# Local Life: Stay Informed Around You, A Scalable Geoparsing and Geotagging
Approach to Serve Local News Worldwide

Deven Santosh Shah
Microsoft
<EMAIL_ADDRESS>
Gosuddin Kamaruddin Siddiqi
Microsoft
<EMAIL_ADDRESS>
Shiying He
Microsoft
<EMAIL_ADDRESS>
Radhika Bansal
Microsoft
<EMAIL_ADDRESS>

###### Abstract

Local news has become increasingly important in the news industry due to its
various benefits. It offers local audiences information that helps them
participate in their communities and interests. It also serves as a reliable
source of factual reporting that can prevent misinformation. Moreover, it can
influence national audiences as some local stories may have wider implications
for politics, environment or crime. Hence, detecting the exact geolocation and
impact scope of local news is crucial for news recommendation systems. There
are two fundamental things required in this process, (1) classify whether an
article belongs to local news, and (2) identify the geolocation of the article
and its scope of influence to recommend it to appropriate users. In this
paper, we focus on the second step and propose (1) an efficient approach to
determine the location and radius of local news articles, (2) a method to
reconcile the user’s location with the article’s location, and (3) a metric to
evaluate the quality of the local news feed. We demonstrate that our technique
is scalable and effective in serving hyperlocal news to users worldwide.

##  1 Introduction

Local news is a vital source of information for users who want to stay updated
on their surroundings or learn about other places. We follow the definition of
local news published by Shah et al. (2023), which states:

> We define local news articles that impact a specific set of users at the
> city/county/state level.

Shah et al. (2023) suggested there are two fundamental things that are
required to keep the users informed about their surroundings, (1) identifying
whether an article is local news, and (2) detecting and recognizing it’s
geolocation and the impact radius, so that we could serve right news articles
to the right audience. In this paper, we focus on the second step assuming the
news articles we get are local.

Detecting the article’s right location and showcasing it to the right audience
not just improves user engagement (Robindro et al., 2017) but it helps keep
the community bind together, preserving it’s culture. It keeps the users
informed of their neighborhood, may that be related to crime, events, new
restaurants opening up, real estate, schools, sports, etc. Showcasing right
local news to the cold start users is beneficial to start engaging them.
Since, geolocation is the only information we get for cold start users,
personalizing the feed based on the user’s geolocation helps them engage
better with the articles and convert them into warm users. For warm users, who
have more preferences and behaviors, we can increase their retention rate by
serving them high-quality local news.

We also review the existing literature on local news detection and geolocation
extraction, and identify the main challenges and gaps in this research area.
The papers (discussed in Section 2) focus on extracting the geolocation
information from the articles however, they fail to mention a way to reconcile
the user’s location with the article’s location to serve the right local
articles. The main challenges that we found in this research area are:

  1. 1.

Acronyms/Teams/Organizations/Highways as locations: We came across multiple
local news articles in which geographical location name wasn’t mentioned
explicitly but was mentioned in the form of an acronym which might be a local
place or a local organization, or a highway like I-5 or even a local sports
team name. The locations of these acronyms are difficult to detect. For
instance: "UW students voice concerns about recent University District
crime"111https://www.msn.com/en-us/news/crime/uw-students-voice-concerns-
about-recent-university-district-crime/ar-AA17wteu, UW for University of
Washington, "Seahawks have 2 of PFF’s top 30 graded safeties for
2022"222https://www.msn.com/en-us/sports/nfl/seahawks-have-2-of-pffs-
top-30-graded-safeties-for-2022/ss-AA17cqCH, "Seahawks representing Seattle or
infact entire Washington state.

  2. 2.

Local news of broader interest: We came across certain local news articles
mentioning a specific city but could also be relevant to neighboring cities or
regions. For instance: “Washington State Fair announces 2X Platinum artist
with tickets on sale Wednesday” 333https://www.msn.com/en-
us/entertainment/news/washington-state-fair-announces-2x-platinum-artist-with-
tickets-on-sale-wednesday/ar-AA16V9bk. Even though Washington State Fair,
Tacoma is mentioned in the article, people living in a radius of 50-80 miles
of Tacoma would still love to see this article. Another example where local
publishers capturing semi-local news articles like, "New WSU study says
grocery stores can trick customers into spending
more"444https://www.msn.com/en-us/money/companies/new-wsu-study-says-grocery-
stores-can-trick-customers-into-spending-more/ar-AA17g8YC

  3. 3.

Reconcile user’s location with the article’s location: These techniques (Bell
et al., 2015; Teitler et al., 2008; Middleton et al., 2018) do not mention how
would they extract the geographical location of the user. There are certain
techniques present that backtraces the IP address of the user to a location
(Alt et al., 2010), however, reconciling these locations with the article’s
location to serve the right local news would be a separate task.

  4. 4.

Multiple locations present in the article: We also came across multiple
articles where there were multiple locations present. These approaches fail to
handle this scenario. For instance: "No. 4 Arizona strives for best execution
against offensively challenged Cal" 555https://www.msn.com/en-
us/sports/ncaabk/no-4-arizona-strives-for-best-execution-against-offensively-
challenged-cal/ar-AA17fxR7, this article mentions two states Arizona and
California. People living in these two states would majorly be interested in
seeing this article.

We showcase that not only our technique helps us understand the local
publishers in an area but also scales worldwide and is effective in serving
relevant hyperlocal news to the users globally. Our technique also helps in
converting cold start users to warm users, and helps retain warm users.

Our major contributions include: (1) a novel efficient ensemble approach to
detect and recognize the geographical location of the article and it’s impact
radius, with high precision and high recall, (2) a technique to reconcile the
user’s location with the geographical location of the article to serve the
right local news, and (3) an evaluation metric to measure the quality of the
local news feed. We showcase that not only our technique helps us understand
the local publishers in an area but also scales worldwide and is effective in
serving relevant hyperlocal news to the users globally. Our technique also
helps in converting cold start users to warm users, and helps retain warm
users.

##  2 Related Work

In the news domain, we realized that majority of the existing literature on
geolocation extraction lie in one or more of the four issues buckets mentioned
in section 1.

Sankaranarayanan et al. (2009) focuses on using user-generated content (UGC)
platforms, such as Twitter, to gather breaking news in the area. They proposed
a system that clusters tweets based on their geolocation name mentioned and
the user’s geolocation and assigns the geolocation foci of each cluster to all
the tweets falling in it. However, this approach has several limitations: (1)
it relies on user’s geolocation to extract the geolocation of the tweet, which
may not always match the tweet’s geolocation, (2) it ignores news impact
radius, which depends on topic and audience, and (3) it requires manual
identification of the users who post the news as tweets, which is hard to
scale worldwide

Tahmasebzadeh et al. (2021) proposed a technique to use geolocation extracted
from an image to showcase local news pertaining to that area. They proposed a
technique that uses ResNets to predict the spatial geolocation and the entity
type of an image, and then matches the entity name with the Wikidata corpus
and the Event Registry to retrieve the relevant news articles. They restrict
the entity types to 12 categories, such as landmarks, monuments, etc. However,
this approach also has several limitations: (1) it misses other important
local news categories, such as crime, sports, real estate, science, etc., (2)
it depends on the quality and completeness of the Wikidata and the Event
Registry, which may not be updated and accurate, and (3) it ignores user’s
location and preferences, which may differ from the image’s location and
entity type.

Bell et al. (2015) uses automatic speech recognition (ASR) to broadcast
summaries of the news provided by the broadcasting channels. They proposed an
ASR technique to generate the speech transcript and NER to extract and
disambiguate geolocation with OpenStreet Map. However, they do not mention the
serving technique for the generated local news summaries.

Teitler et al. (2008) and Middleton et al. (2018) use a three-step process to
get text geolocation: (1) geocoding, which maps the location names to the
latitude and longitude coordinates, (2) geoparsing, which identifies the
location names in text, and (3) geotagging, which assigns a geolocation to the
text. They claim this approach can handle different text types and sources,
and can be easily applied to the structured content like news articles present
with news aggregators. However, this approach also has several limitations:
(1) it depends on the size and scope of the gazetteer, which may not cover all
location names or variations, (2) it ignores the impact radius of the local
news articles, which may vary depending on the topic and the audience, and (3)
they do not mention their approach to reconcile the article’s and user’s
location to serve the right local news, which may differ in distance,
relevance, and preference.

Adar et al. (2017), inspired by personalizing the feed of users, proposed a
tool that personalizes the content of the article based on two properties of
the user: geolocation and demographics inference. They capture the location of
the user by backtracing its IP address, and the location of the article by the
journalist from a list of cities. They claim that this technique can provide
more relevant and engaging news content to the user. However, this approach
also has several limitations: (1) it restricts the location of the article to
a list of cities, which may not cover all the possible local news sources or
topics, and (2) it serves the local news content to the user only if they are
within 50 miles of a nearby big city, which makes the technique difficult to
scale worldwide.

Gonçalves et al. (2021) and Väätäjä et al. (2012) focuses on using crowd-
sourcing techniques to generate local news content from the citizens of an
area. They proposed a framework to encourage local journalism in Portugal and
Finland, respectively, by allowing the citizens to share photos, videos, posts
about news-worthy events in their region, and local journalists to pick up the
story if it seemed important. They claim that this technique can enhance the
quality and diversity of local news content, and foster the community
interaction and engagement. However, this approach also has several
limitations: (1) it does not specify how to extract the location from the
citizen’s posts, which may not always contain explicit or accurate location
names, (2) it uses the user’s IP backtraced location as a proxy for the
location of the post, which may not be the correct representative of the
location of the event, and (3) it does not reconcile the user’s location and
the post’s location to serve the right local news to the users, which may have
different preferences and interests.

In contrast to these existing approaches, our proposed approach addresses all
the issues and limitations mentioned above, and provides a more efficient and
effective way to geoparse and geotag the location and radius of local news
articles, and to match them with the user’s location and preferences. We also
propose an evaluation metric to measure the quality and relevance of the local
news feed, and demonstrate that our approach is scalable and effective in
serving hyperlocal news to users worldwide.

##  3 Methodology

We elaborated the local news serving problem into three subtasks, namely: (1)
Geoparsing, detecting the geolocation and geotagging, recognizing it and
determining the impact radius of the article, (2) reconciling the article and
user’s geolocation and preferences, and (3) delivering the personalized local
news feed to the user.

