (1) deviate from the baseline?

Typically, the coarseness $C(t)$ is taken to be the root-mean-square deviation
from the baseline and is measured by the $L^{2}$ norm of the solution; see the
discussion in section 3 of [15] for more details about coarseness in
deterministic MBE models, or the introduction in [17] for stochastic models.

In the present work, we will consider these questions for models of the form
of equation (1) for surface currents $\mathbf{J}$ satisfying

$\mathbf{J}(\nabla u)=O(|\nabla u|^{q})$

for $q\geq 1$. This class contains many well-known models, such as the model
of Rost and Krug [6] (also some special cases of the model of Villain [16]),

$\mathbf{J}(\nabla u)=\left(1-|\nabla u|^{2}\right)\nabla u;$

and the models of Johnson, Orme, Hunt, Graff, Sudijono, Sander, and Orr [2]
and Siegert and Plischke [12], where $\mathbf{J}$ is given by

$\mathbf{J}(\nabla u)=(u_{x}f(u_{x}),u_{y}f(u_{y}))$

for continuous rational functions $f$. For such models, we will prove the
following global well-posedness result:

######  Theorem 1.

Let $1\geq p$, let $\mathbf{J}$ be a differentiable function such that
$\mathbf{J}(\nabla u)=O(|\nabla u|^{q})$. Then there exists a number
$\epsilon_{0}>0$ such that the Cauchy problem

(2) $\displaystyle u_{t}+\Delta^{2}u+\nabla\cdot\mathbf{J}(\nabla u)=0$
$\displaystyle u(0)=u_{0}.$

is globally well-posed in $W^{1,p}\cap W^{1,\infty}(\mathbb{R}^{d})$ for small
initial data whenever

$p=\frac{d(q-1)}{2}\quad\text{and}\quad q>\frac{5}{3}.$

As we will see, this theorem is stated in a slightly stronger form than
necessary; the smallness assumption applies only to the gradient of solutions,
and not to the solution itself.

To prove this result, we first obtain a local well-posedness result in section
3. We will then show that it is possible to control the norm of the solutions
using the following a priori estimate on the gradient of solutions:

######  Theorem 2.

Let $u$ be the solution to the Cauchy problem

$\displaystyle u_{t}+\Delta^{2}u+\nabla\mathbf{J}(\nabla u)=0$ $\displaystyle
u(x,0)=u_{0}$

with $\mathbf{J}(\nabla u)=O(|\nabla u|^{q})$. Then there exists a constant
$\epsilon_{0}>0$ such that if

$\|\nabla u_{0}\|_{L^{p}}\leq\epsilon_{0},$

then

$\|\nabla u(t)\|_{L^{\infty}_{x}}\leq\frac{C_{1}}{t^{\frac{d}{4p}}}.$

and

$\|\nabla u(t)\|_{L^{p}_{x}}\leq C_{2}.$

for some positive constants $C_{1},C_{2}$, and for

$p=\frac{d(q-1)}{2}.$

This will be proved in section 4. We remark that this result should be
compared to the result of Li, Qiao, and Tang in [13], who obtained a similar
result for the problem on the torus. Once we prove Theorem 2, we may obtain
the necessary control on the norms of the solution. This will be done in
section 5.

Our final main result gives us a maximum rate of growth on the coarseness of
solutions:

######  Theorem 3.

Let $u\in C([0,\infty);W^{1,p}(\mathbb{R}^{d})\cap
W^{1,\infty}(\mathbb{R}^{d}))$ be the global solution to the Cauchy problem
(LABEL:Cauchy) from Theorem 1 for

$\max\left\\{1+\frac{2}{d},\frac{6+d}{2+d}\right\\}<q<1+\frac{4}{d}.$

Then the coarseness

$C(t)=\|u(t)\|_{L^{2}}$

of $u$ satisfies

$C(t)\leq C(1+t)^{\frac{1}{4}-\frac{d}{4p}+\frac{d}{8}}.$

for all $t\in[0,\infty)$.

This will be shown in section 6. We remark that in the special case $d=2$
(which is the one of greatest interest in applications), the above result
reduces to

$\|u(t)\|_{L^{2}}\leq C(1+t)^{\frac{1}{2}-\frac{1}{p}}$

when $2<q<3$.

