text:
Demystifying a legend
Local historian rattles a skeleton in John Bidwell's closet—the Indian Treaty of 1851
By Tyler Ash
Chico historian Michele Shover and her husband, Don Lillibridge, display a photo taken during the Indian Treaty of 1851.
Photo By Tyler Ash
More to come: Michele Shover is compiling much of the information she shared about the Indian Treaty of 1851 into a book to be titled The California Indian Wars on the Butte County Front, 1850-1865.
In the summer of 1851, around 300 Maidu Indians gathered in the front yard of John Bidwell's log cabin, near the spot in what is now Chico where his mansion sits today. They were there to see what would happen to their way of life now that white settlers were encroaching on their territories.
The Maidu hoped to determine whether they should rise up against the foreigners or make a pact with them, based on what the white strangers had promised at a treaty meeting Bidwell held on July 21, a meeting he would later regret.
In fact, Chico's founder went so far as to omit every account of the meeting in his diaries and memoirs, which is why it has taken local historian Michele Shover more than 20 years to dig up Bidwell's buried history.
Last Friday, Chico State's Department of Political Science hosted Shover, a professor emerita in the department, at a forum to disclose her years of research on Bidwell and his participation in the Indian Treaty of 1851.
Shover, one of Chico's foremost historians, has produced numerous publications exploring Butte County's colorful history, including studies of its small African-American population, the women of Chico's past, as well as the anti-Chinese campaign of 1877.
As she began her speech, the audience of around 50 fell silent, waiting for Chico's turbulent history to unfold before them.
Bidwell had no choice about the treaty, she said, but he played the cards he was dealt. He furnished the treaty meeting with translators to allow it to flow as smoothly as possible, even though it would lead him to one of the lowest points in his life in Chico.
The treaty's purpose was to legitimize the government seizure of California Indian lands after the conquest of Mexico, Shover explained.
The Native Americans had no concept of money in the settlers' sense, and there were worries about corruption on the part of the white bureaucrats, so the treaty included a system of Indian reservations set up throughout the state, where recompense could come in the form of provisions and education, she said. This was a prevalent method of handling Indian affairs at that time throughout America.
"With reservations they believed that they could protect the helpless Indians that were trying to adjust," Shover said. "They could contain the dangerous Indians who were determined to get revenge for their losses, they could open more land to settlement, and hoped to acculturate Indians on the reservations."
The treaty stated that the mountain and valley Maidu would live together on a large reservation that comprised much of the foothill land east of present-day Chico. There superintendents would guarantee their safety and education.
This posed a problem in itself because there was a deep rivalry between the mountain and valley Maidu. The mountain Indians were lean, quick, aggressive and suited to their rough environment. The valley Indians, however, lived on fertile ground with plenty of food. Shover described them as "the aristocracy of the tribe."
The two had fought each other for ages, and neither trespassed on the other's grounds without strict negotiations.
"You can imagine that there was a certain amount of tension, a great deal of curiosity between the two groups," she said. "They were hostile and tense, but they were also related by intermarriage."
For Bidwell, the treaty offered opportunities to have more Indian labor available, but it ended up leading the valley Indians who were already working for him, the Mechoopda Maidu, away toward the potential reservation. Ranching wasn't fun for them. They were used to living a more leisurely hunting-and-gathering lifestyle, and they wanted a "Bidwell-free environment," Shover said.
When Bidwell heard of his workers' decision to leave, he was shocked and embarrassed, but resilient. To this day, Mechoopda oral history tells that Bidwell promised his workers everything stated in the treaty, all of the provisions and downtime they would receive on the reservation, if they remained working for him. So they stayed.
The treaty was signed, but as time passed Bidwell didn't follow through with the promised changes to the circumstances his Mechoopda workers were living under. Bidwell did intend to fulfill his promises to them, Shover said, but not anytime soon. He had bigger things to worry about, like his failing Indian trading posts that he set up through the treaty. So the Mechoopda turned on him and left.
Bidwell was losing money and workers because of the treaty, so he secretly began writing senators and political officials back east, convincing them that the treaty was a lost cause. He did everything in his power to ensure the defeat of the treaty in the U.S. Senate, which is what eventually happened.
According to legendsofamerica.com, Indians throughout California were robbed of their reservation land by California politicians secretly telling Congress to reject the 18 Indian treaties signed in 1851, leaving California natives stranded in an increasingly aggressive white society.
With a combination of the short window before the treaties and the white settlers' near genocide of Indians, by 1870 California had lost close to 120,000 of its Native Americans.
The treaties, which would have repaid the Indians to a certain extent for their loss of land and the hardships they endured during the American conquest, were nullified because state politicians, including Bidwell, weren't profiting from them.
Shover's years of research have apparently uncovered a John Bidwell that he didn't want the future to know. A local legend has become a little more human.
More Local Stories » News
Last word on Bird?
Court tells Don Bird to steer clear of Jim Nielsen, but both men claim victory.
Pushing for single-payer
National single-payer advocates show Chico how it's done.
Volunteers locate Lost Park, give it a good grooming
Make a Difference Day targets forgotten section of Lower Park.
A 20-year struggle for reform
Butte County Health Care Coalition celebrates its 'pathological persistence' with a dinner and a forum on—what else?—single-payer health care.
Making his voice heard
Political activist Dr. Michael Parenti and comedian Will Durst will share the Sierra Nevada Brewery Big Room stage Oct. 27.


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Best Churches in Miami
Handpicked Top 3 Churches in Miami, Florida. All of our churches actually face a rigorous 50-Point Inspection, which includes customer reviews, history, complaints, ratings, satisfaction, trust, cost and general excellence. You deserve only the best!
BEST RATED MIAMI, FL CHURCHES
SHRINE OUR LADY OF CHARITY
GESU CHURCH
OUR LADY OF LOURDES CATHOLIC CHURCH
3609 South Miami Avenue, Miami, FL 33133
"SHRINE OUR LADY OF CHARITY" has more than 50 years of presence in South Florida; the National Shrine of "OUR LADY OF CHARITY" stands today as a spiritual and evangelization center. During the first years of the 17th century, a beautiful image of the Mother of Christ appeared floating in the waters of the Bay of Nipe. After three days of storm, she was picked up by three peasants who had gone to look for salt and who tradition remembers as "The Three Juanes." The image was on a small table with a sign: I AM THE VIRGIN OF CHARITY. From the beginning, she has been venerated under the title of "OUR LADY" of Charity, and many pilgrims constantly visit her to pray before her and ask for her protection in times of danger and need. "SHRINE OUR LADY" of Charity is a United, privileged place for prayer and to strengthen trust in divine mercy. The Hermitage is the home of all the children, without distinction of sex, race, or religion.
Mass Schedule, Reserve Masses, Sacraments in the Hermitage, Baptism, Communion and Confirmation, Marriage & Donations
san***@e***.org
ermita.org
SHRINE OUR LADY OF CHARITY REVIEWS
A very beautiful shrine & location. It's a very powerful statement of history regarding our exodus as Cubans fleeing Communist oppression & poverty. It's a Catholic church but it's also a shrine to the Cuban exile community in the United States.
This place is so special, dedicated to the Cubans trying to escape Cuba by boat. It's a peaceful and beautiful and right next to the ocean.
A very blessed place and for cuban mor because it's the patron saint of cuban the Virgen Caridad del Cobre. To thank for being blessed to arrive to this great country and take care of our beloved family and friends we left behind in Cuba. Praying for the freedom of our country.
118 Northeast, 2nd Street, Miami, FL 33132
The Gesu Church is a historic Roman Catholic church in Miami, Florida. Gesu Church was founded in 1567 when Jesuit missionaries arrived at the Miami River's mouth with the Spanish settlement established by Don Pedro Menendez de Avila. The church has a beautiful temple built in Germany with polychrome crystal-led widows that relate events in Jesus and Mary's life. The church has eight hundred parishioners. Gesu Church Senior Citizens Center has a 26 computer seat computer lab. Their seniors are given weekly lessons and can use computers for several tasks. The buildings are outstanding religious architecture and remarkable for their style, craftsmanship, and excellence. Gesu Church adult faith formation is offered through weekly bible study and an RCIA program to prepare adults for Christian initiation. Their support group for married couples meets biweekly. Gesu is the oldest Catholic parish in Florida and has served the community's religious and social needs for over a century.
Mass Schedule, Confessions, Reconciliation, Baptism, Radial Program, First Communion, Architectural Buildings, Marriage, Anointing of the Sick, Sacraments & Events
ges***@y***.com
gesuchurch.org
Mon-Fri & SUn: 8am - 5pm
GESU CHURCH REVIEWS
An amazing historical church in downtown Miami. The outside design is normal, however what it lacks from the outside the inside make up far more. The design and feel reminds me a trip I went to Spain and visited the churchs there.
Beautiful church and friendly people. The children's choir was very impressive in the quality and professionalism. I was there for mother's day and the sermon was heartwarming.
Beautiful and historical Roman Catholic Church. Worth visiting. Definitely one of the must visit attraction in Miami.
11291 Southwest 142nd Avenue, Miami, FL 33186
Our Lady of Lourdes Catholic Church is a Roman Catholic Church of the Archdiocese of Miami located in the heart of West Kendall in Miami Dade County, Florida. Their Lady of Lourdes Catholic Church was built in 1985. Their Lourdes lady is proud to be an engaged community with many Ministries building the structure of their parish. Their Christian formation curriculum prepares an individual for complete baptism into the Catholic Church and a whole, active life in the parish. Their Lady of Lourdes is proud to be a vibrant community with many ministers who build the fabric and their parish. The church ministry begins by helping parents and godparents prepare for the initiation of children and spreads to people of all ages through appropriate catechetical and sacramental training. Their Lady of Lourdes Catholic Church Baptisms usually is celebrated twice each month in English and twice each month in Spanish. They are honored to work with you to give your children a formation and education.
Funerals and Burials, First Communion, Compassion Ministries Division, Marriage, Liturgy, Infant Baptism, Youth Groups, Justice and Charity Division, Sacraments, Events, Ministries, Funeral and Burial, Christian Formation & Confirmation
cri***@o***.org
ololourdes.org
Mass Time:
Mon-Sat: 8am (English)|7pm (Spanish)
MONDAY - FRIDAY: 7pm(Spanish)
Sat Only on the first Sat of the month: 9:30am (French)
Sat Vigil: 5pm (English)|6:30pm (Spanish)
Sun: 7:15am (English)|8:30am (Spanish)|10:15am (English)
|11:45am (English)|1:30pm (Spanish)
|5pm (English)|6:30pm (Spanish)
OUR LADY OF LOURDES CATHOLIC CHURCH REVIEWS
The chapel is big and there are a few areas to light a candle and give offerings. There is also an inside shrine to El Señor de Los Milagros, patron of Peru but the most beautiful sight is Outside towards the school, a beautiful grotto of Our Lady of Lourdes with trees and sitting area to reflect. To me this beautiful grotto is definitely worth a visit even if you live far. There is also a Perpetual Adoration chapel open 24 hrs a day 7 days a week.
I have been here 3 times and the best experience I had. I loved this church it's pretty big and nice. They also have a school inside the church so parking can be hectic during school hours. On Sunday Mass they have police offices guarding the parkings and controlling the traffic. I highly recommend!
I was visiting with a friend for the Spanish service, though my Espanol is sub par. The church service was pleasant with singing and scripture.
Garage door repair in Miami, FL

redpajama_set_name: RedPajamaCommonCrawl


OAK VALLEY, TEXAS. Oak Valley, off Farm Road 2555 four miles southwest of Corsicana in central Navarro County, was developed in the 1970s. Most of the residents worked in nearby Corsicana, and children attended the Corsicana public schools. The community had an estimated population of 200 in 1987, 388 in 1990 and 401 in 2000.