###  3.1 Geolocation as a form of geohash

To match the user’s geolocation with the article’s geolocation, we use
geohashes to represent both the user and the article. A geohash is a string of
letters and numbers that encodes a rectangular area on the Earth’s surface.
The longer the geohash, the smaller the area it covers. We use a single
geohash to represent the user’s geolocation, and a set of geohashes to
represent the article’s impacted geolocation. Table 1 shows the width and
height of the rectangular areas corresponding to different geohash
lengths666https://www.movable-type.co.uk/scripts/geohash.html.

Geohash length | Cell width | Cell height
---|---|---
1 $\leq$ | 5000 km | 5000 km
2 $\leq$ | 1250 km | 625 km
3 $\leq$ | 156 km | 156 km
4 $\leq$ | 39.1 km | 19.5 km
5 $\leq$ | 4.89 km | 4.89 km
6 $\leq$ | 1.22 km | 0.61 km
7 $\leq$ | 153 m | 153 m

Table 1: Rectangular geohash cell height and width coverage for geohashes of
different lengths

###  3.2 Article Location detection

Geoparsing and geotagging the geolocation and determining the impact radius of
the article is a key task in local news serving. We use an ensemble of
geoparsing and geotagging techniques for high precision and recall. The
techniques we use are: (1) Location Table (LT) lookup, which detects and maps
the location names to geohashes, (2) Bing Maps API, which detects and map the
location names from the article and provides the bounding box coordinates to
convert them into geohashes, and (3) Publisher-to-Location affinity mapping,
which assigns geohashes to the article based on its publisher. For each
article, we stamp geohashes of the article’s impacted geolocation by applying
the following rules: (1) if the publisher of the article is in the publisher-
to-location mapping, stamp mapped geohashes of length four, (2) if LT geohash
and publisher-to-location geohash have same prefix of length two, stamp all LT
geohashes of length four, (3) if BMA geohash and publisher-to-location geohash
have same prefix of length two, stamp all BMA geohashes of length four, (4) if
publisher-to-location mapping isn’t available and if LT geohash and BMA
geohash have same prefix of length two, stamp all LT and BMA geohashes of
length four, (5) if publisher-to-location mapping and LT lookup are not
available for the article, use BMA geohashes that are predicted with high
confidence, and (6) if none of the rules apply, stamp no geohash to keep high
precision over recall. We describe each of the techniques below:

####  3.2.1 Location Table Lookup

We compiled a list of locations in the United States, Canada, United Kingdom,
and Australia. We checked if the article contained any location names or
alias, and used the Bing Maps API bounding box coordinates to obtain geohashes
of the location’s city-county/district-state-country geochain. We crowdsourced
the evaluation of this technique using the UHRS hit apps
777https://prod.uhrs.playmsn.com/UHRS/, and got accuracy of 0.86 and recall of
0.38. However, we found that this technique is not scalable worldwide.

####  3.2.2 Bing Maps API Location

There are several APIs that provide location information from a query/text,
for instance, Google geocoding API888
https://developers.google.com/maps/documentation/geocoding, OpenStreetMap
Nominatim999http://wiki.openstreetmap.org/wiki/Nominatim and Bing Maps
API101010https://www.microsoft.com/en-us/maps/choose-your- bing-maps-api. We
use Bing Maps API. Bing Maps has an API to geoparse and geotag location
information from text with confidence levels and location type. The confidence
levels are High, Medium and Low, we use only those locations predicted with
High and Medium confidence. Bing Maps API has different entity types, from
Hospital, Building, Neighborhood to City, States, Monument and National Parks.
We also use the coordinates information, like point coordinate of the
location, which equates to the center point of the location and the bounding
box coordinates that identifies four corners of a box like shape containing
the location. We then extract all the geohashes within these coordinates.

To judge the performance of the Bing Maps API on new articles, we generated
recall metrics. As a ground truth, we assumed each news article is stamped
with a location. To gain maximum recall we considered different parts of the
articles, such as title, snippet, url and body and various combinations.
(Table 2).

Market |  Title |  Snippet |  URL |  Body |  Title + Body |  Title + Snippet + URL |  Title + Snippet + Body
---|---|---|---|---|---|---|---
en-au |  0.75 |  0.45 |  0.75 |  0.73 |  0.88 |  0.85 |  0.90
en-ca |  0.73 |  0.54 |  0.72 |  0.73 |  0.88 |  0.83 |  0.89
en-gb |  0.59 |  0.31 |  0.58 |  0.55 |  0.69 |  0.69 |  0.74
en-us |  0.72 |  0.66 |  0.69 |  0.75 |  0.91 |  0.86 |  0.94
Table 2:  Comparing BMA Location Recall Metrics for news article attributes;
Snippet extracted by in-house trained textrank model; Title, snippet, body
trimmed to starting and ending 10 words each for QPS reduction.

We found that the Title + Snippet + Body query gave the best precision and
recall for the BMA location. We also noticed that URL in query could bias the
BMA to the location name in the provider’s name, which might not be article’s
location. For example, KOMO-TV Seattle reported a news from Sammamish area,
for instance: "Person shot during home invasion in
Sammamish"111111https://www.msn.com/en-us/news/crime/person-shot-during-home-
invasion-in-sammamish/ar-AA17qJG0, but the BMA might pick Seattle as the
article’s location, which would prevent the article from reaching the
Sammamish users, and also annoy the Seattle users who might not care about
Sammamish news. Therefore, we decided to exclude the URL from the query for
BMA, thus avoiding label bias (Shah et al., 2019).

We measured the precision of the recalled locations with crowd-sourced
evaluation using the UHRS hit apps. For each article, we shared, the title,
snippet, body and made the URL available for the user to read the article. We
then asked the users if detected location is correct for the given article.
These numbers aggregated by different locales are listed in Table 3.

Market | Baseline Recall | BMA Recall | BMA Precision
---|---|---|---
en-au | 0.35 | 0.80 | 0.94
en-gb | 0.18 | 0.84 | 0.97
en-ca | 0.25 | 0.85 | 0.93
en-us | 0.66 | 0.84 | 0.91
de-de | 0.41 | 0.73 | 0.95
es-es | NA | 0.69 | 0.96
fr-fr | NA | 0.63 | 0.94
it-it | NA | 0.59 | 0.85
ja-jp | NA | 0.83 | 0.84
zh-cn | NA | 0.86 | 0.92

Table 3: Precision-Recall Metrics comparison between baseline and BMA’s
extraction model.

####  3.2.3 Publisher-Location affinity

We observed that some local news articles did not explicitly mention the
location name in them, such as "Boba exhibit opening at the Chinese American
Museum"121212https://www.msn.com/en-us/video/peopleandplaces/boba-exhibit-
opening-at-the-chinese-american-museum/vi-AA17g8k4, or "Eastbound 520 Bridge
Closure Planned Saturday Night"131313https://www.msn.com/en-
us/news/us/eastbound-520-bridge-closure-planned-saturday-night/ar-AA17gizN, or
"Warriors win against OKC reveals silver lining to Steph Curry’s
absence"141414https://www.msn.com/en-us/sports/nba/warriors-win-against-okc-
reveals-silver-lining-to-steph-curry-s-absence/ar-AA17dbSvwhich were published
by local news providers like “CBS Los Angeles”, “Patch”, and “Mercury News”
respectively. We also noticed that users in nearby cities, such as Seattle and
Bellevue, will be interested in each other’s local news, and local news
providers often covered news from neighboring cities as well. To handle these
cases, we identified the strongly local providers using the method proposed by
Shah et al. (2023). We then computed the publisher-to-location affinity for
these providers using all the LT and BMA locations recognized on their
articles in a month. We used the gap ratio (Shah et al., 2023) to filter out
the remote locations that the local providers rarely covered, and retained the
locations that they frequently covered. We assumed that the articles that did
not have the location name in them were from these frequent locations, which
could be at the city/county/district level. For example, we mapped the
publisher “KOMO-TV Seattle” to the location “King County, Washington, United
States”, suggesting that KOMO-TV Seattle mainly covers news articles from King
County. We created a mapping of publisher-to-location affinity using the
following steps, which are also illustrated in Figure 1:

  * •

Collect all the articles from a specific provider in a given time range.

  * •

Apply Bing Maps API and Location Table lookup to extract the locations from
these articles.

  * •

Use Bing Maps API to get the geohashes from the bounding box of the extracted
locations.

  * •

Use gap ratio on geohashes of length three to filter out the outlier
geohashes, and map the remaining geohashes of length four to their
corresponding locations

  * •

Use gap ratio to filter out outlier counties and states.

  * •

The remaining locations are the high-affinity locations that the provider
covers.

Figure 1: Publisher-to-Location Affinity: Illustrating the steps to obtain the
local publisher-to-location affinity.

The mapped location is converted to a set of geohashes using the Bing Maps API
bounding box coordinates. The publisher-to-location affinity mapping helped
address the issues of "Acronyms as locations", and "Local news of broader
interest" issues discussed in the Introduction section.

###  3.3 Local news serving

We obtain the user location from various sources, such as their IP address or
their input on the weather card. We convert this location, especially the
latitude and longitude, to a geohash of length four, which is used to retrieve
the articles to show to the users. If there are not enough articles in that
geohash, we backfill it with the nearest popular city, assuming that some
users might commute to big cities for work and want to stay informed about
them. The retrieved articles are then ranked by an in-house trained ranker to
display to the users. Converting geolocations to geohashes helped reconcile
the user’s and article’s location to correct serving.

##  4 Evaluation Metrics

We evaluated our End-To-End Local News serving technique, by performing an
online A/B experiment. We measured our technique specifically with the earth’s
distance between the user’s location and identified document location. In
addition to this, we accounted the fact that if a user’s location for example
a city Seattle, Washington matches with document’s location, for example the
same city, we assume the distance as 0 kms. This suggests our ability to serve
hyper-local news at the most granular level such as a city. Similarly, we take
into account for other geographical divisions such as county and state.