##  2\. Preliminaries

In this section, we state some preliminary results which will be used in the
proofs of Theorems 1, 2 and 3. First, we recall some basic facts about Cahn-
Hilliard-type equations. Consider the problem

(3) $\displaystyle u_{t}+\Delta^{2}u=G(x,t),$ $\displaystyle u(x,0)=f(x),$

with $(x,t)\in\mathbb{R}^{d}\times\mathbb{R}_{+}$. By elementary Fourier
analysis, the solution is given by

(4) $u(x,t)=\int_{\mathbb{R}^{d}}k_{t}(x-y)f(y)\
dy+\int_{0}^{t}\int_{\mathbb{R}^{d}}k_{t-s}(x-y)G(y,s)\ dyds,$

where the kernel $k_{t}(x)$ is the function whose spatial Fourier transform is

$\hat{k}_{t}(\xi)=e^{-t|\xi|^{4}}.$

We observe that the expression in equation (4) can be expressed in the more
succinct form

$u(x,t)=[k_{t}*f](x)+\int_{0}^{t}[k_{t-s}*G(\cdot,s)](x)\ ds,$

where $*$ denotes the convolution

$f*g=[f*g](x)=\int_{\mathbb{R}^{d}}f(x-y)g(y)\ dy.$

For later convenience, we introduce the following convention: if
$\mathbf{f}=(f_{1},\cdots,f_{n})$ and $\mathbf{g}=(g_{1},\cdots,g_{n})$ are
vector functions, we write

$\mathbf{f}*\mathbf{g}=\int_{\mathbb{R}^{d}}\mathbf{f}(x-y)\cdot\mathbf{g}(y)\
dy,$

where $\mathbf{f}\cdot\mathbf{g}$ denotes the standard dot product of the
vector functions $\mathbf{f}$ and $\mathbf{g}$.

Applying this to the nonlinear problem (1), we recast our problem in the
integral form

$u(x,t)=[k_{t}*u_{0}](x)+\int_{0}^{t}[k_{t-s}*\nabla\cdot\mathbf{J}(\nabla
u)](x)\ ds.$

Note that by properties of convolutions, this may be rewritten in the slightly
different form

(5) $u(x,t)=[k_{t}*u_{0}](x)+\int_{0}^{t}[\nabla k_{t-s}*\mathbf{J}(\nabla
u)](x)\ ds.$

Following the nomenclature of Tao [3], we call solutions to the integral
equation (5) which belong to the space $C(I;Z)$ for some function space $Z$
and time interval $I$ a _strong_ $Z$ _solution_.

To find solutions to equation (5), we will need to make use of properties of
the function $k_{t}$, which we collect in the following lemma:

######  Lemma 4.

Assume that $\hat{k}_{t}(\xi)=e^{-t|\xi|^{4}}$, $t>0$, and
$\xi,x\in\mathbb{R}^{d}$. Then we have the estimates

(6) $\displaystyle\quad\ \|k_{t}\|_{L^{p}_{x}}\leq
C_{p}t^{-\frac{d}{4}\left(1-\frac{1}{p}\right)},$ (7)
$\displaystyle\|D^{n}k_{t}\|_{L^{p}_{x}}\leq
C_{p,n}t^{-\frac{d}{4}\left(1-\frac{1}{p}\right)-\frac{n}{4}}.$

Here, $C_{p}$ and $C_{p,n}$ are positive constants with $C_{1}=1$, and
$D^{n}k_{t}$ denotes the set of all derivatives of $k_{t}$ of order $n$.

For a proof, see [5], Lemma 2.1.

Next, we state some lemmas which will be useful in establishing the global
theory. The first is the following result, which will be useful in the global
theory:

######  Proposition 5 (Integrable Bihari Inequality).

Let $g$ be continuous on $[a,b]$, let $h(t)$ be integrable on $[a,b]$, and
assume that

$k\geq 0\quad\text{and}\quad M\geq 0.$

Suppose that the function $\omega(u)$ is a non-negative, non-decreasing
continuous function for $u\geq 0$. If $g$ satisfies the inequality

$g(t)\leq k+M\int_{a}^{t}h(s)\omega\left(g(s)\right)\ ds,$

then

$g(t)\leq\Omega^{-1}\left(\Omega(k)+M\int_{a}^{t}h(s)\ ds\right),$

where

$\Omega(u)=\int_{u_{0}}^{u}\frac{dy}{\omega(y)},$

where $u_{0}>0$, $u\geq 0$. Note that the expression $\Omega(k)$ must be well-
defined, and $t$ must belong to some subinterval $[a,b^{\prime}]$ of $[a,b]$
so that the expression $\Omega(k)+M\int_{a}^{t}h(s)\ ds$ is in the domain of
$\Omega^{-1}$.

The proof will require basic result from integration theory (see, for example,
[7], page 107):

######  Lemma 6.

Let $f$ be integrable on $[a,b]$, and suppose that

$F(t)=F(a)+\int_{a}^{t}f(s)\ ds.$

Then $F^{\prime}(t)=f(t)$ for almost every $t\in[a,b]$.