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Beneath a Steel Sky je sci-fi počítačová hra pro PC (DOS) i Amigu, odehrávající se v neurčité budoucnosti po planetární katastrofě. Žánrově se jedná o klasickou grafickou adventuru, kde hráč ovládá hlavního hrdinu. Ten se jmenuje Robert Foster a spolu s droidem Joeym je v úvodu hry zajat autoritou blízkého města, do něhož má být deportován vrtulníkem. Během přepravy ale nastanou problémy, následkem jichž je havárie, ze které se Fosterovi s Joyem podaří zachránit a utéct. Následně Foster odhaluje principy tohoto chladnokrevného světa, setká se s několika jeho obyvateli, několikrát podnikne výlet do virtuální reality tamního informačního systému, než jej na konci čeká tajemné setkání s jeho biologickým otcem.

Hra byla vydána na začátku roku 1994, ale v roce 2003 byla zveřejněna společností Revolution Software jako freeware. Je možné ji najít na stránkách projektu ScummVM. je také dostupná zdarma na Good Old Games.

Externí odkazy

 Revolution Software - vývojářská společnost
 https://web.archive.org/web/20060424154219/http://plnehry.idnes.cz/adventure/beneathasteelsky.html - plná verze ke stažení

Point-and-click adventury
Hry podporované ScummVM
Hry pro Amigu
Hry pro DOS
Videohry z roku 1994
Freewarové videohry
Hry pro CD32
Kyberpunkové videohry
Science fiction videohry
Hry pro iOS
Hry pro macOS
Hry pro Windows
Britské videohry
Videohry pro jednoho hráče
Videohry od Virgin Interactive

redpajama_set_name: RedPajamaWikipedia


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Does CalcOcclusionAndVirtualPositions take GeometrySets into account, or does it only look at rooms and portals?
In short, yes you need to determine yourself if a sound is occluded/obstructed via a ray cast or some other means. We do not yet do emitter-listener visibility checks via the geometry sets, however this is something we may look into for a future release. The purpose of this funciton is mostly to position the sound, when it is behind portals, in the correct place via "virtual positions", but also provides some occlusion values based on the number of enabled portals vs total portals, connected to the room. You call the function after having already failed the visibility check.
In the future, this somewhat awkward function will most likely be deprecated in favor of internal positioning of the sound emitter.
selected Sep 5, 2017 by Evan T.


A South Carolina farm wife called the local phone company to report her telephone failed to ring when her friends called, and that on the few occasions when it did ring, her dog always moaned right before the phone rang.
The telephone repairman proceeded to the scene, curious to see this psychic dog or senile lady. Upon arriving at the residence he climbed the utility pole, plugged in his test set, and dialed the subscriber's phone number.
The phone didn't ring right away, but then the dog moaned and then the telephone began to ring.


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Taylor's Swifties fight back
Xavier Greenwood
#Fairness
#HigherPrices
A group of Taylor Swift fans are suing a ticket company for fraud, price fixing and antitrust violations. It's not the first time Ticketmaster has been criticised
"Good Morning America. It's Taylor. I wanted to tell you something that I've been so excited about for a really long time. I've been planning for ages and I finally get to tell you I'm going back on tour."
Taylor Swift on Good Morning America
They are known for being some of the most devoted fans on the planet.
"The tour is called The Eras tour and it's a journey through all of the musical eras of my career."
So when Taylor Swift announced a 52-date tour across the US, her first since 2018, Swifties – as her fans are known – were ecstatic.
Three and a half million people, that's more than one in every 100 Americans, registered as "verified fans". The promise? Early access to tickets.
But then it all went wrong.
"Fans of Taylor Swift hoping to score tickets to her upcoming tour have met a confusing and chaotic system, prompting outrage from fans and lawmakers alike."
And emotions ran high.
"The Swifties went from freaking out to melting down."
They had reason to be upset. Many people's access codes didn't work. Dynamic pricing meant the cost of a ticket rocketed to meet demand. The public sale, which was meant to follow the pre-sale, was cancelled.
And while ordinary fans missed out, resale sites soon advertised tickets for tens of thousands of dollars.
Ticketmaster, the company selling the seats, went on the defensive.
Here's the chairman of Live Nation Entertainment, which owns the ticket site.
"All the Live Nation team is sympathetic for the long wait times and the fans who couldn't get what they wanted. The reality is it's a function of the massive demand Taylor Swift has… We could have filled 900 stadiums."
Greg Maffei on CNBC
But that did nothing to dampen the fury, not just from Taylor Swift fans, but politicians too.
"New Jersey congressman Bill Pascrell Jr described Ticketmaster and its parent company Live Nation as a monopoly that should be broken up. Alexandria Ocasio-Cortez shared a similar message on social media, saying a lack of competition can drive up prices and worsen inflation."
Taylor Swift herself described it as excruciating "to just watch mistakes happen with no recourse". And she's not the first musician to have taken aim at Ticketmaster.
Back in 1994, Pearl Jam was angry about the fees charged by Ticketmaster, so the band tried to sell tickets for their tour in a different way. That dispute ended up in the US Congress.
"All the members of Pearl Jam remember what it's like to be young and not have a lot of money. Many Pearl Jam fans are teenagers who do not have the money to pay thirty dollars or more that is often charged for tickets today."
Stone Gossard, Pearl Jam guitarist, in US Congress in 1994
But Taylor Swift is arguably an even more powerful force.
"Fans are now suing LiveNation Entertainment, Ticketmaster's parent company, for a laundry list of anti-competitive practices, arguing that led to soaring ticket prices for Swift's The Eras tours."
At the start of December, a group of Swifties filed a lawsuit in Los Angeles. They say that Ticketmaster, which transacts billions of dollars of tickets each month, has a monopoly on ticket sale markets, and the Taylor Swift fans want Ticketmaster to behave differently.
It feeds into a bigger debate about whether Live Nation Entertainment should be broken up altogether.
The company was formed in 2010 from a merger of Ticketmaster and Live Nation. It owns 400 venues around the world and holds tens of thousands of shows. Many major US venues have exclusive contracts with the firm.
Ticketmaster has long been in the crosshairs of politicians and musicians. Even before the Taylor Swift fiasco, the US Department of Justice had opened an antitrust investigation into its parent company.
But now that it faces the ire of Swifties, Ticketmaster may have its work cut out.
Here's Lina Khan, chair of the federal trade commission.
Interviewer: "Your opinion of the Taylor Swift ticket debacle?"
Lina Khan: "I know. I think that one incident ended up converting more Gen Zers into anti-monopolists overnight than anything I could have done."
Wall Street Journal CEO Council Summit
Perhaps that's why Ticketmaster is trying to make amends. It's said some fans who missed out will now have another opportunity to buy tickets.
"Ticketmaster sent out an email earlier today to those who were part of the verified fan pre-sale but who were unable to buy tickets last month. So fans will now get individual invitations to submit requests before December 23."
In the battle against anti-competitive practices across the United States, Taylor Swift fans could end up being an unlikely lodestar.
But by giving an inch, Ticketmaster may survive in its current form to fight another day.
This episode was written by Xavier Greenwood and mixed by Hannah Varrall.


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This season some deliveries are taking longer to make it through the postal system than we would usually expect.
Surprise Reads
Publication Date: 09/07/2020 ISBN: 9781529384482 Category: Books
Eat, Gay, Love
'You've never read a travel memoir like this before'
The Sunday Times, 'Pride Culture Guide'
'Sweet and fun, with real emotional depth and a rousing, feisty spirit'
Matt Cain
In the spring of 2012, Calum finds himself single again after his relationship of six years comes to an end.
Heartbroken, unhappy and unsure of what to do next, he leaves the hometown he has been in all his life to embark on a journey that takes him all around the world, from teaching in a school on the outskirts of Rome to exploring the sex clubs of Berlin, to raising tigers in an animal sanctuary deep in the jungles of Thailand. Along the way, he meets LGBT+ people from all walks of life and every part of the rainbow – from an Italian teenager struggling with a homophobic father to a kathoey navigating life as a trans person in Thailand, to a young HIV-positive man living on the streets of London.
Their individual stories, not only of hardship and sorrow but also of profound strength and hope, show the breadth and depth of queer life and experience, shedding light on themes such as homophobia, sexual violence, marriage equality and gender identity. Through these meetings and friendships, Calum not only finds the encouragement to embrace life after heartbreak, but also discovers a beautiful, loving global community who support and uplift him through the best and worst moments of his time on the road.
A travel memoir with a difference, Eat, Gay, Love is a celebration of the power of community and a personal tribute to the extraordinary lives of LGBT+ people everywhere in the world.
Publisher Review
'Sweet and fun, with real emotional depth and a rousing, feisty spirit' -- Matt Cain
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Home » Portal » Spain » English Expats In Spain
English Expats In Spain
If you are looking to connect with English expats in Spain, then this page brings together all the latest expat blogs Spain written by English bloggers, any interviews, articles and news items we have here at ExpatsBlog.com.
Latest Blogs by English Expats Living in Spain
Random thoughts from a Brit in the North West. Sometimes serious, sometimes not. Quite often curmudgeonly.
Ohla Living
Ohla Living is a place of exploration, inspiration, and creativity. I write about lifestyle, expat life home, travel, family, tech, food and design.
Life in Culebrón
Life in Culebrón is a very British view of life in a small village in Alicante province, my experience of Spain, of Spaniards and sometimes of the other Britons who live nearby.
Peter and Maureen Scargill have had a cavehome in Galera for many years - the blog is a record of our adventures in Spain.
Chilled at sunset
From UK slog to chilled-out blog
Latest Expat Interviews/ Articles
American Expat Living in Spain - Interview with Jiab & Jim
Jiab, born and raised in Thailand, is from one side of the world. Jim, a born and bred Texan, is from the other side. Jiab is quiet, serious, and reserved. She has a no-nonsense attitude, is...
American Expat Living in Spain - Interview with Kate
People say don't fall in love when you go abroad because you won't come back...and based on my experience, they're totally right. After graduating from college with a communications degree I...
American Expat Living in Spain - Interview with Mitchell
Mitchell loves change and had lived, and traveled, all over the United States before moving permanently to Southern Spain in 2011. His goal is to become absorbed into Spanish culture, to study and...
German Expat Living in Spain - Interview with Ina
Ina is a German expat living in Barcelona, Spain. Over the last 15 years she has been living in numerous countries: it all started with a yearlong Erasmus exchange in Grenoble, France, which probably...
British Expat Living in Spain - Interview with Maria
Maria is a thirty something freelance graphic & web designer. She was born in the Philippines but raised in England from the age of 10. Having had enough of the rat race and being fortunate enough to...
Australian Expat Living in Spain - Expat Interview with Brett
Brett is the the author of "The Remade Parent (Why We are Losing Our Children & How We Can Get Them Back.") He is a parent (and was a secondary school teacher for 15 years.) Brett is a regular...
Australian Expat Living in Spain - Interview with Vanessa
Leaving behind her 10 year career in special events and concert touring, Vanessa moved to Menorca in Spain from Sydney Australia to renovate her grandparent's house on the island. Eight years...
US Expat Living in Madrid, Spain - Expat Interview with Jessica
Jessica Wray is the creator and blogger at CuriosityTravels.org. Ever since college graduation she has been on a continuous journey to see the world as an expat, first in South Korea and now in...
New Zealand Expat Living in Spain, Interview with Ocean
Ocean is a kiwi born travel enthusiast, living in Madrid on an internship with the United Nations World Tourism Organization. Having spent the last four years living in various locations globally...
American Expat Teaching in Valencia, Spain - Interviewing Allison
Allison is a food and travel blogger originally from Denver, Colorado. After graduating from Kansas State University with degrees in Spanish and Theater in 2011 she moved to Spain for a year…..and...
US Expat Teaching English in Spain - Interviewing Chelsea
Chelsea is a Buffalo, NY native who is heading into her third year as an English teacher in Spain. Having spent the last two years in Andalucía, she and her (Spanish) boyfriend will be moving three...
Meet Allie, American Expat Who Moved to South of Spain
Allie is an American who moved to the south of Spain last spring to start her married life after a few years of long distance love and a couple of weddings (to the same swarthy Spaniard). She blogs...
American Expat Living in Galicia - Interview With Ali
Ali's the literary lovechild of David Sedaris and Carrie Bradshaw. Born biracial and raised bicoastal, her life has always been unconventional. However, after a series of serious breakups, one...
American Expat Living in Madrid - Interview With Alexis
Alexis hails from Portland, Oregon where she worked for years in event marketing before leaving it behind to follow her dream of living abroad in Spain. She now teaches English to second and third...
British Expat Living in Barcelona - Interview With Jessica
24-year-old Jessica is English-born, California-raised, and living in Barcelona, Spain. She first got a taste Spain as a study abroad student in Sevilla, but really fell in love with Barcelona. After...
Why British Expats Flock To Spain?
Spain has been attracting Britons of all ages since the 1960's. The diverse country consists of mainland, coastal resorts and islands that attract thousands of holiday makers in search of a relaxing...
US Expats Living in Spain - Meet Nancy & Regina
Nancy Todd and Regina Winkle-Bryan are co-editors of The Spain Scoop (see listing here), a savvy, fun guide to Spain. The Spain Scoop includes guest writers from different locations in Spain to lend...
Expat Interview With Lauren - American Expat in Spain
Lauren David has always dreamed of living in Spain and when she was offered a grant from the Spanish Ministry of Education to be a Language and Culture Assistant, she couldn't refuse. What she...
San Francisco to Madrid - Expat Interview With Nicole
Nicole is trading (at least) a year of San Francisco life for Spanish life in Madrid. She's teaching English at an Elementary school, getting a Master's in Bilingual and Multicultural Education...
Expat Interview With Kaley - Indiana to Madrid
Kaley Hendrickson is the blogger at Y Mucho Más (see listing here). Born and raised in Indiana, she fell in love with Spanish in high school and chose her major accordingly. After graduating from...
England to Andalucía - Expat Interview With Marianne
Marianne is a former lawyer, EFL teacher, neophyte blogger, petrol-head, amateur photographer, traveller, English woman and shameless arctophile. For the past seven years she has lived in...
British Expat In Gran Canaria - Expat Interview With Matthew Hirtes
Matthew Hirtes was an unwilling expat at first, hankering for his former London life. To his eternal shame, he had to be talked into relocating to Gran Canaria by his sister-in-law. Little did he...
British Expat In Barcelona - Expat Interview With Simon Harris
Simon Harris has lived in Barcelona for 25 years and after half a lifetime of experiences, feels more Catalan than British. He arrived with a guitar strapped to his back and ended up recording and...
Expat Interview With Fiona - British Expat In Seville
Fiona Flores Watson is a journalist and blogger who lives in Seville, Spain, where she writes about life as a Spanish expat wife and mother; the best places to eat and stay around Andalucia; fun...
Expat Interview With Lisa Sadleir - British Expat In Spain
Lisa Sadleir, aka "Mum" from Family in Spain, is a very happy workaholic mum who lives in southern Spain with her husband and two beautiful children. She blogs about her Family Life In Spain. With...
Expat Interview With Anna Nicholas - British Expat in Mallorca
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RGB (not to be confused with the color format) pours a deep brown/black with two inches of bubbly chocolate frosting. This beer may possibly have the thick lacing I've ever seen on a beer; it's like foamy siding for your beer glass.
The nose on this beer was just glorious. Dark chocolate, smoke, pork and espresso are all present in the nose and easy distinguishable. As the beer warms I detected some chocolate chip cookie dough.
Rauch Geek Breakfast starts out smokey, with a noticeable cured pork flavor. As you would expect when you hear the term "cured pork", there is a saltiness present that kinda dries out your mouth. The taste then transitions into dark chocolate, before fading into a rich coffee flavor. A faint doughy/oat flavor comes through as it warms.
Mouthfeel is medium-to-thick, syrupy and plenty dry. This beer reminded me of a stout meets a rauchbier.
Rauch Geek Breakfast is an interesting beer. One of the most flavorful beers I've had of late, with each flavor standing out and easily detectable. It's not the type of beer you could session or drink frequently, but it's uniqueness makes it one definitely worth trying.