##  5 Results

Figure 2: P50 Distance Metrics: Comparing P50 metrics between treatment with
Local News serving technique and Control  Figure 3: P75 Distance Metrics:
Comparing P75 metrics between treatment with Local News serving technique and
Control

For our baseline, we used the publisher-to-DMA mapping. We manually mapped
strongly local publishers to a DMA, and stamped all the articles from the
local publisher with all the geohashes corresponding to the mapped DMA. DMA
stands for Designated Market Area, which is a geographical region comprising a
group of counties and zip codes defined by Neilsen
151515https://markets.nielsen.com/us/en/contact-us/intl-campaigns/dma-maps/.
There are 210 DMAs in the United States, mostly used by the television and
radio channels to broadcast to a set of users. We used lexicon-based knowledge
constraints (Shah et al., 2021) to tag DMA-based geohashes on the article.
These lexicons were manually created. We compared our ensemble technique of
detecting the article’s location to the DMA tagging, to measure the localness
of the delivered local news articles in terms of distance, and to ensure the
relevance of the local news shown to the users. We conducted on online
experiment to compare the distance between users and document. Treatment group
was exposed to End to End Local News serving technique and we observed
improvement on 50t̂h and 75t̂h Percentile distance. 50t̂h Percentile distance
improved from 15 kms on average to 8 kms on average (Fig:2) and 75t̂h
Percentile distance improved from 120 kms on average to 80 kms on average
(Fig:3). We also observed an improvement in content interactions per user
session for cold as well as warm start users.

##  6 Conclusion

In this paper, we proposed (1) an ensemble approach to determine the
geographical location of the article and it’s impact radius, (2) a technique
to reconcile the user’s location with the geographical impacted location of
the article to serve the right local news, and (3) an offline and online
evaluation metrics to measure the quality of the local news feed. We also
showcased our technique resolves the major issues with local news to showcase
right local articles to the right set of audience. We eventually showcased
that our technique is scalable worldwide, and helps convert cold start users
to warm start users.

## Limitations

Currently we have a rule based ensemble approach to stamp geolocation on an
article. There is a good potential to train a ML model to make the selection
from the models that provide the geolocation. Bing Maps API is a query based
API, and hence one limitation that we currently have in our system is we use
starting and ending 10 words to save on the QPS. Having an in-house or off-
the-shelf NER model to detect the location mention in the article and pass it
as an input to the Bing Maps API would help increase the precision of the BMA
Location extraction.

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# Long-time behavior of scalar conservation laws with critical dissipation

Dallas Albritton  Courant Institute of Mathematical Sciences, New York
University, New York, NY 10012<EMAIL_ADDRESS>and  Rajendra Beekie
Courant Institute of Mathematical Sciences, New York University, New York, NY
10012<EMAIL_ADDRESS>
###### Abstract.

The critical Burgers equation $\partial_{t}u+u\partial_{x}u+\Lambda u=0$ is a
toy model for the competition between transport and diffusion with regard to
shock formation in fluids. It is well known that smooth initial data does not
generate shocks in finite time. Less is known about the long-time behavior for
‘shock-like’ initial data: $u_{0}\to\pm a$ as $x\to\mp\infty$. We describe
this long-time behavior in the general setting of multidimensional critical
scalar conservation laws $\partial_{t}u+\operatorname{div}f(u)+\Lambda u=0$
when the initial data has limits at infinity. The asymptotics are given by
certain self-similar solutions, whose stability we demonstrate with the
optimal diffusive rates.

##  1\. Introduction

Our motivating example is the Burgers equation with critical non-local
dissipation

$\partial_{t}u+u\partial_{x}u+\Lambda u=0$ (1.1)

and ‘shock-like’ initial data:

$u_{0}(x)\to\pm a\text{ as }x\to\mp\infty,$ (1.2)

where $\Lambda=(-\Delta)^{1/2}$ and $a>0$. This equation arises as a toy model
in fluid mechanics. It models the competition between the transport non-
linearity $u\partial_{x}u$, which drives the solution towards a shock, and the
dissipation term $\Lambda u$, whose smoothing effects counteract the tendency
of the non-linearity to form shocks. The equation is _critical_ in the sense
that these two terms are in balance. In PDE terms, the strongest known
monotone quantities, the $L^{\infty}$ norm and total variation, are invariant
under the scaling symmetry

$u\to u(\lambda x,\lambda t),$ (1.3)

which preserves the equation (1.1).

By now, it is well known that solutions of (1.1) evolving from smooth initial
data do not form shocks in finite time. _What happens in infinite time?_ We
answer this question for the critical Burgers equation (1.1) and in the more
general context of scalar conservation laws with critical non-local
dissipation in $\mathbb{R}^{n}$:

$\partial_{t}u+\operatorname{div}f(u)+\Lambda u=0,$ (1.4)

where the initial data has ‘limits at infinity’. The long-time behavior is
described to leading order by certain _self-similar solutions_ , that is,
solutions invariant under the scaling symmetry (1.3).

Let $h\in C^{\infty}(S^{n-1})$ and $u_{0}^{\SS}(x)=h(x/|x|)$. Let
$f\colon\mathbb{R}\to\mathbb{R}^{n}$ belong to $C^{\infty}_{\rm
loc}(\mathbb{R};\mathbb{R}^{n})$. Let $v_{0}\in L^{\infty}(\mathbb{R}^{n})$
with $|v_{0}|\to 0$ as $|x|\to+\infty$, specifically,

$\lVert v\rVert_{L^{\infty}(\mathbb{R}^{n}\setminus B_{R})}\to 0\text{ as
}R\to+\infty.$ (1.5)

Let $u_{0}=u_{0}^{\SS}+v_{0}$ and $\lVert
u_{0}^{\SS}\rVert_{L^{\infty}(\mathbb{R}^{n})},\lVert
u_{0}\rVert_{L^{\infty}(\mathbb{R}^{n})}\leq m$. Let $u^{\SS}$, $u$ be the
unique _entropy solutions_ to (1.4) with initial data $u_{0}^{\SS}$, $u_{0}$,
respectively. In the context of (1.4), the notion of entropy solution was
introduced by Alibaud in [3], and we review it below. Notice that, by virtue
of its uniqueness, $u^{\SS}$ must be self similar.

Here is our main theorem:

######  Theorem 1.1 (Long-time behavior).

The above entropy solution $u$ converges to the self-similar solution
$u^{\SS}$ with the following (diffusive) rates:

$\lVert
u(\cdot,t)-u^{\SS}(\cdot,t)\rVert_{L^{q}(\mathbb{R}^{n})}\lesssim_{m,n}o_{t\to+\infty}(1)t^{\frac{n}{q}-\frac{n}{p}}\lVert
u_{0}-u_{0}^{\SS}\rVert_{L^{p}(\mathbb{R}^{n})}$ (1.6)

for all $1<p\leq q\leq+\infty$. If $p=1$, then (1.6) holds with $O(1)$ instead
of $o(1)$ on the RHS.

When $f\equiv 0$, (1.4) reduces to the fractional heat equation, and the above
diffusive rates are easily seen to be sharp.

In dimension $n=1$, we also have stability in ${\rm BV}$:

######  Theorem 1.2 (${\rm BV}$ convergence).

If also $u_{0}\in{\rm BV}(\mathbb{R})$, then

$\lVert u(\cdot,t)-u^{\SS}(\cdot,t)\rVert_{{\rm TV}(\mathbb{R}^{n})}\to
0\text{ as }t\to+\infty.$ (1.7)

Additionally, $u^{\SS}$ is monotone and satisfies the following spatial
asymptotics:

$C^{-1}\langle x\rangle^{-1}\leq|u^{\SS}(x,1)-u^{\SS}_{0}|\leq C\langle
x\rangle^{-1},\quad|x|\geq C,$ (1.8)

provided that $u^{\SS}_{0}$ is not identically constant.111It may be possible
to obtain more precise spatial asymptotics for $u^{\SS}$ and its derivatives
by analyzing the similarity profile $u^{\SS}(\cdot,1)$, which satisfies a
quasilinear nonlocal elliptic equation. Notice that, when $n\geq 2$,
$u^{\SS}_{0}$ does not generally belong to ${\rm BV}(\mathbb{R}^{n})$, and the
total variation is no longer scaling invariant.

### Comparison with existing literature

The critical Burgers equation (1.1) belongs to the following family of Burgers
equations with fractional diffusion:

$\partial_{t}u+u\partial_{x}u+\Lambda^{s}u=0,$ (1.9)

where $s\in(0,2]$. These models were considered by Biler et al. in [7], where
they are known as _fractal Burgers equations_. One may consider also the
analogous conservation laws with fractional diffusion $\Lambda^{s}$. The
relevant literature is fairly extensive:

Regularity theory. A detailed picture of the regularity theory of (1.9) was
shown by Kiselev, Nazarov, and Shterenberg in [33] in the periodic setting.
When $s\geq 1$, smooth initial data gives global smooth solutions, whereas
when $s<1$, solutions may develop shocks in finite time. In that case,
solutions may be continued uniquely within the class of entropy solutions. The
shock formation was also shown in [4]. The proof of global regularity in [33]
in the critical case follows the method of ‘moduli of continuity’. This method
was introduced in [31] by Kiselev, Nazarov, and Volberg in the context of the
critical SQG equation:222This method has since been generalized to other
models, including the 1d critical Keller-Segel equations [10], the 1d
fractional Euler alignment system [20], and the 2d Muskat problem [12].

$\partial_{t}\theta+\vec{R}^{\perp}\theta\cdot\nabla\theta+\Lambda\theta=0.$
(SQG)

Other proofs of the regularity of (SQG) are contained in [11] (De Giorgi’s
method), [30], [34] (Nash’s method), [18] (‘nonlinear maximum principle’), and
[17]. The above proofs can be categorized as proofs of _smoothing_ [11, 34] or
_propagation of regularity_ [31, 30, 18, 17]. The smoothing proofs notably
‘forget’ that the equation is nonlinear. Alternative proofs of regularity for
(1.1), based on smoothing, were given in [13] (De Giorgi’s method) and [38,
39] (non-divergence form techniques). We rely on these smoothing estimates,
particularly those of Silvestre, in an essential way below.333For
supercritical SQG, global regularity remains open, though it is possible to
show eventual regularity [37, 19, 32, 14]. We mention also the recent
extension of [11] to bounded domains in [41].