###### Proof of Proposition 5.

The proof is a modification of Bihari’s original argument in [1]. Define

$V(t)=k+M\int_{a}^{t}h(s)\omega(g(s))\ ds,$

from which we observe that $V(a)=k$. Since $h$ is integrable and $g$ and
$\omega$ are continuous, then the product $h(s)\omega(g(s))$ is integrable.
Applying Lemma 6, we have that

(8) $V^{\prime}(t)=Mh(t)\omega(g(t))$

for almost every $t\in[a,b]$.

Next, observe that $g(t)\leq V(t)$ by construction. Since $\omega$ is non-
decreasing, then we must have that $\omega(g(t))\leq\omega(V(t))$, which
implies that

$\frac{Mh(t)\omega(g(t))}{\omega(V(t))}\leq Mh(t).$

Applying equation (8), this becomes

$\frac{V^{\prime}(t)}{\omega(V(t))}\leq Mh(t)$

for almost every $t$. Integrating this expression then yields

$\int_{a}^{t}\frac{V^{\prime}(s)}{\omega(V(s))}\ ds\leq M\int_{a}^{t}h(s)\
ds.$

We may rewrite the integral on the left as

$\displaystyle\int_{a}^{t}\frac{V^{\prime}(s)}{\omega(V(s))}\ ds$
$\displaystyle=\int_{V(a)}^{V(t)}\frac{dV}{\omega(V)}$
$\displaystyle=\Omega(V(t))-\Omega(V(a))$
$\displaystyle=\Omega(V(t))-\Omega(k).$

Hence, we obtain

$\Omega(V(t))\leq\Omega(k)+M\int_{a}^{t}h(s)\ ds.$

To conclude the proof, we observe that if $\omega$ is non-decreasing, then so
are $\Omega$ and $\Omega^{-1}$. The desired result follows. ∎

In addition to the Bihari-type inequality above, we will also need the
following result, which is Lemma 3.7 of [14]:

######  Lemma 7.

Let $M(t)$ be a non-negative continuous function of $t$ satisfying the
inequality

$M(t)\leq c_{1}+c_{2}[M(t)]^{\gamma}$

in some interval containing 0, where $c_{1}$ and $c_{2}$ are positive
constants and $\gamma>1$. If $M(0)\leq c_{1}$ and

$c_{1}c_{2}^{(\gamma-1)^{-1}}<(1-\gamma^{-1})\gamma^{-(\gamma-1)^{-1}},$

then in the same interval,

$M(t)<\frac{c_{1}}{1-\gamma^{-1}}.$

To finish our preliminary discussion, we collect some basic results which will
be useful below.

######  Lemma 8.

Let $a<1$ and $b<1$. Then there exists a constant $C_{a,b}$ such that

$\int_{0}^{t}(t-s)^{-a}s^{-b}\ ds=C_{a,b}t^{1-a-b}.$

###### Proof.

By the simple substitution $u=\frac{s}{t}$, we have

$\int_{0}^{t}(t-s)^{-a}s^{-b}\ ds=t^{1-a-b}\int_{0}^{1}(1-u)^{-a}u^{-b}\ du.$

The integral on the right converges when $a<1$ and $b<1$. ∎

######  Lemma 9 (Young’s Inequality).

Let $f\in L^{p}(\mathbb{R}^{d})$ and $g\in L^{q}(\mathbb{R}^{d})$, for $1\leq
p,q\leq\infty$ with

$\frac{1}{p}+\frac{1}{q}\geq 1.$

Then $f*g\in L^{r}(\mathbb{R}^{d})$, where

$\frac{1}{r}=\frac{1}{p}+\frac{1}{q}-1.$

Moreover, we have the estimate

(9) $\|f*g\|_{L^{r}}\leq\|f\|_{L^{p}}\|g\|_{L^{q}}.$

######  Lemma 10 (Gagliardo-Nirenberg Inequality).

Let $1<p<q\leq\infty$, and let $s\geq 0$ be such that

$\frac{1}{q}=\frac{1}{p}-\frac{\theta s}{d}$

for some $0<\theta<1$. Then for any $u\in W^{1,p}(\mathbb{R}^{d})$, we have

$\|u\|_{L^{q}}\leq C\|u\|_{L^{p}}^{1-\theta}\|u\|_{\dot{W}^{s,p}}^{\theta}$

for some $C>0$.