text:
Death of Osama bin Laden Announced by Obama - Video
Obama Announces Death of Osama bin Laden
VOA Kent Klein | White House May 01, 2011
Twenty-five minutes before midnight on a Sunday night, President Obama made an announcement that the American people had waited almost ten years to hear.
"I can report to the American people and to the world, that the United States has conducted an operation that killed Osama bin Laden, the leader of al-Qaida and a terrorist who is responsible for the murder of thousands of innocent men, women and children," Obama said.
Osama bin Laden has been the world's most-wanted terrorist since more than 3,000 people were killed in al-Qaida's attacks on the United States on September 11, 2001.
Mr. Obama said he was briefed last August on a possible lead to bin Laden's location. He said the terrorist leader had been hiding in a compound in Abbottabad, Pakistan.
The president said he authorized a mission last week to "get Osama bin Laden and bring him to justice."
Watch President Obama's Announcement:
Mr. Obama said a small team of Americans Sunday carried out the operation on the compound.
"After a firefight, they killed Osama bin Laden and took custody of his body," Obama said.
The president acknowledged that al-Qaida will continue to pursue attacks against the United States, and he warned Americans to remain vigilant.
"I have made clear, just as President Bush did shortly after 9-11, that our war is not against Islam, because bin Laden was not a Muslim leader. He was a mass murderer of Muslims. Indeed, al-Qaida has slaughtered scores of Muslims in many countries, including our own."
Mr. Obama said counterterrorism cooperation with Pakistan helped lead U.S. forces to bin Laden. He said he called Pakistani President Asif Ali Zardari to notify him of bin Laden's death. The president said Pakistani officials agreed that this was a "good and historic day" for both nations.
Mr. Obama also called former President George W. Bush, who was president on September 11, 2001, and who launched the U.S. war on terror. In a written statement, Mr. Bush called bin Laden's killing a "momentous achievement" and a "victory for America." He said "No matter how long it takes, justice will be done."
Osama bin Laden was born March 10, 1957 to a wealthy family in Saudi Arabia.
When the Soviet Union invaded Afghanistan in 1979, bin Laden joined the Afghan mujahedin Islamic fighters. Several years later, he used his wealth to form his own militia force, later called al-Qaida, Arabic for "the base."
In 1996, bin Laden declared a holy war against the United States, which he accused of looting the natural resources of Muslim nations and helping Islam's enemies.
While hiding in Sudan, bin Laden is said to have plotted attacks on the U.S. military in Somalia and Saudi Arabia. He also orchestrated the 1998 bombing of U.S. embassies in Kenya and Tanzania.
Within weeks after the September 11, 2001 attacks, the United States led a coalition that overthrew Afghanistan's Taliban government, which had refused to turn bin Laden over to the U.S.
For almost ten years, U.S. soldiers and intelligence officers combed the mountainous area along the Afghanistan-Pakistan border, trying to find bin Laden.
Shortly before President Obama announced that bin Laden had been killed, a jubilant crowd gathered outside the White House, chanting, cheering and singing.
Meanwhile, the U.S. State Department has put its embassies on alert and warned Americans of possible al-Qaida reprisal attacks.
Article from VOA
Eastern Shore of MD information Posted by Eastern Shore Magazine - Oxford, St Michaels, Ocean City and more. We invite you to post a comment and enjoy the Chesapeake Bay. Eastern Shore Magazine is our trademark and service mark. at 9:19 AM
Labels: military, Osama Bin Laden, Osama Bin Laden Killed, Pakistan, terrorism, U.S. President Barack Obama, USA, video


13/12/2018�� To make marzipan, start by adding 200 grams of ground almonds and 200 grams of confectioner's sugar to a bowl. Then, add 3 drops of vanilla extract and 2 egg �... This sweet recipe for orange cream cheese frosting and marzipan carrots. It�s the perfect topping for brownies, cake or cookies. It�s the perfect topping for brownies, cake or cookies. This week has been a bit crazy.
Light and moist Carrot Cupcakes with Cream Cheese Frosting. Easy to make and delicious topped with marzipan carrots. Great for birthday parties!... This German carrot cake is not only a great cake for Easter. The German recipe includes how to make a glaze and even how to make the carrot decoration out of marzipan.
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Make tiny indentations in each carrot with cocoa-dusted knife to evoke the vegetable's texture. Carrots will keep, covered, for up to 1 week. Push fronds into top of each carrot before serving. Carrots will keep, covered, for up to 1 week.
I want to make these just to play with marzipan now (though I do love to eat marzipan, too). I�m getting an idea to make those carrots to top off a carrot cake for Mother�s Day coming up�! I love the ideas I can gather from around the blog world!
This sweet recipe for orange cream cheese frosting and marzipan carrots. It�s the perfect topping for brownies, cake or cookies. It�s the perfect topping for brownies, cake or cookies. This week has been a bit crazy.
How to Make Decorative Carrots for your carrot cake or cupcakes. Includes a step-by-step video showing THREE ways to make cute carrots: with buttercream, candy melts, and marzipan.


text:
Orthochtha nadiae är en insektsart som beskrevs av Popov, G.B. och Fishpool 1992. Orthochtha nadiae ingår i släktet Orthochtha och familjen gräshoppor. Inga underarter finns listade i Catalogue of Life.

Källor

Gräshoppor
nadiae


text:
\section{Introduction}

We study the averaging of a system of reaction-diffusion equations
with linear transmission condition posed in a class of highly
heterogeneous media including areas of low and high diffusivity. Our
aim is twofold: on one hand, we wish to justify rigorously the
formal asymptotics expansions performed in \cite{Tycho_Adrian},
while on the other hand we wish to understand the error caused by
replacing a heterogeneous solution by an approximation (averaged)
estimate. To this end, we prove an upper bound for the convergence
rate of the limit procedure $\epsilon\to 0$, where $\epsilon$ is a
suitable scale parameter (see section \ref{microstructure} for the
definition of $\epsilon$). The materials science scenario we have in
view is motivated by a very practical problem: the sulfate corrosion
of concrete. We refer the reader to
 \cite{Beddoe} for a nice and detailed description of the physico-chemical scenario and to  \cite{Tasnim1} for the formal (two-scale) averaging of
 locally-periodic distributions of unsaturated pores attacked by sulphuric
 acid. The reference
 \cite{Tasnim2} contains the rigorous proof of the limiting procedure $\epsilon\to0$ treating the uniformly periodic case of
 the same corrosion scenario. In \cite{Tasnim2}, the main working tools involve the two-scale convergence concept in the sense of Nguetseng and
 Allaire\footnote{For an introduction to two-scale convergence, see
 \cite{Lukkassen}.}
 combined with  a periodic unfolding of the oscillatory boundary.
 Here, we use an energy-type method.