Long-time behavior. The long-time behavior of (1.9) is perhaps less well
studied than its regularity. When $s\in(0,2)$ and the initial data is well
localized, the non-linearity of (1.9) is ‘irrelevant’, in the sense of [9],
for the long-time dynamics. When $s=1$, Iwabuchi [24, 25] demonstrated that
all solutions with $u_{0}\in L^{1}\cap\dot{B}^{0}_{\infty,1}$ converge to the
Poisson kernel. When $s=2$, the spaces $L^{1}$ and $\mathcal{M}$ (finite
measures) are critical, and it is classical that the long-time behavior is
given by a self-similar solution, sometimes called a _diffusion wave_. This
case and its precise asymptotic behavior can be illuminated by the Cole-Hopf
transformation [15, 35, 29, 6].

What about non-decaying solutions? The current best results in this direction
concern ‘rarefaction-like’ initial data, that is, $a<0$ in (1.2). In [28], it
was shown that such solutions converge to an inviscid rarefaction wave when
$s>1$. In [5], it was shown that when $s=1$, the solutions converge to a
certain self-similar solution, and when $s<1$, the non-linearity is
‘irrelevant’ in the long-time asymptotic expansion. Notably, in the
rarefaction case, the potential term in the energy estimates for the
linearized equation appears with a good sign.

In this paper, we analyze the case of ‘shock-like’ initial data, which is less
clear. Initially, one might wonder whether (i) solutions converge to a smooth
traveling or standing wave, known as a ‘viscous shock’, or perhaps (ii)
solutions form a shock in infinite time. Regarding (i), it was already shown
in [7] that traveling wave solutions satisfying reasonable regularity
conditions do not exist when $s\in(0,1]$. Regarding (ii), one might
additionally wonder whether the standing waves constructed in the subcritical
case $s>1$ in [16] converge to a shock as $s\to 1^{+}$. This is apparently
also not the case, as we show in Theorem 1.1.

It is tempting to conjecture that, in the subcritical case $s>1$, shock-like
solutions of (1.9) behave as in the classical case $s=2$, where there is a
unique viscous shock, whose global asymptotic stability was shown by Ilín and
Oleĭnik in [23]. See [36, 27, 44] and many others for further developments and
precise asymptotics. For $s\in(1,2)$, the uniqueness, spatial asymptotics, and
global asymptotic stability of the monotone viscous shocks constructed in [16]
seem to be unknown, although local asymptotic stability was recently
demonstrated in [1].444It is also interesting to examine the convergence of
viscous solutions to shocks as $\nu\to 0^{+}$. In the case $s>1$, this was
done in [2].

Self-similarity and (non-)uniqueness. The two-dimensional Navier-Stokes
equations

$\partial_{t}\omega+u\cdot\nabla\omega-\Delta\omega=0,\quad
u=\nabla^{\perp}\Delta^{-1}\omega$ (NS)

exhibit a family of self-similar solutions known as the _Oseen vortices_ :
$\omega(x,t)=\alpha\Gamma(x,t/\nu)$, where $\Gamma$ is the heat kernel and
$\alpha=\int\omega_{0}\,dx$ is the circulation. In [21], Gallay and Wayne
famously showed that all localized solutions converge to Oseen vortices as
$t\to+\infty$, and, moreover, the vortex solutions are unique within a natural
solution class. Our situation is analogous, with the circulation $\alpha$
corresponding to the jump parameter $a$ in (1.2). By contrast, self-similar
solutions of the three-dimensional Navier-Stokes equations are expected to be
non-unique [26, 22]. For (SQG), this is investigated in forthcoming work of
Bradshaw and the first author. While the entropy solutions of (1.1) are
unique, there may be a different class of self-similar solutions with
potential non-uniqueness, for example, with $u_{0}\sim\log x$, so that $\nabla
u_{0}$ is $-1$-homogeneous.

### Main idea

Our starting point is the existence and uniqueness of $L^{\infty}$ entropy
solutions to (1.4), due to Alibaud [3], which immediately gives the existence
and uniqueness of a self-similar solution $u^{\SS}$. Let $v=u-u^{\SS}$ be the
difference between an entropy solution $u$ and the self-similar solution.
Consider a sequence $(v^{(k)})_{k\in\mathbb{N}}$ obtained by ‘zooming out’ on
$v$ using the scaling symmetry (1.3). Then establishing $v(\cdot,t)\to 0$ as
$t\to+\infty$ is the same as establishing $v^{(k)}\to 0$ on
$\mathbb{R}^{n}\times(1/2,1)$ as $k\to+\infty$. To analyze the new problem, we
exploit a key (standard) observation about viscous scalar conservation laws,
namely, that $v$ satisfies the _viscous continuity equation_

$\partial_{t}v+\operatorname{div}(gv)+\Lambda v=0,$ (1.10)

where

$g=\frac{f(u)-f(u^{\SS})}{u-u^{\SS}}\in
L^{\infty}(\mathbb{R}^{n}\times(0,+\infty)).$ (1.11)

In our setting, _the main difficulty is that at the initial time, $g$ is no
better than bounded, since $u^{\SS}_{0}$ is not continuous._ This is an
essential feature of the problem, and we handle it using two tools:

  1. (1)

_smoothing for drift-diffusion equations_. By the known regularity theory, the
solution, which is initially merely bounded, instantaneously becomes
$C^{\alpha}$-in-$x$. This may be bootstrapped to higher regularity. The key
point is then to move the problem past $t=0$, which is done by the

  2. (2)

_controlled speed of propagation_. Solutions of (1.10) have finite propagation
speed _up to the effect of the diffusion_. This allows us to keep the initial
spatial decay of the solution for small positive times and exploit (1.10) with
smooth coefficients and smooth, decaying initial data.

The above tools, due to [38, 39] and [3], respectively, are key to our
arguments. We encounter a further, technical difficulty in that the controlled
speed of propagation only allows us to propagate $L^{1}$-based quantities.
This requires the use of special norms
$\lVert\cdot\rVert_{\ell^{q}_{k}L^{p}_{x}(\mathbb{R}^{n})}$, for example,

$\lVert
v\rVert_{\ell^{\infty}_{k}L^{1}_{x}(\mathbb{R}^{n})}=\sup_{k\in\mathbb{Z}^{n}}\int_{k+(-1/2,1/2)^{n}}|v(x)|\,dx.$
(1.12)

After the initial time, we use the smoothing effect to estimate more standard
quantities, such as $\lVert v\rVert_{L^{q}(\mathbb{R}^{n})}$, in terms of
these special norms.555Similar norms appear in the Navier-Stokes literature.
See [8] and the references therein. Apparently, these spaces are known as
_Wiener amalgam spaces_. For this, we use pointwise estimates for fundamental
solutions of non-local parabolic equations with subcritical lower order terms,
due to Xie and Zhang [43]. When $f$ is merely Lipschitz, we offer less precise
asymptotics, see Remark 3.2.

##  2\. Preliminaries

In the sequel, constants in the $C$, $\lesssim$ notation may implicitly depend
on $n\geq 1$, $f\in W^{1,\infty}_{\rm loc}(\mathbb{R};\mathbb{R}^{n})$.

Recall that the Poisson kernel $P$ is given by

$P(x,t)=c_{n}\frac{t}{(|x|^{2}+t^{2})^{\frac{n+1}{2}}},$ (2.1)

where $c_{n}>0$ is chosen to satisfy $\int P(x,t)\,dx=1$ for all $t>0$.

In [3, Definition 2.3], Alibaud introduced the notion of _entropy solution_ to
the critical scalar conservation law (1.4). We summarize only the facts we
need about entropy solutions. See Section 3 of [3]. For each $u_{0}\in
L^{\infty}(\mathbb{R}^{n})$, there exists a unique entropy solution $u$ of
(1.4). This solution exists globally and satisfies the maximum principle

$\lVert u\rVert_{L^{\infty}_{t,x}(\mathbb{R}^{n}\times(0,+\infty))}\leq\lVert
u_{0}\rVert_{L^{\infty}(\mathbb{R}^{n})}.$ (2.2)

The PDE (1.4) is satisfied in the distributional sense. Finally, $u$ belongs
to $C([0,T];L^{1}(K))$ for each $T>0$ and compact $K\subset\mathbb{R}^{n}$.

The following proposition is contained in [3, Theorem 3.2]:

######  Proposition 2.1 (Controlled speed of propagation).

Let $u_{0},\widetilde{u}_{0}\in L^{\infty}(\mathbb{R}^{n})$. Consider
$u,\widetilde{u}$ entropy solutions to (1.4) with initial conditions $u_{0}$
and $\widetilde{u}_{0}$, respectively. Then for all $x_{0}\in\mathbb{R}^{n}$,
all $t>0$ and all $R>0$,

$\int_{B(x_{0},R)}|u(x,t)-\widetilde{u}(x,t)|\,dx\leq\int_{B(x_{0},R+Lt)}P(\cdot,t)\ast|u_{0}-\widetilde{u}_{0}|\,dx$
(2.3)

where $L$ is the Lipschitz constant of $f$ on $[-m,m]$, with
$m=\max(\|u_{0}\|_{L^{\infty}(\mathbb{R}^{n})},\|\widetilde{u}_{0}\|_{L^{\infty}(\mathbb{R}^{n})})$.

We use Proposition 2.1 to establish

######  Proposition 2.2 (Controlled ${\rm BV}$).

If $u_{0}\in{\rm BV}(\mathbb{R}^{n})$, then $u(\cdot,t)\in{\rm
BV}(\mathbb{R}^{n})$ with $\lVert u(\cdot,t)\rVert_{{\rm
TV}(\mathbb{R}^{n})}\leq\lVert u_{0}\rVert_{{\rm TV}(\mathbb{R}^{n})}$ for all
$t>0$. Let $\psi\in C^{\infty}_{0}(\mathbb{R}^{n})$ be non-negative and radial
with $\psi\equiv 1$ in a neighborhood of the origin. Let
$\psi(x,t)=\psi(x-xLt/|x|)$ when $|x|\geq Lt$ and $\psi\equiv 1$ otherwise.
Then, for all $x_{0}\in\mathbb{R}^{n}$, all $t>0$, and $k=1,\ldots,n$,

$\int_{\mathbb{R}^{n}}\psi(x-x_{0})|\omega_{k}(x,t)|\,dx\leq\int_{\mathbb{R}^{n}}\psi(x-x_{0},t)P(\cdot,t)\ast|\omega_{k,0}|\,dx,$
(2.4)

where $\omega_{k}=\partial_{k}u$ and $\omega_{k,0}=\partial_{k}\omega_{0}$ are
finite measures.666This is why we require integration against continuous
$\psi$ on the LHS.