##  3\. Local Well-Posedness

We now turn to the proof of Theorem 1. As is standard practice, we begin by
first obtaining a local result. We reformulate our local result in the
following more explicit (and more general) form:

######  Proposition 11.

Let $1\leq p<\infty$, and assume that the surface current satisfies
$\mathbf{J}(\nabla u)\leq C|\nabla u|^{q}$ for some $q\geq 1$. Then for any

$u_{0}^{*}\in W^{1,p}(\mathbb{R}^{d})\cap W^{1,\infty}(\mathbb{R}^{d}),$

there exists a time $\delta>0$ and an open ball $B\subset
W^{1,p}(\mathbb{R}^{d})\cap W^{1,\infty}(\mathbb{R}^{d})$ containing
$u_{0}^{*}$, such that for each $u_{0}\in B$ there exists a unique strong
$W^{1,p}(\mathbb{R}^{d})\cap W^{1,\infty}(\mathbb{R}^{d})$ solution $u$ to the
integral equation (5) in the interval $I=[0,\delta)$. Furthermore, the
solution map $u_{0}\mapsto u$ is continuous from $B$ to $C(I;W^{1,p}\cap
W^{1,\infty}(\mathbb{R}^{d}))$.

We prove this by a standard iteration argument. Fix $u_{0}^{*}\in
W^{1,p}(\mathbb{R}^{d})$, and let $B=B(u_{0}^{*},R)$ be the open ball of
radius $R$ centered at $u_{0}^{*}$. Define a sequence
$\\{u^{n}\\}_{n=0}^{\infty}$ by

$\displaystyle u^{0}_{t}+\Delta^{2}u^{0}=0$ $\displaystyle
u^{n}_{t}+\Delta^{2}u^{n}+\nabla\cdot\mathbf{J}(\nabla u^{n-1})=0$
$\displaystyle u^{0}(0)=u_{0}$ $\displaystyle u^{n}(0)=u_{0}.$

We will show that this sequence converges in $C\left([0,\delta);W^{1,p}\cap
W^{1,\infty}\right)$ for sufficiently small $\delta$. As a first step, we show
it is bounded.

######  Proposition 12.

If $\delta>0$ is sufficiently small, then

$\|u^{n}\|_{L_{t}^{\infty}(W^{1,p}\cap W^{1,\infty})}\leq
2\|u_{0}\|_{W^{1,p}\cap W^{1,\infty}}$

for each $n\geq 0$.

###### Proof.

By induction. For $n=0$, we apply equation (4) with $G=0$ to obtain

$\|u^{0}(t)\|_{W^{1,p}}=\|k_{t}*u_{0}\|_{W^{1,p}}\leq\|k_{t}\|_{L^{1}}\|u_{0}\|_{W^{1,p}}.$

If we now apply Lemma 4, this simplifies to

$\|u^{0}(t)\|_{W^{1,p}}\leq\|u_{0}\|_{W^{1,p}},$

which implies the desired result for $n=0$.

Now assume the result holds for $n=m$. Then applying equation (4) once again,
we see that

(10) $\displaystyle\|u^{m+1}\|_{W^{1,p}}$
$\displaystyle\leq\|k_{t}*u_{0}\|_{W^{1,p}}+\int_{0}^{t}\|k_{t-s}*\nabla\cdot\mathbf{J}(\nabla
u^{m})\|_{W^{1,p}}\ ds$
$\displaystyle\leq\|u_{0}\|_{W^{1,p}}+\int_{0}^{t}\|\nabla
k_{t-s}\|_{W^{1,1}}\|\mathbf{J}(\nabla u^{m})\|_{L^{p}}\ ds$

Using the assumption on $J$, we have that

(11) $\displaystyle\|\mathbf{J}(\nabla u^{m}(s))\|_{L^{p}}$
$\displaystyle\leq\|\nabla u^{m}(s)\|_{L^{\infty}}^{q-1}\|\nabla
u^{m}(s)\|_{L^{p}}$
$\displaystyle\leq\|u(s)\|_{W^{1,\infty}}^{q-1}\|u(s)\|_{W^{1,p}}$
$\displaystyle\leq\|u(s)\|_{W^{1,p}\cap W^{1,\infty}}^{q}$
$\displaystyle\leq(2\|u_{0}\|_{W^{1,p}\cap W^{1,\infty}})^{q}$

Using the properties of $k_{t-s}$ from Lemma 4, we see that

(12) $\|\nabla
k_{t-s}\|_{W^{1,1}}\lesssim(t-s)^{-\frac{1}{2}}+(t-s)^{-\frac{1}{4}}.$

Combining equations (10), (11), and (12), we see that

$\|u^{m+1}(t)\|_{W^{1,p}}\leq\|u_{0}\|_{W^{1,p}}+C(t^{\frac{1}{2}}+t^{\frac{3}{4}})(2\|u_{0}\|_{W^{1,p}\cap
W^{1,\infty}})^{q}.$

Taking the supremum in $I=[0,\delta)$ gives us

$\|u^{m+1}\|_{L^{\infty}_{t}W^{1,p}_{x}}\leq\|u_{0}\|_{W^{1,p}}+C(\delta^{\frac{1}{2}}+\delta^{\frac{3}{4}})(2\|u_{0}\|_{W^{1,p}\cap
W^{1,\infty}})^{q}.$

The desired result follows for $\delta$ sufficiently small. To complete the
proof, we simply observe that the computations above also hold in the special
case $p=\infty$ with only a minor modification. ∎

######  Proposition 13.

The sequence $\\{u^{n}\\}_{n=0}^{\infty}$ satisfies

$\|u^{n}-u^{n-1}\|_{L^{\infty}_{t}(W^{1,p}\cap W^{1,\infty})}\leq
C\delta^{\frac{n}{2}}$

for some $C>0$.