The framework that we tackle in this context is essentially a
deterministic one. We assume that a distribution of the {\em
locally-periodic array of microstructures}\footnote{Here, we deviate
from the purely periodic setting. It is worth noting also that the
assumption of statistically homogeneity of distribution of
microstructures, which is a crucial restriction of stochastic
homogenization, does not cover
 all possible configurations of locally-periodic geometries. Also, as we formulate the working framework, we cannot treat neither randomly placed microstructures
 nor stochastic distributions of microgeometries. Consequently, the exact connection between these two averaging techniques is hard to make precise.} is known {\em a priori}. We refer the reader to \cite{Capriz} for related discussion of continuum mechanical descriptions of balance laws in continua with microstructure.  At the mathematical level, we succeed to
combine successfully  the philosophy of getting correctors as
explained in the analysis by Chechkin and Piatnitski \cite{Chechkin}
with the intimate two-scale structure of our system; see also
\cite{CAM,CF} for related settings where similar averaging
strategies are used. For methodological hints on how to get corrector estimates, we
refer the reader to the analysis shown in
\cite{Eck_correctors} for the case of a reaction-diffusion phase
field-like system posed in fixed domains. For a higher level of discussion, see the monograph
\cite{CPS} which is a nicely written introductory text on homogenization
methods and applications. It is worth noting that the periodic
homogenization of linear transmission problems is a well-understood
subject (cf. \cite{Conca0,Ene,AG,fast}, e.g.), however much less is
known if one steps away from the periodic setting even if one stays
within the deterministic case. If the microstructures are
distributed in a suitable random fashion, then concepts like random
fields (see \cite{Belyaev_Ef} or \cite{Mikelic_random_fields} and
references cited therein; \cite{Heida2009} and intimate connection
with the stochastic geometry of the perforations) can turn to be
helpful to getting averaged equations (eventually also capturing new
memory terms), but corrector estimates seem to be very hard to get;
cf., e.g., \cite{Bal}. Disordered media for which the stationarity
and/or ergodicity assumptions on the random measure do not hold or
situations where at a given time scale length scales are
non-separated are typical situations that cannot be averaged with
existing techniques (so it makes no sense to search here for correctors).

This paper is organized as follows:
In Section \ref{problem} we introduce the necessary notation, the
microscale and the two-scale limit model.
Section \ref{main_result} contains the main result of our paper --
Theorem \ref{MR}. In Section \ref{technical} we introduce the
technical assumptions needed to obtain the correctors. At this
point, we also collect the known well-posedness and regularity
results for both the microscopic model and the two-scale limit
model. The rest of the paper consists of the proof of the
convergence rate and is the subject of Section \ref{Proofs}.

\section{Statement of the problem}\label{problem}
In this section we introduce our notation, the microscale model and the two-scale limit problem. We start off with the definition of the
locally periodic heterogeneous medium.

\subsection{Definition of the locally-periodic micro medium}\label{microstructure}

\begin{figure}[hbpt]
\begin{center}
\includegraphics[width=8cm]{locperpic.pdf}
\end{center}
\caption{Schematic representation of a locally-periodic heterogeneous medium.
The centers of the gray circles are on a grid with width $\epsilon$.
These circles represent the areas of low diffusivity and their radii may vary.
\label{fig1}}
\end{figure}

We consider a heterogenous medium consisting of areas of
high and low diffusivity.
The medium is in the present paper represented by a two dimensional domain. For the definition of the locally-periodic medium with inclusions, we follow in main lines \cite{Chechkin}.
We denote the two dimensional bounded domain by $\Omega\subset \mathbb{R}^2$, with boundary $\Gamma$.
Denote
\begin{align*}
&J^\epsilon:=\{j\in\mathbb{Z}^2\,|\,\mbox{dist}(\epsilon j,\Gamma)\geq\epsilon\sqrt{2}\},\\
&{\cal U}:=\{y\in \mathbb{R}^2\,|\, -1/2\leq y_i \leq 1/2\,\, \mbox{for}\,\, i=1,2\}).
\end{align*}
A convenient way to parameterize the interface $\Gamma^\epsilon$ between the high and low diffusivity areas, is to use a level set function, which we denote by
$S^\epsilon(x)$:
\begin{align*}
x\in \Gamma^\epsilon \Leftrightarrow S^\epsilon(x)=0,
\end{align*}
Since we allow the size and shape of the perforations to vary with the
macroscopic variable $x$, we use the following characterization
of $S^\epsilon$:
\begin{align}
S^\epsilon(x):=S(x,x/\epsilon),
\label{locper}
\end{align}
where $S:\Omega\times {\cal U} \rightarrow \mathbb{R}$ is 1-periodic in its second variable, and where $S$ is independent of $\epsilon$.
We assume that $S(x,0)<const.<0$ and $S(x,y)|_{y\in\partial U}>const.>0$ for all $x\in\Omega$, so that the areas of low diffusivity in each unit cell do not touch each other.
We set
\begin{align*}
Q_j^\epsilon:=\{x\in\epsilon({\cal U}+j)\,|\,S(x,x/\epsilon)<0\},
\end{align*}
and introduce the area of low diffusivity $\Omega_l^\epsilon$ as follows:
\begin{align*}
\Omega_l^\epsilon:=\bigcup_{j\in J^\epsilon}Q_j^\epsilon,
\end{align*}
and the area of high diffusivity $\Omega_h^\epsilon$ as:
\begin{align*}
\Omega_h^\epsilon:=\Omega\backslash\Omega_l^\epsilon.
\end{align*}

We call a medium of which the geometry is specified with a level set function of the type that is given in \eqref{locper} a locally periodic medium \cite{Chechkin}. In Fig.\ \ref{fig1} a schematic
picture is given of how such a medium might look like for a given $\epsilon>0$. Note that by construction the area of high diffusivity $\Omega_h^\epsilon$ is connected and the area of low diffusivity $\Omega_l^\epsilon$ is disconnected.
The interface between high and low diffusivity areas $\Gamma^\epsilon$ is now given by $\Gamma^\epsilon=\partial \Omega_l^\epsilon$ and the boundary of $\Omega_h^\epsilon$ is given by $\partial \Omega_h^\epsilon=\Gamma^\epsilon\cup \Gamma$.

Furthermore, we use the notation $\Omega_1^\epsilon:=\Omega\backslash\bigcup_{j\in J^\epsilon}(\epsilon({\cal U}+j))$, and we introduce for later use the smooth cut-off function $\chi_\epsilon(x)$ that satisfies
$0\leq \chi_\epsilon(x)\leq 1$, $\chi_\epsilon(x)=0$ if $x\in\Omega_1^\epsilon$ and $\chi_\epsilon(x)=1$ if $\mbox{dist}(x,\Omega_1^\epsilon)\geq\mbox{dist}(\Gamma^\epsilon,\Omega_1^\epsilon)$.
Moreover, $\epsilon|\nabla \chi_\epsilon|\leq C$ and $\epsilon^2|\Delta \chi_\epsilon|\leq C$, with $C$ independent of $\epsilon$,
and also
\begin{align}
\|1-\chi_\epsilon\|_{L^2(\Omega)}\leq \epsilon^{1/2} C,\nonumber\\
\|\nabla \chi_\epsilon\|_{L^2(\Omega)}\leq \epsilon^{-1/2} C,\label{boundschi}\\
\|\Delta \chi_\epsilon\|_{L^2(\Omega)}\leq \epsilon^{-3/2} C, \nonumber
\end{align}
with $C$ again independent of $\epsilon$ (see e.g.\ \cite{Chechkin,Eck_correctors}).

In addition, we need to expand the normal $\nu^\epsilon$ to $\Gamma^\epsilon$ in a power series in $\epsilon$.
This can be done in terms of the level set function $S^\epsilon$, which we assume to be sufficiently regular so that all the following computations make sense (see assumption (B1) in section \ref{technical}):
\begin{align}
\nu^\epsilon=\frac{\nabla S^\epsilon(x)}{|\nabla S^{\epsilon}(x)|}=\frac{\nabla S(x,x/\epsilon)}{|\nabla S(x,x/\epsilon)|}=\frac{\nabla_x S+\frac{1}{\epsilon}\nabla_yS}{|\nabla_x S+\frac{1}{\epsilon}\nabla_yS|}
\,\,\, \mbox{at}\,\,\, x\in \Gamma^\epsilon.
\end{align}
First we expand $|\nabla S^{\epsilon}|$. Using
the Taylor series of the
square-root function, we obtain
\begin{eqnarray}
|\nabla S^{\epsilon}|&=&
\frac{1}{\epsilon}|\nabla_{y} S|+O(\epsilon^0). \label{nabsex}
\end{eqnarray}
In the same fashion, we get
\begin{eqnarray*}
\nu^{\epsilon}=\nu_{0}+\epsilon \nu_{1}+O(\epsilon^{2}),
\end{eqnarray*}
where
\begin{eqnarray*}
\nu_{0}:=\frac{\nabla_{y}S}{|\nabla_{y}S|}
\end{eqnarray*}
and
\begin{eqnarray}
\nu_{1}&:=&\frac{\nabla_{x}S}{|\nabla_{y}S|}-
\frac{(\nabla_{x}S\cdot \nabla_{y}S)}{|\nabla_{y}S|^{2}}\frac{\nabla_{y}S}{|\nabla_{y}S|}. \label{nu11}
\end{eqnarray}
If we introduce the normalized tangential vector $\tau_{0}$, with
$\tau_0\perp\nu_{0}$,
we can rewrite $\nu_1$ as
\begin{eqnarray}
\nu_{1}&=&\tau_{0}\frac{\tau_{0}\cdot(\nabla_{x}S)}{|\nabla_{y}S|}.\label{nu1}
\end{eqnarray}
Later on we will also use the notation
\begin{align*}
\nu_1^\epsilon=\epsilon^{-1}(\nu^\epsilon-\nu_0)=\nu_1+O(\epsilon).
\end{align*}

\subsection{Microscopic equations and their upscaled form}

We focus on the following microscopic model
\begin{eqnarray}
&&\begin{cases}
\partial_t u_{\epsilon}=\nabla\cdot(D_h \nabla u_{\epsilon}-q_{\epsilon}u_{\epsilon}) &\\
\end{cases}
\,\,\,\,\,\,\mbox{   in   } \Omega_h^{\epsilon},\label{dimlesseq1}
\\
&&
\begin{cases}
\partial_t v_{\epsilon}=\epsilon^2\nabla\cdot(D_l \nabla v_{\epsilon}) &
\,\,\,\,\,\,\mbox{   in   } \Omega_l^{\epsilon},
\end{cases}\label{dimlesseq1b}
\\
&&\begin{cases}\label{transmission}
\nu^\epsilon\cdot(D_h \nabla u_{\epsilon})=
\epsilon^2\nu^\epsilon\cdot(D_l \nabla v_{\epsilon}) & \\
u_\epsilon=v_\epsilon & \\
\end{cases}
\,\,\,\,\,\, \mbox{   on   } \Gamma^{\epsilon}, \label{dimlesseq2}\\
&&\begin{cases}
u_\epsilon(x,t)=u_b(x,t) & \\
\end{cases} \,\,\,\,\,\,\, \mbox{on}\,\,\, \Gamma,\label{bcg}\\
&&\begin{cases}
u_{\epsilon}(x,0)=u_\epsilon^{I}(x)&\mbox{   in   } \Omega_h^{\epsilon},\\
v_{\epsilon}(x,0)=v_\epsilon^{I}(x)&\mbox{   in   } \Omega_l^{\epsilon}.
\end{cases}
\label{dimlesseq3}
\end{eqnarray}
Here we denoted the tracer concentration in the high diffusivity area by
$u_\epsilon$, the concentration in the low diffusivity area by $v_\epsilon$, and the
velocity of the fluid phase by $q_\epsilon$. Furthermore,
$D_h$ denotes the diffusion coefficient in the high diffusivity
region, $D_l$ the diffusion coefficient in the low diffusivity regions, $\nu^\epsilon$ denotes the unit normal to the boundary $\Gamma^\epsilon(t)$, where  $u_b$ denotes the Dirichlet boundary data for
the concentration $u^\epsilon$, and where
$u^I_\epsilon$ and $v^I_\epsilon$ are the initial values for the active concentrations
$u_\epsilon$ and $v_\epsilon$, respectively.