By approximation, if also $\nabla u(\cdot,t)\in L^{1}(\mathbb{R}^{n})$ for a
given $t>0$, then $\psi=\mathbf{1}_{B_{R}}$ with $R>0$ is an admissible weight
function.

###### Proof.

The global ${\rm BV}$ bound is directly from [3, Proposition 3.4]. Let us
justify (2.4) with $x_{0}=0$ when $\nabla u_{0}\in L^{1}(\mathbb{R}^{n})$ is
compactly supported. First, Alibaud’s formula (2.3) holds with integration
against weight $\psi$ as in (2.4). This is shown by integrating (2.3)
according to the principle $\int_{\mathbb{R}^{n}}\psi
F\,dx=\int_{0}^{\infty}\int_{\\{\psi>\lambda\\}}F\,dx\,d\lambda$. Let
$D_{k}^{\varepsilon}$ be the different quotient operator
$D_{k}^{\varepsilon}u=(u(x)-u(x-\varepsilon\vec{e}_{k}))/\varepsilon$. Letting
$\widetilde{u}=u(\cdot-\varepsilon\vec{e}_{k},t)$ in (2.3) with weight $\psi$,
and dividing by $\varepsilon$, we have

$\int_{\mathbb{R}^{n}}\psi(x)|D_{k}^{\varepsilon}u(x,t)|\,dx\leq\int_{\mathbb{R}^{n}}\psi(x,t)P(\cdot,t)\ast|D_{k}^{\varepsilon}u_{0}|\,dx.$
(2.5)

We have $D_{k}^{\varepsilon}u_{0}\to\omega_{k,0}$ strongly in
$L^{1}(\mathbb{R}^{n})$. Then $P(\cdot,t)\ast|D_{k}^{\varepsilon}u_{0}|\to
P(\cdot,t)\ast|\omega_{k,0}|$ in $L^{1}(\mathbb{R}^{n})$ also. This implies
that the LHS remains bounded as $\varepsilon\to 0^{+}$. Hence, $\nabla
u(\cdot,t)$ actually belongs to $L^{1}(\mathbb{R}^{n})$, and the LHS converges
to $\int_{\mathbb{R}^{n}}\psi(x)|\omega_{k}(x,t)|\,dx$ as $\varepsilon\to
0^{+}$. To complete the proof for general $u_{0}\in{\rm BV}(\mathbb{R}^{n})$,
we approximate $u_{0}$ in $L^{\infty}(\mathbb{R}^{n})$ by $u^{(i)}_{0}$,
$i\in\mathbb{N}$, and we approximate $\nabla u_{0}$ weakly-$\ast$ in
$\mathcal{M}(\mathbb{R}^{n})$ by $\nabla u_{0}^{(i)}$ compactly supported,
belonging to $L^{1}(\mathbb{R}^{n})$, and satisfying
$|\omega_{k,0}^{(i)}|\overset{\ast}{\rightharpoonup}|\omega_{k,0}|$ in the
sense of measures. Then one may verify, using the Lebesgue dominated
convergence theorem and kernel estimates, that
$P(\cdot,t)\ast|\omega_{k,0}^{(i)}|\to P(\cdot,t)\ast|\omega_{k,0}|$ strongly
in $L^{1}(\mathbb{R}^{n})$. The LHS is handled by lower semicontinuity. This
completes the proof. ∎

Proposition 2.1 only allows us to propagate $L^{1}$-based quantities, which
then smooth to $L^{q}$-based quantities, $q\geq 1$, after the initial time:

Let $\ell>0$ and $\ell\,\square(k)$ be the open cube with center at $k$ and
side length $\ell$. That is, $\ell\,\square(k)=k+(-\ell/2,\ell/2)^{n}$. We
write $\square(k)=1\square(k)$. Define

$\lVert f\rVert_{\ell^{p}_{k}L^{q}_{x}(\mathbb{R}^{n})}=\left\lVert\left\lVert
f\right\rVert_{L^{q}_{x}(\square(k))}\right\rVert_{\ell^{p}_{k}(\mathbb{Z}^{n})}.$
(2.6)

When $p=+\infty$, the space $\ell^{\infty}_{k}L^{q}_{x}(\mathbb{R}^{n})$ is
known in the literature as $L^{q}_{\mathrm{uloc}}(\mathbb{R}^{n})$. We have
$L^{p}(\mathbb{R}^{n})=\ell^{p}_{k}L^{p}_{x}(\mathbb{R}^{n})$ with equality of
norms. We also have the obvious embeddings

$\lVert f\rVert_{\ell^{p}_{k}L^{q_{1}}_{x}(\mathbb{R}^{n})}\leq\lVert
f\rVert_{\ell^{p}_{k}L^{q_{2}}_{x}(\mathbb{R}^{n})}$ (2.7)

when $q_{1}\leq q_{2}$, and

$\lVert f\rVert_{\ell^{p_{2}}_{k}L^{q}_{x}(\mathbb{R}^{n})}\leq\lVert
f\rVert_{\ell^{p_{1}}_{k}L^{q}_{x}(\mathbb{R}^{n})}$ (2.8)

when $p_{1}\leq p_{2}$. The short-time and small-distance behavior of these
spaces is akin to that of $L^{q}(\mathbb{R}^{n})$, whereas the large-distance
behavior is more closely akin to that of $L^{p}(\mathbb{R}^{n})$.

We will require the following smoothing estimates when $q=1$ or $p=q$.
However, it is no more effort to prove the general estimates:

######  Lemma 2.3 (Smoothing for the heat equation).

Let $p,q_{1},q_{2}\in[1,+\infty]$ with $q_{1}\leq q_{2}$. Let
$w_{0}\in\ell^{p}_{k}L^{q_{1}}_{x}(\mathbb{R}^{n})$. Define

$w(\cdot,t)=P(\cdot,t)\ast w_{0}.$ (2.9)

Then for $t\leq 1$,

$\lVert w(\cdot,t)\rVert_{\ell^{p}_{k}L^{q_{2}}_{x}(\mathbb{R}^{n})}\lesssim
t^{\frac{n}{q_{2}}-\frac{n}{q_{1}}}\lVert
w_{0}\rVert_{\ell^{p}_{k}L^{q_{1}}_{x}(\mathbb{R}^{n})}.$ (2.10)

###### Proof.

Let $k\in\mathbb{Z}^{n}$. We decompose $\mathbb{R}^{n}$ into near-to-$k$ and
far-from-$k$ regions:

$\displaystyle\lVert
w(x,t)\rVert_{L^{q_{2}}_{x}(\square(k))}\leq\underbrace{\left\lVert\int_{y\in
3\square(k)}P(x-y,t)|w_{0}|(y)\,dy\right\rVert_{L^{q_{2}}_{x}(\square(k))}}_{=I_{1}(k)}$
(2.11)
$\displaystyle\quad+\underbrace{\left\lVert\int_{y\in\mathbb{R}^{n}\setminus
3\square(k)}P(x-y,t)|w_{0}|(y)\,dy\right\rVert_{L^{q_{2}}_{x}(\square(k))}}_{=I_{2}(k)}.$

Eventually, we will sum in $\ell^{p}_{k}(\mathbb{Z}^{n})$. First, we estimate
$I_{1}(k)$:

$I_{1}(k)\leq\left\lVert\sum_{|j|_{\infty}\leq
1}P(\cdot,t)\ast(\mathbf{1}_{\square(k+j)}|w_{0}|)\right\rVert_{L^{q_{2}}_{x}(\mathbb{R}^{n})}\lesssim
t^{\frac{n}{q_{2}}-\frac{n}{q_{1}}}\sum_{|j|_{\infty}\leq 1}\lVert
w_{0}\rVert_{L^{q_{1}}_{x}(\square(k+j))},$ (2.12)

by Young’s convolution inequality. We now sum in
$\ell^{p}_{k}(\mathbb{Z}^{n})$. By the triangle inequality, and since there
are only a finite number of boxes (specifically, $3^{n}$) in the $j$ sum, we
have

$\lVert I_{1}(k)\rVert_{\ell^{p}_{k}(\mathbb{Z}^{n})}\lesssim
t^{\frac{n}{q_{2}}-\frac{n}{q_{1}}}\lVert
w_{0}\rVert_{\ell^{p}_{k}L^{q_{1}}_{x}(\mathbb{R}^{n})}.$ (2.13)

Now we estimate $I_{2}(k)$:

$\displaystyle I_{2}(k)$
$\displaystyle\leq\sum_{|j|_{\infty}>1}\int_{\square(k-j)}|w_{0}(y)|\,\lVert
P(x-y,t)\rVert_{L^{q_{2}}_{x}(\square(k))}\,dy$ (2.14)
$\displaystyle\leq\sum_{|j|_{\infty}>1}\left[\sup_{x\in\square(k)}\sup_{y\in\square(k-j)}P(x-y,t)\right]\int_{\square(k-j)}|w_{0}(y)|\,dy,$

where we used Hölder’s inequality in $x$ and $|\square(k)|=1$. When
$x\in\square(k)$ and $y\in\square(k-j)$, we have $|x-y|\geq|j|-1$. Recall that
$P(z,t)\lesssim t/|z|^{n+1}\lesssim 1/|z|^{n+1}$ for $t\leq 1$. Hence,

$\sup_{x\in\square(k)}\sup_{y\in\square(k-j)}P(x-y,t)\lesssim\frac{1}{(|j|-1)^{n+1}},$
(2.15)

and

$I_{2}(k)\lesssim\sum_{|j|_{\infty}>1}\frac{1}{(|j|-1)^{n+1}}\int_{\square(k-j)}|w_{0}(y)|\,dy.$
(2.16)

One may recognize (2.16) as a discrete convolution with a summable-in-$j$
kernel. Applying $\lVert\cdot\rVert_{\ell^{p}_{k}(\mathbb{Z}^{n})}$, we have