###### Proof.

By induction on $n$. For $n=1$, we apply a similar computation as in the proof
of Proposition 12 to say that

$\|u^{1}-u^{0}\|_{L^{\infty}_{t}W^{1,p}_{x}}\leq
C(\delta^{\frac{1}{2}}+\delta^{\frac{3}{4}})(2\|u_{0}\|_{W^{1,p}\cap
W^{1,\infty}})^{q}.$

If $\delta$ is sufficiently small, this reduces to

$\|u^{1}-u^{0}\|_{L^{\infty}_{t}W^{1,p}_{x}}\leq
2^{q+1}C\delta^{\frac{1}{2}}\|u_{0}\|_{W^{1,p}\cap W^{1,\infty}}^{q}.$

This proves the result for $n=1$.

Now assume the result holds for $n=m$. Applying equations (4) and (6), we
obtain that

$\|u^{m+1}-u^{m}\|_{W^{1,p}}\leq\int_{0}^{t}\|\nabla
k_{t-s}\|_{W^{1,1}}\|\mathbf{J}(\nabla u^{m})-\mathbf{J}(\nabla
u^{m-1})\|_{L^{p}}\ ds.$

Using the fact that the sequence is uniformly bounded, and the fact that
$\mathbf{J}$ is differentiable, we obtain

$\|\mathbf{J}(\nabla u^{m})-\mathbf{J}(\nabla u^{m-1})\|_{L^{p}}\leq C\|\nabla
u^{m}-\nabla u^{m-1}\|_{L^{p}}$

for some constant $C$ by the Mean Value Theorem. Using the inductive
hypothesis and equation (7), we obtain

$\displaystyle\|u^{m+1}-u^{m}\|_{W^{1,p}}$ $\displaystyle\leq
C\delta^{\frac{m}{2}}\int_{0}^{t}(t-s)^{-\frac{1}{2}}+(t-s)^{-\frac{1}{4}}\
ds$ $\displaystyle\leq
C\delta^{\frac{m}{2}}\left(\delta^{\frac{1}{2}}+\delta^{\frac{3}{4}}\right)$
$\displaystyle\leq C\delta^{\frac{m+1}{2}},$

as desired. As before, we observe that the above computations also hold for
$p=\infty$. This completes the proof of the proposition. ∎

######  Corollary 14 (Existence).

The sequence $\\{u_{n}\\}_{n=0}^{\infty}$ converges to a solution $u$ of
equation (5).

The next step is to show uniqueness.

######  Proposition 15 (Uniqueness).

The strong $W^{1,p}\cap W^{1,\infty}$ solution to equation (1) is unique.

###### Proof.

Suppose $u$ and $v$ are solutions to equation (5) with initial data $u_{0}$.
By proceeding as we did above, we may use equation (5) and the properties of
$\mathbf{J}$ and $k_{t}$ to obtain the estimate

$\displaystyle\|u(t)-v(t)\|_{W^{1,p}\cap W^{1,\infty}}$
$\displaystyle\qquad\qquad\quad\leq\int_{0}^{t}\left[(t-s)^{-\frac{1}{2}}+(t-s)^{-\frac{1}{4}}\right]\|u(s)-v(s)\|_{W^{1,p}\cap
W^{1,\infty}}\ ds.$

Applying Proposition 5, we obtain that

$\|u-v\|_{L^{\infty}_{t}(W^{1,p}\cap W^{1,\infty})}\leq 0.$

Uniqueness follows. ∎

To complete the proof of Proposition 11, we must also show that the solution
map is continuous.

######  Proposition 16.

The solution map $u_{0}\mapsto u$ is continuous.