In order to formulate the upscaled equations and obtain closed-formulae for the effective transport coefficients, we use the notations
\begin{equation}
B(x):=\{y\in {\cal U}: S(x,y)<0\},
\end{equation}
and
\begin{equation}
Y(x):={\cal U}-B(x),
\end{equation}
and we define the following $x$-dependent cell problem:\footnote{See, e.g.\ \cite{CP,hornung}, for the role of cell problems in formulating averaged equations.}
\begin{eqnarray}
\begin{cases}
\Delta_{y}M_{j}(x,y)=0 & \mbox{for all}\,\,\, x\in\Omega,\, y\in Y(x),\\
\nu_{0}\cdot\nabla_{y}M_{j}(x,y)=-\nu_0\cdot e_{j}& \mbox{for all }\ x\in\Omega,\, y\in \partial B(x),\\
\mbox{$M_j(x,y)$ $y$-periodic},&
\end{cases} \label{cellpu}
\end{eqnarray}
for $j\in\{1,2\}$. To ensure the uniqueness of weak solutions to this cell problem, suitable conditions on the spatial averages of the cell functions need to be added.

The solution to this cell problems allows us to write the results of the
formal homogenization procedure in the form of the following
distributed-microstructure (two-scale) model
\begin{eqnarray}\label{upsceq}
&&\begin{cases}
\partial_{t}v_{0}(x,y,t)=D_l \Delta_y v_0(x,y,t)& \mbox{for}\,\,\,y\in B(x),\,\,x\in\Omega,\\
\partial_{t}\left(\theta(x)u_{0}+\int_{|y|<r(x)} v_0\,dy\right)= \\
\hspace{4cm}\nabla_{x}\cdot
({\cal D}(x)\nabla_{x}u_{0}-\bar{q}u_0)&\mbox{for}\,\,\,x\in
\Omega,
\end{cases} \label{upsc1}\\
&&\begin{cases}
v_0(x,y,t)=u_0(x,t)&\mbox{for}\,\,\,y\in\partial B(x),\\
u_0(x,t)=u_b(x,t)&\mbox{for}\,\,\,x\in\Gamma,
\end{cases}\label{upsc2}\\
&&\begin{cases}
u_0(x,0)=u_I(x)&\mbox{for}\,\,\,x\in
\Omega,\\
v_0(x,y,0)=v_I(x,y)& \mbox{for}\,\,\,y\in B(x),\,\,x\in\Omega.
\end{cases}\label{upsc3}
\end{eqnarray}
where the porosity $\theta(x)$ of the medium is given by
\begin{align*}
\theta(x):=|Y(x)|=1-|B(x)|,
\end{align*}
and where the effective diffusivity tensor ${\cal D}(x)$ is defined by
\begin{eqnarray*}
{\cal D}(x):=D_h\int_{Y(x)}(I+\nabla_yM(x,y))\,dy,
\end{eqnarray*}
with $M=(M_1,M_2)$.

\section{Main result}\label{main_result}

Recall that  $(u_\epsilon, v_\epsilon)$ is the  solution vector for the micro problem and $(u_0, v_0)$ is the solution vector for the two-scale problem. We introduce now the macroscopic reconstructions\footnote{We borrowed this terminology from \cite{Eck_correctors}. Note however that the concept of reconstruction operators appears in various other frameworks like for the heterogeneous multiscale method \cite{HMM}.} $u_0^\epsilon, v_0^\epsilon, u_1^\epsilon$,
which are defined as follows
\begin{eqnarray}
u_0^\epsilon(x,t)&:=&u_0(x,t) \mbox{ for all } x\in \Omega_h^\epsilon,\\
v_0^\epsilon(x,t)&:=&v_0(x,x/\epsilon,t) \mbox{ for all } x\in \Omega_l^\epsilon,\\
u_1^\epsilon(x,t)&:=&u_0^\epsilon(x,t)+\epsilon M(x,x/\epsilon)\nabla u_0^\epsilon(x,t) \mbox{ for all } x\in \Omega_h^\epsilon,
\end{eqnarray}
In the same spirit, we introduce the reconstructed flow velocity $q^\epsilon_0(x)=q(x)$ for all  $x\in \Omega_h^\epsilon$ and the corresponding reconstructions $u_{0I}^\epsilon,v_{0I}^\epsilon$ for the macroscopic initial data $u_I$ and $v_I$, respectively.

The main result of our paper is stated in the next Theorem. The
applicability of this result confines to our working assumptions
(A1), (A2), (B1), (B2), and (B3) that we introduce in Section
\ref{technical}.

\begin{theorem}\label{MR}
Assume (A1), (A2), (B1), (B2), and (B3). Then the following
convergence rate holds
\begin{eqnarray}
||u_\epsilon-u_0^\epsilon||_{L^\infty(I, L^2(\Omega_h^\epsilon))}+||v_\epsilon-v_0^\epsilon||_{L^\infty(I, L^2(\Omega_l^\epsilon))}+\nonumber\\
||u_\epsilon-u_1^\epsilon||_{L^\infty(I, H^1(\Omega_h^\epsilon))}+\epsilon||v_\epsilon-v_0^\epsilon||_{L^\infty(I, H^1(\Omega_l^\epsilon))}\leq c\sqrt{\epsilon},
\end{eqnarray}
where $I=(0,T]$, and where the constant $c$ is independent of $\epsilon$.
\end{theorem}

The remainder of the paper is concerned with the proof of Theorem \ref{MR}.

\section{Technical preliminaries}\label{technical}

\subsection{Function Spaces. Assumptions. Known results}

\subsubsection{Functional setting}

As space of test functions for the microscopic problem, we take the Sobolev space
$$H^1(\Omega_\epsilon;\partial\Omega):=\left\{\varphi\in H^1(\Omega_\epsilon) \mbox{ with } \varphi=0 \mbox{ at }\partial\Omega\right\}.$$
Since the formulation of the upscaled problem involves two distinct spatial variables $x\in\Omega$ and $y\in B(x)$ (with $B(x)\subset \Omega$),  we need to introduce the following spaces:
\begin{align}
&V_1:=H_0^1(\Omega),\\
&V_2:=L^2(\Omega;H^2(B(x))),\\
&H_1:=L^2_\theta(\Omega),\\
&H_2:=L^2(\Omega;L^2(B(x))).
\end{align}
The spaces $H_2$ and $V_2$ make sense, for instance, as indicated in \cite{sebam_equadiff}.

\subsubsection{Assumptions for the microscopic model}

{\bf Assumption (A1)}:  We assume the following restrictions on data and parameters:  Take $D_h,D_l\in ]0,\infty[$, $u_\epsilon^I\in H^1(\Omega_h^\epsilon)$, $v_\epsilon^I\in H^1(\Omega_l^\epsilon)$, and $u_b\in H^1(I;H^3(\Gamma))$. Furthermore, we assume
\begin{eqnarray*}
||u_\epsilon^I-u_{0I}^\epsilon||_{L^\infty(\Omega_h^\epsilon)}=\mathcal{O}(\epsilon^{\theta_1}) \mbox{ with } \theta_1\geq \frac{1}{2} \\
||v_\epsilon^I-v_{0I}^\epsilon||_{L^\infty(\Omega_l^\epsilon)}=\mathcal{O}(\epsilon^{\theta_2}) \mbox{ with } \theta_2\geq \frac{1}{2}
\end{eqnarray*}
\newline
{\bf Assumption (A2)}: $q_\epsilon\in H^2(\Omega_h^\epsilon;\mathbb{R}^d)\cap L^\infty(\Omega_h^\epsilon;\mathbb{R}^d)$ with $\nabla\cdot q_\epsilon=0$ a.e. in $\Omega_h^\epsilon$. Additionally, we assume that
\begin{equation}\label{st2}
||q_\epsilon-\frac{1}{\theta}\bar{q}||_{L^2(\Omega_h^\epsilon;\mathbb{R}^d)}=\mathcal{O}(\epsilon^{\theta_3})
\mbox{ with } \theta_3\geq \frac{1}{2}.\end{equation}

 \begin{remark} If one wishes to replace $q_\epsilon$ with the stationary Stokes or Navier-Stokes equations, then a few additional things have to be taken into account. One of the most striking facts is that the exponent
 $\theta$ arising in (\ref{st2}) seems to be restricted to $\theta_3=\frac{1}{6}$.  Consequently, this worsens essentially the convergence rate;
 see, for instance, \cite{Marusic} (Theorem 1) for a discussion of homogenization of the periodic case.
 \end{remark}

\subsubsection{Assumptions for the two-scale model}

{\bf Assumption (B1)}:
The level set function $S:\Omega\times{\cal U}\rightarrow \mathbb{R}$ is 1-periodic in its second variable and is in
$C^2(\Omega\times{\cal U})$.

{\bf Assumption (B2)}:  We assume the following restrictions on data and parameters:
\begin{align*}
\begin{cases}
\theta,\, D \in L^\infty_+(\Omega)\cap H^2(\Omega),&\\
\bar{q}\in H^2(\Omega;\mathbb{R}^d)\cap L^\infty(\Omega;\mathbb{R}^d)\,\,\mbox{with}\,\, \nabla\cdot \bar{q} =0 \mbox{ a.e. in } \Omega,&\\
u_b\in L_+^\infty(\Omega\times I)\cap H^1(I;H^3(\Gamma)),&\\
\partial_tu_b\leq 0\,\, \mbox{a.e.}\,\,(x,t)\in\Omega\times I,&\\
u_I\in L_+^\infty(\overline{\Omega})\cap H_1\cap H^2(\Omega),&\\
v_I(x,\cdot)\in L_+^\infty(B(x))\cap H_2\,\,\mbox{for a.e.}\,\,x\in\overline{\Omega}.&
\end{cases}
\end{align*}

{\bf Assumption (B3)}:
\begin{align}
&H_1:=H_0^1(\Omega),\\
&H_2:=L^2(\Omega;H^2(B(x))),\\
&V_1:=H^2_\theta(\Omega),\\
&V_2:=L^2(\Omega;H^3(B(x))).
\end{align}

\begin{remark}Following the lines of \cite{sebam_equadiff} and \cite{Show_walk}, Assumption (B1)  implies in particular that the measures $|\partial B(x)|$ and $|B(x)|$ are bounded away from zero (uniformly in $x$). Consequently, the following direct Hilbert integrals (cf. \cite{Dix} (part II, chapter 2), e.g.)
\begin{eqnarray}
L^2(\Omega;H^1(B(x)))&:=&\{u\in L^2(\Omega;L^2(B(x))): \nabla_y u\in L^2(\Omega;L^2(B(x)))\}\nonumber\\
L^2(\Omega;H^1(\partial B(x)))&:=& \{u:\Omega\times \partial B(x)\to \mathbb{R} \mbox{ measurable  such that }  \int_\Omega ||u(x)||^2_{L^2(\partial B(x))}<\infty\} \nonumber
\end{eqnarray}
are well-defined separable Hilbert spaces and, additionally, the {\em distributed trace}
$$\gamma: L^2(\Omega;H^1(B(x)))\to L^2(\Omega, L^2(\partial B(x)))$$ given by
\begin{equation}\label{g}
\gamma u(x,s):=(\gamma_x U(x))(s), \ x\in \Omega, s\in \partial B(x),u\in L^2(\Omega;H^1(B(x)))
\end{equation}
is a bounded linear operator.  For each fixed $x\in \Omega$, the map $\gamma_x$, which is arising in (\ref{g}), is the standard trace operator from $H^1(B(x))$ to $L^2(\partial B(x))$. We refer the reader to \cite{sebam_PhD} for more details on the construction of these spaces and to \cite{sf} for the definitions of their duals as well as for  a less regular condition (compared to (B1)) allowing to define these spaces in the context of a certain class of anisotropic Sobolev spaces.
\end{remark}

For convenience, we also introduce the evolution triple $(\mathbb{V},\mathbb{H},\mathbb{V}^*)$,
where
\begin{align}
&\mathbb{V}:=\{(\phi,\psi)\in V_1\times V_2\,|\,\phi(x)=\psi(x,y)\, \mbox{for}\,
x\in \Omega,\, y\in \partial B(x)\},\\
&\mathbb{H}:=H_1\times H_2,
\end{align}