$\lVert
I_{2}(k)\rVert_{\ell^{p}_{k}(\mathbb{Z}^{n})}\lesssim\sum_{|j|_{\infty}>1}\frac{1}{(|j|-1)^{n+1}}\lVert
w_{0}\rVert_{\ell^{p}_{k}L^{1}_{x}(\mathbb{R}^{n})}\lesssim\lVert
w_{0}\rVert_{\ell^{p}_{k}L^{q_{1}}_{x}(\mathbb{R}^{n})},$ (2.17)

where we use the trivial embedding (2.7). This completes the proof. ∎

We now justify that the entropy solutions immediately become Hölder continuous
and better:

######  Proposition 2.4 (Regularity).

Let $u$ be the unique entropy solution of (1.4) with initial data satisfying
$\lVert u_{0}\rVert_{L^{\infty}(\mathbb{R}^{n})}\leq m$. Suppose also that
$f\in C^{\infty}_{\rm loc}(\mathbb{R};\mathbb{R}^{n})$. There exists
$\alpha=\alpha(m,n)\in(0,1)$ such that $u\in L^{\infty}_{t,{\rm
loc}}C^{2,\alpha}_{x}(\mathbb{R}^{n}\times(0,+\infty))$ and

$\operatorname*{ess\,sup}_{t>0}\,t\lVert\nabla
u(\cdot,t)\rVert_{L^{\infty}(\mathbb{R}^{n})}+t^{2}\lVert\nabla^{2}u(\cdot,t)\rVert_{L^{\infty}(\mathbb{R}^{n})}+t^{2+\alpha}[\nabla^{2}u(\cdot,t)]_{C^{\alpha}(\mathbb{R}^{n})}\lesssim_{m}1.$
(2.18)

###### Proof.

The estimate

$t^{\alpha}[u(\cdot,t)]_{C^{\alpha}(\mathbb{R}^{n})}\lesssim_{m}1$ (2.19)

follows from a direct application of the $L^{\infty}_{x}\to C^{\alpha}_{x}$
smoothing estimates developed by Silvestre in [39, Theorem 1.1] and [38] for
bounded ‘solutions’ of non-local drift-diffusion equations

$\partial_{t}u+b\cdot\nabla u+\Lambda u=g,$ (2.20)

where $b,g$ are bounded. Notably, _$b$ may be large and not necessarily
divergence free._ In our situation, $b(x,t)=f^{\prime}(u(x,t))$ and $g=0$. The
notion of ‘solution’ is in quotation marks because, here, $b$ is allowed to be
discontinuous, so the notion of viscosity solution may not be directly
applicable.777This is discussed in Section 5 of Silvestre’s paper [39], see
also Section 3 of [40]. Silvestre mentions that, if viscosity solutions are
unavailable, then one may justify the estimates at the level of the vanishing
viscosity approximation
$\partial_{t}u^{\varepsilon}+b^{\varepsilon}\cdot\nabla
u^{\varepsilon}+\Lambda u^{\varepsilon}=\varepsilon\Delta u^{\varepsilon}$
(2.21) with $\varepsilon\to 0^{+}$. In principle, this is possible. However,
in our setting, the construction in [3] was by an operator splitting method,
rather than regularization by $\varepsilon\Delta u^{\varepsilon}$, so we argue
differently. To employ Silvestre’s estimates rigorously, one may mollify the
initial data, argue at the level of classical solutions, and pass to the
limit.

To bootstrap $C^{\alpha}_{x}\to C^{1,\alpha}_{x}$, we apply linear estimates
due to Silvestre in [40, Theorem 1.1] for the drift-diffusion equation (2.20).
It is also possible to proceed more directly, as in Appendix B of [11] or in
[18, 17]. Since $b=f^{\prime}(u)$ is $\alpha$-Hölder continuous in
$\mathbb{R}^{n}\times(1/2,1)$ with bounds depending only on $m$, Theorem 1.1
in [40] gives

$\lVert
u\rVert_{L^{\infty}_{t}C^{1,\alpha}_{x}(\mathbb{R}^{n}\times(1/2,1))}\lesssim_{m}1.$
(2.22)

Hence, $b=f^{\prime}(u)$ satisfies the same bounds. Next, we apply
$\partial_{k}$, $1\leq k\leq n$, to the PDE. This gives

$\partial_{t}\partial_{k}u+\Lambda\partial_{k}u+b\cdot\nabla\partial_{k}u=-\partial_{k}b\cdot\nabla
u.$ (2.23)

We regard $g=-\partial_{k}b\cdot\nabla u$ as a forcing term belonging to
$L^{\infty}_{t}C^{\alpha}_{x}(\mathbb{R}^{n}\times(1/2,1))$ Finally, Theorem
1.1 in [40] gives

$\lVert\partial_{k}u\rVert_{L^{\infty}_{t}C^{1,\alpha}_{x}(\mathbb{R}^{n}\times(3/4,1))}\lesssim_{m}1.$
(2.24)

Scaling invariance gives the sharp dependence on $t$. One could also proceed
to higher derivatives. ∎

Consider the linear PDE

$\partial_{t}u+\Lambda u+b\cdot\nabla u+cu=0$ (2.25)

where $b\in L^{\infty}_{t}C^{1,\alpha}_{x}(Q_{1})$ and $c\in
L^{\infty}_{t}C^{\alpha}_{x}(Q_{1})$ with $\lVert
b\rVert_{L^{\infty}_{t}C^{1,\alpha}_{x}(Q_{1})}+\lVert
c\rVert_{L^{\infty}_{t}C^{\alpha}_{x}(Q_{1})}\leq M$. Here,
$Q_{T}=\mathbb{R}^{n}\times(0,T)$.

######  Proposition 2.5 (Fundamental solution estimates).

There exists a continuous function $\Gamma=\Gamma(x,t;y,s)$,
$x,y\in\mathbb{R}^{n}$ and $0\leq s<t\leq 1$, satisfying the following
properties:

  * •

(Pointwise upper and lower bounds) For all $0\leq S\leq s<t\leq T\leq 1$,

$C_{0}^{-1}P(x,t;y,s)\leq\Gamma(x,t;y,s)\leq C_{0}P(x,t;y,s),$ (2.26)

where $C_{0}=C_{0}(T-S,M)>0$ and $P$ is the Poisson kernel.

  * •

(Maximum principle) If $c\equiv 0$, then
$\int_{\mathbb{R}^{n}}P(x,t;y,s)\,dy=1$.

  * •

(Representation formula) If $w\in L^{\infty}_{t,x}(Q_{1})$, with $w\in
L^{\infty}_{t,{\rm loc}}C^{1,\beta}_{x}(Q_{1})$ for some $\beta\in(0,1)$, is a
solution of (2.25) on $Q_{1}$ and
$w(\cdot,t)\overset{\ast}{\rightharpoonup}w_{0}$ in
$L^{\infty}(\mathbb{R}^{n})$ as $t\to 0^{+}$, then

$w(x,t)=\int_{\mathbb{R}^{n}}\Gamma(x,t;y,0)w_{0}(y)\,dy.$ (2.27)

Solutions given by the representation formula belong to the above class.

Proposition 2.5 was obtained in the paper [43] of Xie and Zhang by E. E.
Levi’s parametrix method, _except for uniqueness_ , which we sketch below. In
[43], the authors work with more general assumptions: $b$ in the subcritical
space $L^{\infty}_{t}C^{\alpha}_{x}(Q_{1})$ and $c$ in a critical Kato space.

###### Proof of uniqueness.

Let $u_{0}\in L^{2}(\mathbb{R}^{n})$. Define

$v(x,t)=\int_{\mathbb{R}^{n}}\Gamma(x,t;y,0)u_{0}(y)\,dy.$ (2.28)

Let $L=\Lambda+b\cdot\nabla+c$ and
$L^{*}=\Lambda-b\cdot\nabla+(c-\operatorname{div}b)$. Under our additional
regularity assumptions, it is possible to show that $v$ is a weak
solution888Due to the quite general conditions in [43], the authors avoided
classical solutions and space-time distributional solutions. Instead, they
connect the fundamental solution to the PDE via the ‘generator’ notion. of the
PDE in the sense that

$\int\kern-3.39996pt\int v(x,t)(-\partial_{t}+L^{*})\varphi\,dx\,dt=0$ (2.29)

for all $\varphi\in C^{\infty}_{0}(\mathbb{R}^{n}\times(0,1))$. Additionally,
we have $v\in L^{\infty}_{t}L^{2}_{x}(Q_{1})$ and $v\in L^{2}_{t,{\rm
loc}}H^{1/2}_{x}(\mathbb{R}^{n}\times(0,1])$, among many other spaces, and
$v(\cdot,t)\to u_{0}$ in $L^{2}(\mathbb{R}^{n})$ as $t\to 0^{+}$. This follows
from the pointwise upper bounds of the fundamental solution and its first
derivatives (see [43, Theorem 1.1 (v)]) and the convergence result in [43,
Theorem 1.1 (ii)]. One may show, via energy estimates, that the above solution
is unique in its class and, additionally, belongs to
$C([0,1];L^{2}(\mathbb{R}^{n}))\cap L^{2}_{t}H^{1/2}_{x}(Q_{1})$.999It is
important for the energy estimates that $b\in
L^{\infty}_{t}C^{1/2}_{x}(Q_{1})$.

Assume now that $u_{0}\in L^{1}\cap L^{\infty}(\mathbb{R}^{n})$. Then the
above solution $v$ also belongs to $L^{\infty}_{t}L^{1}_{x}\cap
L^{\infty}_{t,x}(Q_{1})$. By uniqueness within the energy class, the solution
$v$ may be obtained by vanishing viscosity:

$\partial_{t}u^{\varepsilon}+\Lambda u^{\varepsilon}+b\cdot\nabla
u^{\varepsilon}+cu^{\varepsilon}=\varepsilon\Delta u^{\varepsilon}.$ (2.30)

According to Silvestre’s estimates, we have that $v\in L^{\infty}_{t,{\rm
loc}}C^{1,\alpha}_{x}(\mathbb{R}^{n}\times(0,1])$ for some $\alpha\in(0,1)$
with estimates depending only on $\lVert
u_{0}\rVert_{L^{\infty}(\mathbb{R}^{n})}$ and the coefficients. By
approximation, we have that when $u_{0}\in L^{\infty}(\mathbb{R}^{n})$, $v$
satisfies the same _a priori_ estimates.