###### Proof.

It suffices to show that there exists a constant $C>0$ such that

$\|u-v\|_{L^{\infty}_{t}(W^{1,p}\cap W^{1,\infty})_{x}}\leq
C\|u_{0}-v_{0}\|_{W^{1,p}}$

for solutions $u,v$ of equation (5) corresponding to initial data
$u_{0},v_{0}$ respectively. Since $u$ and $v$ satisfy equation (5), we have

$\displaystyle\|u-v\|_{W^{1,p}}$
$\displaystyle\leq\|k_{t}*(u_{0}-v_{0})\|_{W^{1,p}}$
$\displaystyle\quad+\left\|\int_{0}^{t}\nabla k_{t-s}*(\mathbf{J}(\nabla
u)-\mathbf{J}(\nabla v))\ ds\right\|_{W^{1,p}}.$

Proceeding as above, we have

$\|k_{t}*(u_{0}-v_{0})\|_{W^{1,p}}\leq\|u_{0}-v_{0}\|_{W^{1,p}}$

and

$\left\|\int_{0}^{t}\nabla k_{t-s}*(\mathbf{J}(\nabla u)-\mathbf{J}(\nabla
v))\ ds\right\|_{W^{1,p}}\leq
C(t^{3/4}+t^{1/2})\|u-v\|_{L^{\infty}_{t}W^{1,p}_{x}}.$

We may thus conclude that

$\|u-v\|_{L^{\infty}_{t}W^{1,p}_{x}}\leq\|u_{0}-v_{0}\|_{W^{1,p}}+C(\delta^{3/4}+\delta^{1/2})\|u-v\|_{L^{\infty}_{t}W^{1,p}_{x}},$

which we rewrite as

(13)
$\|u-v\|_{L^{\infty}_{t}W^{1,p}_{x}}\leq\frac{1}{1-C(\delta^{3/4}+\delta^{1/2})}\|u_{0}-v_{0}\|_{W^{1,p}}.$

Of course, this requires that

$C(\delta^{3/4}+\delta^{1/2})<1$

so that the right-hand side of equation (13) is well-defined. In this case, we
may conclude that the solution map $u_{0}\mapsto u$ is continuous. ∎

Proposition 11 follows immediately from Corollary 14 and Propositions 15 and
16.

##  4\. Gradient Bounds on Solutions

In this section, we will prove Theorem 2. To begin, define

$M=\sup_{t\in[0,T]}\left\\{\|\nabla
u(t)\|_{L^{p}_{x}}+t^{\frac{d}{4p}}\|\nabla u(t)\|_{L^{\infty}_{x}}\right\\}.$

We will show that if the gradient of $u_{0}$ is sufficiently small, we will
have

$M<\infty,$

from which the desired result will follow.

Observe that

$\|\nabla u(t)\|_{L^{\infty}_{x}}=\|u(t)\|_{\dot{W}^{1,\infty}_{x}},$

so that

$\displaystyle\|\nabla u\|_{L^{\infty}_{x}}$
$\displaystyle=\|k_{t}*u_{0}\|_{\dot{W}^{1,\infty}_{x}}+\int_{0}^{t}\|\nabla
k_{t-s}*\mathbf{J}(\nabla u(s))\|_{\dot{W}^{1,\infty}_{x}}\ ds$
$\displaystyle\leq\|k_{t}\|_{L^{p^{\prime}}}\|\nabla
u_{0}\|_{L^{p}}+\int_{0}^{t}\|D^{2}k_{t-s}\|_{L^{1}}\|\nabla
u(s)\|_{L^{\infty}_{x}}^{q}\ ds.$

Here, $p^{\prime}$ is the conjugate exponent to $p$:

$\frac{1}{p}+\frac{1}{p^{\prime}}=1$

Thus, the above inequaility simplifies to

$\displaystyle\|\nabla u\|_{L^{\infty}_{x}}$ $\displaystyle\leq
Ct^{-\frac{d}{4p}}\|\nabla
u_{0}\|_{L^{p}}+C\int_{0}^{t}(t-s)^{-\frac{1}{2}}\|\nabla
u(s)\|_{L^{\infty}_{x}}^{q}\ ds.$

Multiplying by $t^{\frac{d}{4p}}$ yields

$\displaystyle t^{\frac{d}{4p}}\|\nabla u\|_{L^{\infty}_{x}}$
$\displaystyle\leq C\|\nabla
u_{0}\|_{L^{p}}+Ct^{\frac{d}{4p}}\int_{0}^{t}(t-s)^{-\frac{1}{2}}\|\nabla
u(s)\|_{L^{\infty}_{x}}^{q}\ ds$ $\displaystyle\leq C\|\nabla u_{0}\|_{L^{p}}$
$\displaystyle\quad+Ct^{\frac{d}{4p}}\left(\int_{0}^{t}(t-s)^{-\frac{1}{2}}s^{-\frac{dq}{4p}}\
ds\right)\left(\sup_{s\in[0,T]}s^{\frac{d}{4p}}\|\nabla
u(s)\|_{L^{\infty}_{x}}\right)^{q}.$