\subsubsection{Analysis of microscopic equations}\label{analysis_micro}

\begin{definition}\label{def_weak_micro}
Assume (A1), (A2) and (B1). The pair $(u_\epsilon,v_\epsilon)$, with $u_\epsilon=U_\epsilon+u_b$ and
where $(U_\epsilon,v_\epsilon)\in H^2(\Omega_h^\epsilon;\partial \Omega)\times H^2(\Omega_l^\epsilon)$, is a weak solution of the problem $(P_\epsilon)$
if the transmission conditions (\ref{transmission}) are fulfilled and the following identities hold
\begin{align}
&\int_{\Omega_h^\epsilon}\partial_t(U_\epsilon+u_b)\phi\, dx+\int_{\Omega_h^\epsilon}
(D_h\nabla (U_\epsilon+u_b)-q_\epsilon(U_\epsilon+u_b))\cdot \nabla \phi \, dx=\nonumber\\
&\hspace{8cm}-\int_{\Gamma_\epsilon} \nu^\epsilon\cdot (\epsilon^2D_l\nabla v_\epsilon)\phi ds,\\
&\int_{\Omega_l^\epsilon} \partial_t v_\epsilon \psi \,dydx +
\int_{\Omega_l^\epsilon}  \epsilon^2 D_l \nabla v_\epsilon\cdot \nabla \psi \,dx =
\int_{\partial \Gamma_\epsilon}\nu^\epsilon \cdot (\epsilon^2D_l\nabla v_\epsilon)\psi \, ds
dx,
\end{align}
for all $(\phi,\psi)\in H^1(\Omega_h^\epsilon;\partial \Omega)\times H^1(\Omega_l^\epsilon)$ and $t\in I$.
\end{definition}
\begin{theorem}\label{main_result_micro} Assume (A1), (A2) and (B1).
Problem ($P_\epsilon$) admits  a unique global-in-time weak solution in the
sense of Definition \ref{def_weak_micro}.
\end{theorem}
\begin{proof} Since we deal here with a linear transmission problem, the proof of the Theorem can be done with  standard techniques  (see \cite{Evans}, e.g.).
\end{proof}

\subsubsection{Analysis of two-scale equations}\label{analysis_macro}

This section contains basic results concerning the well-posedness of the two-scale problem, which we reformulate here as:
\begin{align*}
(P)\begin{cases}
\theta(x)\partial_tu_0-\nabla_x\cdot({\cal D}(x)\nabla_x u_0 -q u_0)=-\int_{\partial B(x)}
\nu_0\cdot(D_l\nabla_y v_0)\,d\sigma & \mbox{in}\,\, \Omega,\\
\partial_tv_0-D_l \Delta_y v_0=0 & \mbox{in}\,\, B(x
,\\
u_0(x,t)=v_0(x,y,t)& \mbox{at}\,\, (x,y)\in \Omega\times \partial B(x),\\
u_0(x,t)=u_b(x,t) & \mbox{at}\,\, x\in \partial \Omega,\\
u_0(x,0)=u_I(x) & \mbox{in}\,\,\overline{\Omega},\\
v_0(x,y,0)=v_I(x,y) & \mbox{at}\,\,(x,y)\in \overline{\Omega}\times\overline{B(x)}.
\end{cases}
\end{align*}

Before starting to discuss the existence and uniqueness of weak solutions to problem $(P)$, we denote $U:=u-u_b$ and notice that $U=0$ at $\partial \Omega$.

\begin{definition}\label{def_weak}
Assume (B1) and (B2). The pair $(u,v)$, with $u=U+u_b$
where $(U,v)\in\mathbb{V}$, is a weak solution of the problem $(P)$
if the following identities hold
\begin{align}
&\int_\Omega \theta \partial_t(U+u_b)\phi\, dx+\int_\Omega
(D\nabla_x(U+u_b)-q(U+u_b))\cdot \nabla_x \phi \, dx=\nonumber\\
&\hspace{8cm}-\int_\Omega \int_{\partial B(x)}\nu_0\cdot (D_l\nabla_y v)\phi \, d\sigma
dx,\\
&\int_{\Omega} \int _{B(x)}\partial_t v \psi \,dydx +
\int_{\Omega} \int _{B(x)} D_l \nabla_y \cdot \nabla_y \psi \,dydx =
\int_\Omega \int_{\partial B(x)}\nu_0\cdot (D_l\nabla_y v)\phi \, d\sigma
dx,
\end{align}
for all $(\phi,\psi)\in \mathbb{V}$ and $t\in I$.
\end{definition}

\begin{proposition}[Uniqueness] Assume (B1) and (B2). Problem ($P$) admits at most one weak solution in the sense of Definition \ref{def_weak}.
\end{proposition}
\begin{proof}
Since Problem ($P$) is linear, the uniqueness follows in the
standard way and can be done directly in the $x$-dependent function
spaces: One takes two different weak solutions to ($P$) satisfying
the same initial data. Testing with their difference and using the
Gronwall's inequality conclude the proof.
\end{proof}
\begin{theorem}\label{main_result_macro}
Assume (B1) and (B2). Problem (P) admits at least a global-in-time weak solution in the
sense of Definition \ref{def_weak}.
\end{theorem}
\begin{proof} See the proof of Theorem 5.11 in \cite{Tycho_Adrian}.
\end{proof}
To get the correctors estimates stated in Theorem \ref{MR} we need more two-scale regularity for the macroscopic reconstruction of the concentration field $v_0$. We state this fact in the following result:

\begin{lemma}[Additional two-scale regularity] \label{addreg}
Assume (B1) and (B2). Then
\begin{equation}
v_0^\epsilon\in L^2(I;H^2(\Omega;H^2(B(x))))).
\end{equation}
\end{lemma}
\begin{proof}
The proof of this result is similar to the regularity lift proven in Claim 5.10 in \cite{Tycho_Adrian}. The main ingredients are fixing of the boundary and testing with difference quotients  in the weak formulation posed in fixed domains. We omit to show the details.
\end{proof}
\begin{theorem}[On strong solutions] \label{strongsol}
Assume (B1), (B2), and (B3). Problem (P) admits one  global-in-time strong solution.
\end{theorem}
\begin{proof}
Under the assumptions (B1) and (B2), Theorem \ref{main_result_macro} guarantees the existence of global-in-time weak solutions. Relying on the assumption (B3), we can lift the regularity until getting a strong solution. A similar calculation is done in  \cite{Evans} (Theorem 5, pp. 360--364). In particular, (B3) allows for a regularity lift such that
$$||\nabla \partial_t u_0||_{L^2(I\times\Omega)}+||\nabla \partial_t v_0||_{L^2(I\times\Omega\times Y)}\leq c.$$
For our purpose, we only need
\begin{eqnarray}
||\nabla \partial_t u_0^\epsilon||_{L^2(I\times\Omega_h^\epsilon)}\leq c,\\
||u_0^\epsilon||_{L^2(I;H^3(\Omega_h^\epsilon))}\leq c\label{regi}
\end{eqnarray}
where $u_0^\epsilon$ is the macroscopic reconstruction of $u_0$. Quite probably (\ref{regi}) could be relaxed to $||u_0^\epsilon||_{L^2(I;H^{2+\theta}(\Omega_h^\epsilon))}$ for some $\theta>0$, but we don't address here this issue.  Since we rather wish that the reader focusses on our strategy of getting the correctors,  we omit to show the proof details for this regularity result.
\end{proof}

\section{Proof of Theorem \ref{MR}}\label{Proofs}
In this section, we give the proof of the main result of our paper,
i.e. of Theorem \ref{MR}. The proof uses the auxiliary results
stated in the lemmas below. They mainly concern integral estimates for rapidly oscillating functions with prescribed average; Related estimates can be found, for instance, in  \cite{Chechkin} and Section 1.5 in \cite{CPS}.
\begin{lemma}\label{lem1} Assume the hypothesis of Theorem \ref{MR}
to hold. Then
\begin{align*}
\int_{Y(x)}\Big(\nabla_x\cdot((I+\nabla_yM)\nabla_xu_0)-\frac{1}{\theta(x)}\nabla_x\cdot\int_{Y(x)}(I+\nabla_yM)\nabla_xu_0\,dy\Big)\,dy=\\
-\int_{\partial B(x)}\nu_1\cdot(I+\nabla_yM)\nabla_xu_0\,d\sigma.
\end{align*}
\end{lemma}
\begin{proof}
We compute
\begin{align*}
\int_{Y(x)}\Big(\nabla_x\cdot((I+\nabla_yM)&\nabla_xu_0)-\frac{1}{\theta(x)}\nabla_x\cdot\int_{Y(x)}(I+\nabla_yM)\nabla_xu_0\,dy\Big)\,dy=\\
&\int_{Y(x)}\nabla_x\cdot((I+\nabla_yM)\nabla_xu_0\,dy-\nabla_x\cdot\int_{Y(x)}(I+\nabla_yM)\nabla_xu_0\,dy.
\end{align*}
Reynolds's transport theorem (see for instance \cite{Gurtin}) gives
\begin{align*}
\nabla_x\cdot\int_{Y(x)}(I+\nabla_yM)\nabla_xu_0\,dy=\int_{Y(x)}\nabla_x\cdot((I+\nabla_yM)\nabla_xu_0)\,dy\\
+\int_{\partial B(x)}\frac{\nabla_x S}{|\nabla_yS|}(I+\nabla_y M)\nabla_xu_0\,d\sigma.
\end{align*}
By the boundary condition in \eqref{cellpu}, we have
\begin{align*}
\nu_0(I+\nabla_yM)=\frac{\nabla_yS}{|\nabla_yS|}(I+\nabla_yM)=0\,\,\,\mbox{on}\,\,\,\partial B(x),
\end{align*}
so that we can write, using \eqref{nu11},
\begin{align*}
\frac{\nabla_x S}{|\nabla_yS|}(I+\nabla_y M)\nabla_xu_0&=
\Bigg(
\frac{\nabla_x S}{|\nabla_yS|}-
\frac{\nabla_xS\cdot\nabla_yS}{|\nabla_yS|^2}
\frac{\nabla_yS}{|\nabla_yS|}
\Bigg)(I+\nabla_y M)\nabla_xu_0\\
&=\nu_1\cdot(I+\nabla_yM)\nabla_xu_0
\end{align*}
on $\partial B(x)$.
Combining the above expressions proves the conclusion of this lemma.
\end{proof}