We now demonstrate the following uniqueness theorem by duality: _If $u\in
L^{\infty}_{t,{\rm loc}}C^{1,\alpha}_{x}(\mathbb{R}^{n}\times(0,1])$ is a
solution of the linear PDE (2.25) with
$u(\cdot,t)\overset{\ast}{\rightharpoonup}0$ in $L^{\infty}(\mathbb{R}^{n})$
as $t\to 0^{+}$, then $u\equiv 0$._101010This argument is modeled off a
similar argument in a forthcoming work by the first author and Zachary
Bradshaw.

Let $T\in(0,1)$ and $\psi_{0}\in L^{1}\cap L^{\infty}(\mathbb{R}^{n})$. The
above analysis demonstrated that there exists $\psi\in
L^{\infty}_{t}L^{1}_{x}\cap L^{\infty}_{t,x}(Q_{T})$ with $\psi\in
L^{\infty}_{t,{\rm loc}}C^{1,\alpha}_{x}(\mathbb{R}^{n}\times[0,T))$ and
satisfying the adjoint problem

$-\partial_{t}\psi+L^{*}\psi=0$ (2.31)

with $\psi(T)=\psi_{0}$. Let $0<t_{0}<t_{1}<T$ and $R,\varepsilon>0$. Let
$\chi\in C^{\infty}_{0}(B_{2})$ with $\chi\equiv 1$ on $B_{1}$ and
$\chi_{R}=\chi(x/R)$. Let $\varphi^{t_{0},t_{1}}_{\varepsilon}$ be a
mollification of the indicator function $\mathbf{1}_{(t_{0},t_{1})}$ at scale
$\varepsilon\ll 1$. We test (2.25) against
$\psi\chi_{R}\varphi^{t_{0},t_{1}}_{\varepsilon}$ and omit
$t_{0},t_{1},R,\varepsilon$ from the notation as convenient:

$\displaystyle\int\kern-3.39996pt\int\underbrace{\partial_{t}u+Lu}_{=0}\psi\chi\varphi\,dx\,dt=\int\kern-3.39996pt\int\underbrace{-\partial_{t}\psi+L^{*}\psi}_{=0}u\varphi\,dx\,dt$
(2.32) $\displaystyle\quad+\int\kern-3.39996pt\int(-\partial_{t}\varphi)\chi
u\psi+\varphi(-b\cdot\nabla\chi)u\psi+\varphi u[\Lambda,\chi]\psi\,dx\,dt.$

Upon sending $\varepsilon\to 0^{+}$, we have

$\int\chi_{R}u(x,t_{1})\psi(x,t_{1})\,dx-\int\chi_{R}u(x,t_{0})\psi(x,t_{0})\,dx=\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{n}}b\cdot\nabla\chi_{R}u\psi+u[\Lambda,\chi_{R}]\psi\,dx\,dt$
(2.33)

for a.e. $t_{0},t_{1}\in(0,1)$. Moreover, (2.33) is valid for all
$t_{0},t_{1}\in[0,T]$, since $u\colon[0,1]\to L^{\infty}(\mathbb{R}^{n})$ is
weak-$\ast$ continuous and $\psi\in C([0,T];L^{2}(\mathbb{R}^{n}))$. We focus
on $t_{0}=0$ and $t_{1}=T$. Upon sending $R\to+\infty$, we have

$\left|\int\kern-3.39996pt\int u[\Lambda,\chi_{R}]\psi\,dx\,dt\right|\lesssim
R^{-1}R^{n/p}\lVert
u\rVert_{L^{\infty}_{t,x}(Q_{1})}\lVert\psi\rVert_{L^{\infty}_{t}L^{p^{\prime}}_{x}(Q_{1})}\to
0$ (2.34)

when $p>n$, since $\lVert[\Lambda,\chi_{R}]\rVert_{L^{p}(\mathbb{R}^{n})\to
L^{p}(\mathbb{R}^{n})}\lesssim_{p}R^{-1}$ for all $p\in(1,+\infty)$. The term
containing $b\cdot\nabla\chi_{R}$ is $O(R^{-1})$, since $b,u\in
L^{\infty}_{t,x}(Q_{1})$ and $\psi\in L^{\infty}_{t}L^{1}_{x}(Q_{T})$. Hence,
(2.33) becomes

$\int u(x,T)\psi_{0}\,dx=0,$ (2.35)

for all $T\in(0,1)$ and $\psi_{0}\in L^{1}\cap L^{\infty}(\mathbb{R}^{n})$.
Therefore, $u\equiv 0$ on $Q_{1}$. ∎

##  3\. Proof of main results

### Proof of Theorem 1.1

Let $u_{0}\in L^{\infty}$ and $u$ be the corresponding entropy solution. For
each $v_{0}$, we consider the solution $\widetilde{u}=u+v$ with initial data
$\widetilde{u_{0}}=u_{0}+v_{0}\in L^{\infty}$.

Let $m>0$ and $\lVert
u_{0}\rVert_{L^{\infty}},\lVert\widetilde{u_{0}}\rVert_{L^{\infty}}\leq m$.

We prove continuity with respect to $v_{0}$.

######  Proposition 3.1 (Continuity estimate).

Let $1\leq p\leq q\leq+\infty$. If $v_{0}\in L^{p}(\mathbb{R}^{n})$, we have

$\lVert
v(\cdot,t)\rVert_{L^{q}(\mathbb{R}^{n})}\lesssim_{m,p,q}t^{\frac{n}{q}-\frac{n}{p}}\lVert
v_{0}\rVert_{L^{p}(\mathbb{R}^{n})}.$ (3.1)

###### Proof of Proposition 3.1.

By scaling invariance, it is enough to demonstrate (3.1) with $t=1$.

Step 1. Propagation of localization. First, we demonstrate that, for all
$t\in(0,1/2]$, we have

$\lVert
v(\cdot,t)\rVert_{\ell^{p}_{k}L^{1}_{x}(\mathbb{R}^{n})}\lesssim_{m,p}\lVert
v_{0}\rVert_{\ell^{p}_{k}L^{1}_{x}(\mathbb{R}^{n})}.$ (3.2)

Using Proposition 2.1 (Controlled speed of propagation), we have

$\displaystyle\int_{\square(k)}|v(t,x)|dx$
$\displaystyle\leq\int_{B(k,\sqrt{2n}/2)}|v(x,t)|dx$ (3.3)
$\displaystyle\leq\int_{B(k,\sqrt{2n}/2+Lt)}P\ast|v_{0}|(x)\,dx$
$\displaystyle=\sum_{|j|\leq R}\int_{\square(k+j)}P\ast|v_{0}|(x)\,dx$

where $j\in\mathbb{Z}^{n}$ and $R=R(n,L)>0$. We apply
$\lVert\cdot\rVert_{\ell^{p}_{k}(\mathbb{Z}^{n})}$ to each side of (3.3). By
the triangle inequality, and since the sum in $j$ has only finitely many
boxes, we have

$\lVert
v(\cdot,t)\rVert_{\ell^{p}_{k}L^{1}_{x}(\mathbb{R}^{n})}\lesssim_{R}\lVert
P\ast|v_{0}|\rVert_{\ell^{p}_{k}L^{1}_{x}(\mathbb{R}^{n})}.$ (3.4)

Now Lemma 2.3 (Smoothing for the heat equation) with $q_{1}=q_{2}=1$ gives
(3.2).

Step 2. Smoothing. Second, we demonstrate that, for all $t\in(3/4,1]$, we have

$\lVert v(\cdot,t)\rVert_{L^{q}(\mathbb{R}^{n})}\lesssim_{m,p,q}\lVert
v(\cdot,1/2)\rVert_{\ell^{p}_{k}L^{1}_{x}(\mathbb{R}^{n})}.$ (3.5)

By Proposition 2.4 (Regularity), $u$ and $\widetilde{u}$ belong to
$L^{\infty}_{t}C^{2,\alpha}_{x}(\mathbb{R}^{n}\times(1/2,1))$ with bounds
depending only on $m$. Hence, $v=\widetilde{u}-u$ belongs to the same space.
Let $w(\cdot,t)=v(\cdot,t+1/2)$ when $t\in(0,1/2]$. Let $w_{0}=v(\cdot,1/2)$.
Then

$\partial_{t}w+\operatorname{div}(g(x,t)w)+\Lambda w=0,$ (3.6)

where

$g(x,t)=\frac{f(\widetilde{u})-f(u)}{\widetilde{u}-u}$ (3.7)

and

$\lVert
g\rVert_{L^{\infty}_{t}C^{1,\alpha}_{x}(\mathbb{R}^{n}\times(0,1/2))}\lesssim_{m}1.$
(3.8)

Therefore, we may use the representation formula from Proposition 2.5
(Fundamental solution estimates):

$w(x,t)=\int_{\mathbb{R}^{n}}\Gamma(x,t;y,0)w_{0}(y)\,dy.$ (3.9)

In particular, the pointwise upper bound in Proposition 2.5 gives

$|w(x,t)|\lesssim_{m}\int_{\mathbb{R}^{n}}P(x-y,t)|w_{0}|(y)\,dy.$ (3.10)

Then Lemma 2.3 (Smoothing for the heat equation) with $q_{1}=1$ and $q_{2}=q$,
along with the embedding $\ell^{p}_{k}L^{q}_{x}(\mathbb{R}^{n})\hookrightarrow
L^{q}(\mathbb{R}^{n})$, gives (3.5).

Finally, we combine (3.2) and (3.5) to complete the proof. ∎

When $v_{0}\in L^{1}(\mathbb{R}^{n})$, the propagation of localization comes
‘for free’ from the $L^{1}$-contraction property, which was shown in [3].