Applying Lemma 8, this reduces to

$\displaystyle t^{\frac{d}{4p}}\|\nabla u\|_{L^{\infty}_{x}}$
$\displaystyle\leq C\|\nabla
u_{0}\|_{L^{p}}+Ct^{\frac{1}{2}-\frac{d(q-1)}{4p}}\left(\sup_{s\in[0,T]}s^{\frac{d}{4p}}\|\nabla
u(s)\|_{L^{\infty}_{x}}\right)^{q}$ $\displaystyle\leq C\|\nabla
u_{0}\|_{L^{p}}+Ct^{\frac{1}{2}-\frac{d(q-1)}{4p}}M^{q}.$

Observe that the right-hand side of this inequality will be independent of $t$
if

$q=1+\frac{2p}{d}.$

This then becomes

(14) $t^{\frac{d}{4p}}\|\nabla u(t)\|_{L^{\infty}_{x}}\leq C\|\nabla
u_{0}\|_{L^{p}}+CM^{1+\frac{2p}{d}}.$

By a similar computation, it is easy to show that

$\|\nabla u(t)\|_{L^{p}_{x}}\leq\|\nabla
u_{0}\|_{L^{p}}+C\int_{0}^{t}(t-s)^{-\frac{1}{2}}\||\nabla
u(s)|^{q}\|_{L^{p}_{x}}\ ds.$

From the definition of $q$, we have

$\||\nabla u(s)|^{q}\|_{L^{p}_{x}}\leq\|\nabla
u(s)\|_{L^{\infty}_{x}}^{\frac{2p}{d}}\|\nabla u(s)\|_{L^{p}_{x}}.$

Thus, we obtain the estimate

$\|\nabla u(t)\|_{L^{p}_{x}}\leq\|\nabla
u_{0}\|_{L^{p}}+C\int_{0}^{t}(t-s)^{-\frac{1}{2}}\|\nabla
u(s)\|_{L^{\infty}_{x}}^{\frac{2p}{d}}\|\nabla u(s)\|_{L^{p}_{x}}\ ds.$

Note that the integral in the right-hand side can be estimated by

$\displaystyle\int_{0}^{t}(t-s)^{-\frac{1}{2}}\|\nabla
u(s)\|_{L^{\infty}_{x}}^{\frac{2p}{d}}\|\nabla u(s)\|_{L^{p}_{x}}\ ds$
$\displaystyle\qquad\qquad=\int_{0}^{t}(t-s)^{-\frac{1}{2}}s^{-\frac{1}{2}}\left(s^{\frac{d}{4p}}\|\nabla
u(s)\|_{L^{\infty}_{x}}\right)^{\frac{2p}{d}}\|\nabla u(s)\|_{L^{p}_{x}}\ ds$
$\displaystyle\qquad\qquad\leq\left(\int_{0}^{t}(t-s)^{-\frac{1}{2}}s^{-\frac{1}{2}}\
ds\right)M^{1+\frac{2p}{d}}.$

Applying Lemma 8, we thus obtain the estimate

(15) $\|\nabla u(t)\|_{L^{p}}\leq\|\nabla
u_{0}\|_{L^{p}}+CM^{1+\frac{2p}{d}}.$

Combining equations (14) and (15), we have

$\|\nabla u(t)\|_{L^{p}}+t^{\frac{d}{4p}}\|\nabla u(t)\|_{L^{\infty}}\leq
C_{1}\|\nabla u_{0}\|_{L^{p}}+C_{2}M^{1+\frac{2p}{d}}.$

Taking the supremum over $t\in[0,T]$ gives us

$M\leq C_{1}\|\nabla u_{0}\|_{L^{p}}+C_{2}M^{1+\frac{2p}{d}}.$

It then follows from Lemma 7 that

$M<C$

for some constant $C>0$ whenever $\|\nabla u_{0}\|_{L^{p}}$ is sufficiently
small. An immediate consequence of this is that

(16) $\|\nabla
u(t)\|_{L^{\infty}_{x}}\leq\frac{C}{t^{\frac{d}{4p}}}\quad\text{and}\quad\|\nabla
u(t)\|_{L^{p}_{x}}\leq C$

for $t>0$. This completes the proof of Theorem 2.

######  Corollary 17.

Assume that $u$ is a solution to equation (1) for $q>1$, and let
$p<p_{\theta}<\infty$ be such that

$\frac{1}{p_{\theta}}=\frac{1-\theta}{p}.$

Then

$\|\nabla u(t)\|_{L^{p_{\theta}}}\leq Ct^{-\frac{d\theta}{4p}}.$

###### Proof.

This result can be obtained by interpolating the two estimates in equation
(16), which hold under the smallness assumption on $\nabla u_{0}$. ∎

##  5\. Global Well-posedness

In this section, we show that the local solutions constructed in section 3
exist globally. Let $u$ be such a solution, and suppose that $T^{*}$ is the
maximal time of existence. If $T^{*}=+\infty$, there is nothing to prove, so
suppose that $T^{*}<+\infty$. By a standard argument, it must be the case that

$\sup_{t\in[0,T^{*}]}\|u(t)\|_{W^{1,p}\cap W^{1,\infty}}=+\infty.$

We will show that this is not the case.