\begin{lemma} \label{lem2} Assume the hypothesis of Theorem \ref{MR}
to hold. Let $Q(x,y)\in L^2(\Omega;L^2(B(x)))$ and $p\in
 L^2(\Omega;L^2(\partial B(x)))$. Furthermore, suppose that
$\int_{Y(x)}Q(x,y)\,dy=\int_{\partial Y(x)}p(x,y)\,d\sigma$. Then
the inequality
\begin{align*}
\left|\int_{\Omega^\epsilon_h}Q(x,x/\epsilon)\phi(x)\, dx-\epsilon\int_{\Gamma^\epsilon} p(x,x/\epsilon)\phi(x)\, ds \right|\leq C\epsilon\|\phi\|_{H^1(\Omega^\epsilon_h)}
\end{align*}
holds for every $\phi\in H^1(\Omega^\epsilon_h;\partial\Omega)$. The
constant $C$ does not depend on the choice of $\epsilon$.
\end{lemma}
\begin{proof}
The problem
\begin{align*}
&\Delta_y\Psi(x,y)=Q(x,y)\,\,\,\,\mbox{in}\,\,\,Y(x) \\
&\nu_0\cdot{\nabla_y}\Psi=p(x,y)\,\,\,\,\mbox{on} \,\,\,\partial B(x)
\end{align*}
has a 1-periodic in $y$ solution that is unique up to an additive constant.
We multiply the first equation above with $\phi$ and integrate over $\Omega^\epsilon_h$ and use the thus obtained equality to get:
\begin{align*}
&\left|\int_{\Omega^\epsilon_h}Q(x,x/\epsilon)\phi(x)\, dx-\epsilon\int_{\Gamma^\epsilon} p(x,x/\epsilon)\phi(x)\, ds
\right|=\\
&\left|\int_{\Omega^\epsilon_h}\Delta_y\Psi(x,y)|_{y=\frac{x}{\epsilon}}\phi(x)\,dx-\epsilon\int_{\Gamma^\epsilon} p(x,x/\epsilon)\, ds\right|=\\
&\left|\epsilon\int_{\Omega^\epsilon_h}\Big(\nabla_x[\nabla_y\Psi(x,y)|_{y=\frac{x}{\epsilon}}]-\nabla_x\nabla_y\Psi(x,y)|_{y=\frac{x}{\epsilon}}\Big)\phi(x)\,dx-\epsilon\int_{\Gamma^\epsilon} p(x,x/\epsilon)\, ds\right|=\\
&\left|\epsilon\int_{\Gamma^\epsilon}(\nu_0+\epsilon\nu^\epsilon_1)\cdot\nabla_y\Psi(a,y)|_{y=\frac{x}{\epsilon}}\phi(x)\,ds
-\epsilon\int_{\Omega^\epsilon_h}\nabla_y\Psi|_{y=\frac{x}{\epsilon}}\nabla_x\phi(x)\,dx\right.\\
&\,\,\,\,\,\,\,\left.-\epsilon\int_{\Omega^\epsilon_h}\nabla_x\nabla_y\Psi(x,y)|_{y=\frac{x}{\epsilon}}\phi(x)\,dx-\epsilon\int_{\Gamma^\epsilon} p(x,x/\epsilon)\, ds\right|\leq\\
&\epsilon^2\left|\int_{\Gamma^\epsilon}\nu^\epsilon_1\cdot\nabla_y\Psi(a,y)|_{y=\frac{x}{\epsilon}}\phi(x)\,ds\right|+\epsilon\left|\int_{\Omega^\epsilon_h}\nabla_y\Psi|_{y=\frac{x}{\epsilon}}\nabla_x\phi(x)\,dx\right|\\
&\,\,\,\,\,\,\,+\epsilon\left|\int_{\Omega^\epsilon_h}\nabla_x\nabla_y\Psi(x,y)|_{y=\frac{x}{\epsilon}}\phi(x)\,dx\right|\leq\epsilon C
\|\phi\|_{H^1(\Omega^\epsilon_h)}
\end{align*}
The lemma is now proved.
\end{proof}

The last auxiliary lemma is a special case of Lemma 4 in \cite{Chechkin}, and therefore we will state it here without proof.

\begin{lemma}\label{lemch}
Let $\Pi_\epsilon$ be a subset of $\{x\in\Omega\,|\,\mbox{dist}(x,\partial\Omega)\leq
\frac{\sqrt{2}}{2}\epsilon\}$. Then the following inequality
\begin{align*}
\left|\int_{\Pi_\epsilon}\nabla_xu_0\phi\,dx\right|\leq C \epsilon^{3/2}
\|\phi\|_{H^1(\Omega_h^\epsilon)}
\end{align*}
holds for all $\phi\in H^1(\Omega_h^\epsilon;\partial\Omega)$. The constant $C$ does not depend on $\epsilon$.
\end{lemma}

\begin{proof}(of Theorem \ref{MR})
We define
\begin{align*}
&z_\epsilon(x,t):=u_0(x,t)+\epsilon \chi_\epsilon(x) M(x,x/\epsilon)\nabla u_0(x,t)-u_\epsilon(x,t),\\
&w_\epsilon(x,t):=v_0(x,x/\epsilon,t)-v_\epsilon(x,t).
\end{align*}
By, e.g., Theorem 4 in \cite{Lukkassen}, the functions $M(x,x/\epsilon)$ and $v_0(x,x/\epsilon,t)$ are well-defined functions in $H^1(\Omega_h^\epsilon)$ and $L^2(I;H^1(\Omega^\epsilon_l))$, respectively, and furthermore, we know
by construction that there exists a $C>0$ such that
\begin{eqnarray}
&\|z_\epsilon\|_{L^2(I;H^1(\Omega^\epsilon_h))}\leq C,\label{EB1}\\
&\|w_\epsilon\|_{L^2(I;H^1(\Omega^\epsilon_l))}\leq \epsilon
C,\label{EB2}
\end{eqnarray}
where the constant $C$ is independent of the choice of $\epsilon$.
In the following we will use the notation $u_1(x,y,t)=M(x,y)\nabla u_0(x,t)$.
We compute:
\begin{align*}
\Delta z_\epsilon(x,t)=&\Delta u_0(x,t)+\epsilon \chi_\epsilon \Delta_xu_1(x,y,t)|_{y=\frac{x}{\epsilon}} + 2 \chi_\epsilon \nabla_x\cdot\nabla_y u_1(x,y,t)|_{y=\frac{x}{\epsilon}} \\
&
+ \frac{1}{\epsilon}\chi_\epsilon \Delta_y
u_1(x,y,t)|_{y=\frac{x}{\epsilon}}+\epsilon\Delta\chi_\epsilon u_1(x,y,t)|_{y=\frac{x}{\epsilon}}-\Delta u_\epsilon(x,t)\\
&+2\epsilon \nabla \chi_\epsilon \cdot \nabla_x u_1(x,y,t)|
_{y=\frac{x}{\epsilon}}
+ 2\nabla \chi_\epsilon \cdot \nabla_y u_1(x,y,t)|_{y=\frac{x}{\epsilon}}  \\
\Delta_x w_\epsilon(x,\dfrac{x}{\epsilon},t)=&\Delta_xv_0(x,t)
+ \epsilon^{-1}\nabla_x\cdot\nabla_y v_0(x,y,t)|_{y=\frac{x}{\epsilon}}\\
&+\epsilon^{-1}\nabla_y\cdot\nabla_x v_0(x,y,t)|_{y=\frac{x}{\epsilon}}+ \frac{1}{\epsilon^2}\Delta_y
v_0(x,y,t)|_{y=\frac{x}{\epsilon}}-\Delta_xv_\epsilon(x,y,t)|_{y=\frac{x}{\epsilon}}.
\end{align*}
We use that
\begin{align*}
\theta(x)\partial_tu_0-\nabla_x\cdot({\cal D}(x)\nabla_x u_0 -\bar{q}u_0)=-\int_{\partial B(x)}
\nu_0\cdot(D_l\nabla_y v_0)\,d\sigma & \mbox{in}\,\, \Omega
\end{align*}
and $\Delta_y u_1(x,y)=0$ and $\partial_t u_\epsilon = \nabla\cdot(D_h \nabla u_\epsilon-q_\epsilon u_\epsilon)$ to obtain
\begin{align*}
D_h\Delta_x z_\epsilon-\partial_t z_\epsilon=
&D_h \Delta_xu_0
+ \epsilon \chi_\epsilon D_h\Delta_x u_1
+2\chi_\epsilon D_h\nabla_x\cdot\nabla_y u_1
+\epsilon D_h \Delta\chi_\epsilon u_1\\
&+2\epsilon D_h\nabla \chi_\epsilon \cdot \nabla_x u_1
+ 2 D_h \nabla \chi_\epsilon \cdot \nabla_y u_1\\
&-\frac{1}{\theta}\Big(\nabla_x\cdot({\cal D}(x)\nabla_x u_0-\bar{q}u_0)-\int_{\partial B(x)}
\nu_0\cdot(D_l\nabla_yv_0)\,d\sigma\Big)-\epsilon\chi_\epsilon\partial_t u_1\\
&-\nabla\cdot(q_\epsilon u_\epsilon),\\
\epsilon^2D_l\Delta_x w_\epsilon-\partial_t w_\epsilon=&\epsilon^2D_l \Delta_xv_0 +\epsilon D_l(\nabla_x\cdot\nabla_y v_0+\nabla_y\cdot\nabla_x v_0).
\end{align*}
On $\Gamma^\epsilon$ we have
\begin{align*}
\nu^\epsilon\cdot \nabla z_\epsilon=&-\nu^\epsilon\cdot \nabla u_\epsilon+\nu^\epsilon\cdot\nabla_x u_0+\epsilon \nu^\epsilon\cdot \nabla_x u_1+\nu^\epsilon\cdot \nabla_y u_1\\
=&-\epsilon^2\frac{D_l}{D_h}\nu^\epsilon\cdot\nabla v_\epsilon
+\nu^\epsilon\cdot\nabla_x u_0+\epsilon \nu^\epsilon\cdot \nabla_x u_1+\nu^\epsilon\cdot \nabla_y u_1,\\
\nu^\epsilon\cdot \nabla w_\epsilon=&-\nu^\epsilon\cdot \nabla v_\epsilon+\nu^\epsilon\cdot\nabla_x v_0+\epsilon^{-1}\nu^\epsilon\cdot \nabla_y v_0.
\end{align*}
Now, we multiply with $\phi$ and integrate by parts to get
\begin{eqnarray}\label{weakz}
\int_{\Omega^\epsilon_h}\partial_t z_\epsilon \phi \,dx&+&
\int_{\Omega^\epsilon_h}(D_h\nabla z_\epsilon \cdot \nabla \phi+q^\epsilon\cdot\nabla z_\epsilon \phi) \,dx=\epsilon\int_{\Omega^\epsilon_h}\chi_\epsilon \partial_t u_1\phi\,dx
-\int_{\Omega^\epsilon_h}D_h \Delta_xu_0\phi\,dx\nonumber\\
&-&\epsilon\int_{\Omega^\epsilon_h}D_h\chi_\epsilon \Delta_x u_1\phi\,dx
-2\int_{\Omega^\epsilon_h} D_h \chi_\epsilon \nabla_x\cdot\nabla_y u_1 \phi\,dx\nonumber\\
&+&\int_{\Omega^\epsilon_h}\frac{1}{\theta}\Big(\nabla_x\cdot({\cal D}(x)\nabla_x u_0)-\int_{\partial B(x)}
\nu_0\cdot(D_l\nabla_yv_0)\,d\sigma\Big)\phi\,dx \\
&+&\epsilon^2\int_{\Gamma^\epsilon}D_l\nu^\epsilon\cdot\nabla v_\epsilon \phi\,ds
-D_h\int_{\Gamma^\epsilon}\nu^\epsilon\cdot\nabla_x u_0\phi\,ds\nonumber\\
&-&\epsilon D_h\int_{\Gamma^\epsilon}\nu^\epsilon\cdot \nabla_x u_1\phi\,ds-D_h\int_{\Gamma^\epsilon}\nu^\epsilon\cdot \nabla_y u_1\phi\,ds\nonumber \\
&-&\int_{\Omega_h^\epsilon}(\epsilon D_h \Delta\chi_\epsilon u_1+2\epsilon D_h\nabla \chi_\epsilon \cdot \nabla_x u_1
+2 D_h \nabla \chi_\epsilon \cdot \nabla_y u_1)\phi\,dx\nonumber\\
&-&\int_{\Omega_h^\epsilon}(\frac{1}{\theta}\bar{q}-q_\epsilon)\cdot\nabla u_0\phi\,dx
-\epsilon\int_{\Omega_h^\epsilon}\chi_\epsilon u_1 q_\epsilon \cdot \nabla \phi\,dx,
\end{eqnarray}
and
\begin{eqnarray}\label{weakw}
\int_{\Omega^\epsilon_l}\partial_t w_\epsilon \psi \,dx&+&
\epsilon^2 \int_{\Omega^\epsilon_l}D_l\nabla w_\epsilon \cdot \nabla \psi \,dx=-\epsilon^2\int_{\Omega^\epsilon_l}D_l \Delta_xv_0\psi\,dx\nonumber\\
&-&\epsilon\int_{\Omega^\epsilon_l} D_l(\nabla_x\cdot\nabla_y v_0+\nabla_y\cdot\nabla_x v_0) \psi\,dx
-\epsilon^2\int_{\Gamma^\epsilon}D_l\nu^\epsilon\cdot\nabla v_\epsilon \psi\,ds\nonumber\\
&+&\epsilon^2\int_{\Gamma^\epsilon}D_l\nu^\epsilon\cdot\nabla_x v_0 \psi\,ds+\epsilon\int_{\Gamma^\epsilon}D_l\nu^\epsilon\cdot\nabla_y v_0 \psi\,ds.
\end{eqnarray}
We take into account the identity
\begin{align*}
\big(\nabla_y\cdot \nabla_x u_1(x,y)\big)|_{y=x/\epsilon}=
\epsilon \nabla_x\cdot(\nabla_x u_1(x,x/\epsilon))-\epsilon\big(\Delta_x u_1(x,y)\big)|_{y=x/\epsilon},
\end{align*}
which gives
\begin{align*}
\epsilon D_h\int_{\Gamma^\epsilon}\nu^\epsilon\cdot\nabla_xu_1|_{y=x/\epsilon} z_\epsilon\,ds=&\epsilon D_h\int_{\Omega_h^\epsilon} \nabla_x u_1|_{y=x/\epsilon} \cdot \nabla (\chi_\epsilon z_\epsilon)
+ \chi_\epsilon z_\epsilon \nabla_x\cdot(\nabla_xu_1|_{y=x/\epsilon}) \,dx\\
=&\epsilon D_h\int_{\Omega_h^\epsilon} \nabla_x u_1 \cdot \nabla
(\chi_\epsilon z_\epsilon) +\chi_\epsilon z_\epsilon\Delta_xu_1 \,dx\\
&+D_h\int_{\Omega_h^\epsilon}\chi_\epsilon z_\epsilon \nabla_y\cdot\nabla_xu_1\, dx,
\end{align*}
and also the boundary condition $\nu_0 \cdot \nabla_yu_1 =-\nu_0\cdot \nabla_xu_0$ for $y\in \partial B(x)$, which holds also on $\Gamma^\epsilon$, we add the two equations \eqref{weakz} and \eqref{weakw}, substitute $\phi=z_\epsilon$ and $\psi=w_\epsilon$ and obtain
\begin{align*}
&\frac{1}{2}\partial_t\|z_\epsilon\|^2_{L^2(\Omega_h^\epsilon)}+\frac{1}{2}\partial_t\|w_\epsilon\|^2_{L^2(\Omega_l^\epsilon)}
+\int_{\Omega_h^\epsilon}(D_h\nabla z_\epsilon\cdot \nabla z_\epsilon+q^\epsilon\cdot\nabla z_\epsilon)\,dx
+\epsilon^2
D_l\|\nabla w_\epsilon\|^2_{L^2(\Omega_l^\epsilon)} =\\
&-\int_{\Omega^\epsilon_h}D_h \Delta_xu_0z_\epsilon\,dx
-\int_{\Omega^\epsilon_h} \chi_\epsilon D_h(\nabla_x\cdot\nabla_y u_1) z_\epsilon \,dx+\int_{\Omega^\epsilon_h}\frac{1}{\theta}\nabla_x\cdot({\cal D}(x)\nabla_x u_0)z_\epsilon\,dx\\
&-\epsilon D_h\int_{\Gamma^\epsilon}\nu^\epsilon_1\cdot(\nabla_x u_0+\nabla_yu_1)z_\epsilon\,ds\\
&-\int_{\Omega^\epsilon_h}(\int_{\partial B(x)}
\nu_0\cdot(D_l\nabla_yv_0)\,d\sigma)z_\epsilon\,dx+\epsilon\int_{\Gamma^\epsilon}D_l\nu^\epsilon\cdot\nabla_y v_0 z_\epsilon\,ds\\
&+\epsilon\int_{\Omega^\epsilon_h}\chi_\epsilon \partial_t u_1z_\epsilon\,dx-\epsilon\int_{\Omega^\epsilon_h}D_h\nabla_x u_1 \cdot\nabla(\chi_\epsilon z_\epsilon)\,dx\\
&-\epsilon^2\int_{\Omega^\epsilon_l}D_l \Delta_xv_0w_\epsilon\,dx
-\epsilon\int_{\Omega^\epsilon_l} D_l(\nabla_x\cdot\nabla_y v_0+\nabla_y\cdot\nabla_x v_0) w_\epsilon\,dx
+\epsilon^2\int_{\Gamma^\epsilon}D_l\nu^\epsilon\cdot\nabla_x v_0 w_\epsilon\,ds\\
&+\epsilon^2\int_{\Gamma^\epsilon}D_l\nu^\epsilon\cdot(\epsilon\nabla v_\epsilon-\nabla_yv_0)u_1\,ds -\int_{\Omega_h^\epsilon}(\epsilon D_h \Delta\chi_\epsilon u_1+2\epsilon D_h\nabla \chi_\epsilon \cdot \nabla_x u_1
+2 D_h \nabla \chi_\epsilon \cdot \nabla_y u_1)z_\epsilon \,dx\\
&-\int_{\Omega_h^\epsilon}(\frac{1}{\theta}\bar{q}-q_\epsilon)\cdot\nabla u_0z_\epsilon\,dx
-\epsilon\int_{\Omega_h^\epsilon}\chi_\epsilon u_1 q_\epsilon \cdot \nabla z_\epsilon\,dx,
\end{align*}
where we have also used that $\phi-\psi=z_\epsilon-w_\epsilon=\epsilon u_1$ on $\Gamma^\epsilon$ and $\phi=\epsilon u_1$ on $\partial\Omega$.