###### Proof of Theorem 1.1.

Our goal is to demonstrate the $o_{t\to+\infty}(1)$ improvement over the
conclusion of Proposition 3.1 (Continuity estimate) when $p>1$. We approximate
$v_{0}$ strongly in $L^{p}(\mathbb{R}^{n})$ by $v_{0}^{(k)}$ belonging to
$L^{1}\cap L^{\infty}(\mathbb{R}^{n})$ and satisfying the decay condition
(1.5), $|v_{0}^{(k)}|\leq|v_{0}|$, and $\lVert
v_{0}^{(k)}\rVert_{L^{\infty}(\mathbb{R}^{n})}\leq 2m$. Let $v^{(k)}$ be the
solution corresponding to the initial data $v_{0}^{(k)}$. The
$o_{t\to+\infty}(1)$ improvement is obvious for $v^{(k)}$, which satisfies a
faster decay rate because its initial data belongs to $L^{1}(\mathbb{R}^{n})$.
Next, the triangle inequality and Proposition 3.1 yield

$\displaystyle\lVert v(\cdot,t)\rVert_{L^{q}(\mathbb{R}^{n})}$
$\displaystyle\leq\lVert v^{(k)}(\cdot,t)\rVert_{L^{q}(\mathbb{R}^{n})}+\lVert
v(\cdot,t)-v^{(k)}(\cdot,t)\rVert_{L^{q}(\mathbb{R}^{n})}$ (3.11)
$\displaystyle\lesssim_{m}t^{\frac{n}{q}-\frac{n}{p}}o_{t\to+\infty}(1)\underbrace{\lVert
v_{0}^{(k)}\rVert_{L^{p}(\mathbb{R}^{n})}}_{\leq\lVert
v_{0}\rVert_{L^{p}(\mathbb{R}^{n})}}+t^{\frac{n}{q}-\frac{n}{p}}\underbrace{\lVert
v_{0}-v^{(k)}_{0}\rVert_{L^{p}(\mathbb{R}^{n})}}_{\to 0\text{ as
}k\to+\infty}.$

This completes the proof. ∎

###  ${\rm BV}$ convergence and spatial asymptotics

###### Proof of Theorem 1.2.

Let $t_{k}\to+\infty$ with $t_{k}\geq 1$. It will be convenient to work with
the rescaled solutions

$u^{(k)}(y,s)=u(t_{k}y,t_{k}s),$ (3.12)

with $\omega^{(k)}=\partial_{y}u^{(k)}$. Then

$\lVert\omega^{(k)}(\cdot,1)-\omega^{\SS}(\cdot,1)\rVert_{L^{1}(\mathbb{R})}=\lVert\omega(\cdot,t_{k})-\omega^{\SS}(\cdot,t_{k})\rVert_{L^{1}(\mathbb{R})}.$
(3.13)

By Proposition 2.4 (Regularity) we can bootstrap the decay of
$\|u^{(k)}(\cdot,1)-u^{\SS}(\cdot,1)\|_{L^{\infty}(\mathbb{R})}$ given by
Theorem 1.1 to get

$\lVert\omega^{(k)}(\cdot,1)-\omega^{\SS}(\cdot,1)\rVert_{L^{\infty}(B(R))}\to
0\text{ as }k\to+\infty$ (3.14)

for all $R\geq 1$. Therefore, it suffices to show that there is no mass of
$\omega^{(k)}$ escaping to infinity. Let $R\geq L+10$. By Alibaud’s ${\rm BV}$
formula (2.4), and covering $\mathbb{R}\setminus B(R)$ by an appropriate
sequence of balls $B(x_{0},1)$, we have

$\int_{\mathbb{R}\setminus
B(R)}|\omega^{(k)}(x,1)|\,dx\lesssim_{m}\int_{\mathbb{R}\setminus
B(R-L)}P(\cdot,1)\ast|\omega_{0}^{(k)}|\,dx.$ (3.15)

It is not difficult to show that the quantity on the RHS is
$o_{R\to+\infty}(1)$ uniformly in $k$. ∎

###### Proof of (1.8).

First, we remark that $u^{\SS}$ is monotone because the evolution of $\omega$
preserves its sign. This is true at the level of entropy solutions, as can be
seen from their construction by the splitting argument in Alibaud’s paper.

In the following, we allow the constant $C$ to depend on $u_{0}=u_{0}^{\SS}$
and $\widetilde{u}_{0}$. Let $a,b\in\mathbb{R}$ represent the limits of
$u_{0}$ as $x\to\mp\infty$.

Step 1. Asymptotics for smooth approximation $\widetilde{u}$. Let
$\widetilde{u}_{0}\in C^{\infty}(\mathbb{R})$ with $\widetilde{u}_{0}\equiv
u_{0}$ outside of $B_{1}$. Let $\widetilde{u}$ be the corresponding entropy
solution, which may be shown to belong to
$L^{\infty}_{t}C^{2,\alpha}_{x}(\mathbb{R}^{n}\times(0,1))$ by combining
local-in-time well-posedness111111See the expository blog post [42] of Tao on
quasilinear well-posedness. with the estimates in Proposition 2.4
(Regularity).

By Proposition 2.5 (Fundamental solution estimates), we have

$\widetilde{u}(x,t)-u_{0}(x)=\int_{\mathbb{R}^{n}}P(x,t;y,0)[\widetilde{u}_{0}(y)-u_{0}(x)]\,dy,$
(3.16)

since $\int P(x,t;y,0)\,dy=1$ when $c\equiv 0$. Let $x\leq-1$. Hence,

$\widetilde{u}(x,t)-u_{0}(x)=\underbrace{\int_{B_{1}}P(x,t;y,0)[\widetilde{u}_{0}(y)-u_{0}(x)]\,dy}_{=I_{1}(x)}+\underbrace{(a-b)\int_{y\geq
1}P(x,t;y,0)\,dy}_{=I_{2}(x)}.$ (3.17)

Since $[\widetilde{u}_{0}-u_{0}(x)]\mathbf{1}_{B_{1}}$ is compactly supported,
we have that $|I_{1}(x)|\lesssim\langle x\rangle^{-2}$. On the other hand,
when $a\neq b$, we have

$C^{-1}\langle x\rangle^{-1}\leq I_{2}(x)/(a-b)\leq C\langle x\rangle^{-1}.$
(3.18)

A similar argument holds for $x\geq 1$. The $I_{2}$ term will dominate when
$|x|\geq C$.

Step 2. Faster decay for the difference $v$. Let $v=u^{\SS}-\widetilde{u}$. We
will exploit that $v_{0}=v(\cdot,1)$ is supported in $B_{1}$ to demonstrate

$|v(x,1)|\leq C\langle x\rangle^{-2}.$ (3.19)

This will complete the proof, since the $I_{2}$ term will dominate $|v|$ when
$|x|\geq C$. We follow the scheme of propagation of localization and smoothing
as in the proof of Proposition 3.1. Let $k\in\mathbb{Z}$ with $|k|\geq 10$. By
Alibaud’s formula and the decay of the Poisson kernel, we have

$\int_{\square(k)}|v(\cdot,1/2)|\,dx\leq
C\int_{B(k,\sqrt{2}/2+L)}P(\cdot,1/2)\ast|v_{0}|\,dx\leq C\langle
k\rangle^{-2}.$ (3.20)

The difference $v$ also satisfies this estimate when $|k|<10$. Next, we
consider $w(\cdot,t)=v(\cdot,t+1/2)$ and analyze its representation formula
when $t\in(1/4,1/2]$:

$\displaystyle|w(x,t)|\leq\int_{\mathbb{R}}\Gamma(x,t;y,0)|w_{0}|\,dy$
$\displaystyle\leq C\sum_{k\in\mathbb{Z}}\langle
k\rangle^{-2}\lVert\Gamma(x,t;\cdot,0)\rVert_{L^{\infty}_{y}(\square(k))}$
(3.21) $\displaystyle\leq C\sum_{k\in\mathbb{Z}}\langle k\rangle^{-2}\langle
x-k\rangle^{-2}$ $\displaystyle\leq C\langle x\rangle^{-2}.$

The proof is complete. ∎

######  Remark 3.2 (Rough $f$).

Suppose that $f$ is locally Lipschitz and $n\geq 1$. It is possible to show
that, for each $R>0$, we have

$\lVert u-u^{\SS}\rVert_{L^{\infty}(B(Rt))}\to 0\text{ as }t\to+\infty.$
(3.22)

That is, $u$ converges to $u^{\SS}$ _locally uniformly_ in self-similar
coordinates $y=x/t$, $s=\log(t/t_{0})$ where $t_{0}>0$ is a reference time.
Indeed, consider any sequence of rescaled solutions $u^{(k)}$ as above. Since
$u_{0}^{(k)}\to u_{0}^{\SS}$ in $L^{1}_{\mathrm{uloc}}(\mathbb{R}^{n})$,
Alibaud’s formula (2.3) gives that $u^{(k)}(\cdot,1)$ converges in
$L^{1}_{\mathrm{uloc}}(\mathbb{R}^{n})$ to $u^{\SS}(\cdot,1)$. By the _a
priori_ Hölder estimates (2.19) and the Ascoli-Arzelá theorem,121212To justify
(2.19) with Lipschitz $f$, one could mollify $f$ or apply a parabolic
regularization $\varepsilon\Delta u^{\varepsilon}$, justify the estimates at
the regularized level, and pass to the limit. $u^{(k)}(\cdot,1)$ converges in
$L^{\infty}_{\rm loc}(\mathbb{R}^{n})$, and its limit must be
$u^{\SS}(\cdot,1)$.

If $n=1$ and $u_{0}\in{\rm BV}(\mathbb{R})$, then we may choose $R=+\infty$ in
(3.22), since the ${\rm BV}(\mathbb{R})$ norm manages the behavior in
$L^{\infty}(\mathbb{R}\setminus B_{R})$ for $R\gg 1$ according to Alibaud’s
${\rm BV}$ formula (2.4). If $f\in C^{1,\alpha}_{\rm loc}(\mathbb{R})$, then
it is possible to upgrade to ${\rm BV}(\mathbb{R})$ convergence, since
Silvestre’s estimates in [40] allow the solution to be bootstrapped from
$C^{\alpha}_{x}(\mathbb{R})$ to $C^{1,\alpha}_{x}(\mathbb{R})$.

####  3.0.1. Acknowledgments

DA thanks Vladimír Šverák for mentioning this problem and helpful discussions.
We also thank Montie Avery and Vlad Vicol for comments on a preliminary
version. DA was supported by the NDSEG Fellowship and the NSF Postdoctoral
Fellowship Grant No. 2002023. RB was supported by the NSF Graduate Fellowship
Grant No. 1839302.

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