By the local theory in the previous section, it is a simple matter to see that
$u=0$ is the unique strong $W^{1,p}\cap W^{1,\infty}$ solution to the Cauchy
problem (LABEL:Cauchy) with $u_{0}=0$. Thus, we may assume that $u_{0}\neq 0$.
In light of Theorem 2, we need only estimate the growth of $\|u(t)\|_{L^{p}}$
and $\|u(t)\|_{L^{\infty}}$.

By proceeding as we did to obtain equation (13), it is easy to see that $u$
satisfies

$\|u(t)\|_{L^{p}}\leq\|u_{0}\|_{L^{p}}+C\int_{0}^{t}(t-s)^{-\frac{1}{4}}\|\nabla
u(s)\|_{L^{pq}}^{q}\ ds.$

To this, we apply Corollary 17 with $p_{\theta}=pq>p$ to obtain

$\|\nabla u(s)\|_{L^{pq}}\leq Ct^{-\frac{d}{4p}\left(1-\frac{1}{q}\right)}.$

Thus, we obtain that

$\|u(t)\|_{L^{p}}\leq\|u_{0}\|_{L^{p}}+C\int_{0}^{t}(t-s)^{-\frac{1}{4}}s^{-\frac{d}{4p}(q-1)}\
ds.$

Since we assume that

$q=1+\frac{2p}{d},$

then this estimate simplifies to

$\|u(t)\|_{L^{p}}\leq\|u_{0}\|_{L^{p}}+C\int_{0}^{t}(t-s)^{-\frac{1}{4}}s^{-\frac{1}{2}}\
ds.$

Applying Lemma 8, we obtain

(17) $\|u(t)\|_{L^{p}}\leq\|u_{0}\|_{W^{1,p}}+Ct^{\frac{1}{4}},$

which we may express as

(18) $\|u(t)\|_{L^{p}}\leq C(1+t)^{\frac{1}{4}}.$

Now, we turn to estimating the $L^{\infty}$ norm of $u$. Proceeding as we have
before, we have the estimate

$\displaystyle\|u(t)\|_{L^{\infty}_{x}}$
$\displaystyle\leq\|u_{0}\|_{L^{\infty}}+C\int_{0}^{t}\|\nabla
k_{t-s}\|_{L^{p^{\prime}}}\||\nabla u|^{q}\|_{L^{p}}$
$\displaystyle\leq\|u_{0}\|_{W^{1,\infty}}+C\int_{0}^{t}(t-s)^{-\frac{d}{4p}-\frac{1}{4}}\|\nabla
u(s)\|_{L^{pq}}^{q}\ ds$
$\displaystyle\leq\|u_{0}\|_{W^{1,\infty}}+C\int_{0}^{t}(t-s)^{-\frac{d}{4p}-\frac{1}{4}}s^{-\frac{1}{2}}\
ds$ $\displaystyle\leq\|u_{0}\|_{W^{1,\infty}}+Ct^{\frac{3}{4}-\frac{d}{4p}}.$

Of course, this requires that

$p>\frac{d}{3},$

which is satisfied when

$q>\frac{5}{3}.$

This completes the proof of Theorem 1.

##  6\. Coarseness

In this section, we prove Theorem 3. Begin by applying Lemma 10 to obtain the
estimate

$\|u(t)\|_{L^{2}}\leq C\|\nabla u(t)\|_{L^{p}}^{1-\theta}\|\nabla
u(t)\|_{L^{p}}^{\theta}$

for

(19) $\theta=d\left(\frac{1}{p}-\frac{1}{2}\right).$

For this to be valid, we require

$0<d\left(\frac{1}{p}-\frac{1}{2}\right)<1$

and

$1<p<2,$

with $p$ as in Theorem 1. This is satisfied when

$\max\left\\{\frac{6+d}{2+d},1+\frac{2}{d}\right\\}<q<1+\frac{4}{d}.$

In this case, we apply inequality (18) and Theorem 2 to see that

$\|u(t)\|_{L^{2}}\leq C(1+t)^{\frac{1-\theta}{4}}.$

Substituting $\theta$ from equation (19), we obtain

$\|u(t)\|_{L^{2}}\leq C(1+t)^{\frac{1}{4}-\frac{d}{4p}+\frac{d}{8}},$

as desired. This completes the proof of Theorem 3.

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