We know that there exist $\beta>0$ and $\gamma\geq 0$ such that
\begin{align*}
\beta \|z_\epsilon\|_{H^1(\Omega_h^\epsilon)}\leq \int_{\Omega_h^\epsilon}(D_h\nabla z_\epsilon\cdot \nabla z_\epsilon+q^\epsilon\cdot\nabla z_\epsilon)\,dx+\gamma
\|z_\epsilon\|_{L^2(\Omega_h^\epsilon)},
\end{align*}
and we use this to estimate
\begin{align*}
&\frac{1}{2}\partial_t\|z_\epsilon\|^2_{L^2(\Omega_h^\epsilon)}+\frac{1}{2}\partial_t\|w_\epsilon\|^2_{L^2(\Omega_l^\epsilon)}
+\beta\|z_\epsilon\|^2_{H^1(\Omega_h^\epsilon)}+\epsilon^2
D_l\|\nabla w_\epsilon\|^2_{L^2(\Omega_l^\epsilon)} \leq \gamma\|z_\epsilon\|_{L^2(\Omega_h^\epsilon)}+ I_1+I_2+...+I_{11}
\end{align*}
where
\begin{align*}
I_1=&\left|\int_{\Omega^\epsilon_h}D_h \nabla_x\cdot(\nabla_xu_0+\nabla_y u_1)z_\epsilon\,dx
-\int_{\Omega^\epsilon_h}\frac{1}{\theta}\nabla_x\cdot({\cal D}(x)\nabla_x u_0)z_\epsilon\,dx\right.\\
&\left. +\epsilon D_h\int_{\Gamma^\epsilon}\nu_1\cdot(\nabla_x u_0+\nabla_yu_1)z_\epsilon\,ds\right|,\\
I_2=&\left|\int_{\Omega^\epsilon_h}(\int_{\partial B(x)}
\nu_0\cdot(D_l\nabla_yv_0)\,d\sigma)z_\epsilon\,dx-\epsilon\int_{\Gamma^\epsilon}D_l\nu_0\cdot\nabla_y v_0 z_\epsilon\,ds\right|,\\
I_3=&\left|\int_{\Omega^\epsilon_h} (1-\chi_\epsilon) D_h(\nabla_x\cdot\nabla_y u_1) z_\epsilon \,dx\right|,\\
I_4=&\left|\epsilon^2\int_{\Gamma^\epsilon}\Big(D_h(\nu_2+O(\epsilon))\cdot(\nabla_x u_0-\nabla_yu_1)+D_l\nu^\epsilon_1\cdot\nabla_yv_0\Big)z_\epsilon\,ds\right|,\\
I_5=&\left|\int_{\Omega_h^\epsilon}(\epsilon D_h \Delta\chi_\epsilon u_1+2\epsilon D_h\nabla \chi_\epsilon \cdot \nabla_x u_1
+2 D_h \nabla \chi_\epsilon \cdot \nabla_y u_1)z_\epsilon \,dx\right|,\\
I_6=&\left|\epsilon\int_{\Omega^\epsilon_h}\chi_\epsilon \partial_t u_1z_\epsilon\,dx\right|,\\
I_7=&\left|\epsilon\int_{\Omega^\epsilon_h}D_h\nabla_x u_1 \cdot \nabla(\chi_\epsilon z_\epsilon)\,dx\right|,\\
I_8=&\left|\epsilon^2\int_{\Omega^\epsilon_l}D_l \Delta_xv_0w_\epsilon\,dx
+\epsilon\int_{\Omega^\epsilon_l} D_l(\nabla_x\cdot\nabla_y v_0+\nabla_y\cdot\nabla_x v_0) w_\epsilon\,dx
-\epsilon^2\int_{\Gamma^\epsilon}D_l\nu^\epsilon\cdot\nabla_x v_0 w_\epsilon\,ds\right|,\\
I_9=&\left|\epsilon^2\int_{\Gamma^\epsilon}D_l\nu^\epsilon\cdot(\epsilon\nabla v_\epsilon-\nabla_yv_0)u_1\,ds\right|,\\
I_{10}=&\left|\int_{\Omega_h^\epsilon}(\frac{1}{\theta}\bar{q}-q_\epsilon)\cdot\nabla u_0z_\epsilon\,dx\right|,\\
I_{11}=&\left|\epsilon\int_{\Omega_h^\epsilon}\chi_\epsilon u_1 q_\epsilon \cdot \nabla z_\epsilon\,dx\right|.
\end{align*}
For $I_1$ we use that $u_1=M\nabla u_0$ and that ${\cal D}(x):=D_h\int_{Y(x)}(I+\nabla_yM)\,dy$. We set
\begin{align*}
&Q(x,y)=\nabla_x\cdot((I+\nabla_yM)\nabla_xu_0)-\frac{1}{\theta(x)}\nabla_x\cdot\int_{Y(x)}(I+\nabla_yM)\nabla_xu_0\,dy,\\
&p(x,y)=-\nu_1\cdot(I+\nabla_yM),
\end{align*}
and use Lemma \ref{lem2} to obtain $I_1\leq \epsilon C
\|z_\epsilon\|_{H^1(\Omega_h^\epsilon)}$. Lemma \ref{lem1} asserts
that the conditions of Lemma \ref{lem2} are satisfied for these
choices of $Q$ and $p$.

For $I_2$ we also apply Lemma \ref{lem2}, this time for the choice
\begin{align*}
&Q(x,y)=\frac{1}{\theta(x)}\int_{\partial B(x)}\nu_0\cdot\nabla_yv_0\,d\sigma,\\
&p(x,y)=\nu_0\cdot\nabla_yv_0,
\end{align*}
and we again get
$I_2\leq \epsilon C \|z_\epsilon\|_{H^1(\Omega_h^\epsilon)}$
with $C$ independent of $\epsilon$. For $I_3$ we have $I_3\leq \sqrt{\epsilon} C \|z_\epsilon\|_{H^1(\Omega_h^\epsilon)}$.
Application of the regularity results in Lemma \ref{addreg} and Theorem \ref{strongsol} results in
$I_{4}+I_6\leq C\epsilon\|z_\epsilon\|_{H^1(\Omega_h^\epsilon)}$.
With the use of Lemma \ref{lemch} and the properties of $\chi_\epsilon$ we
estimate $I_5+I_7\leq \sqrt{\epsilon} C \|z_\epsilon\|_{H^1(\Omega_h^\epsilon)}$. Another application of the regularity results gives $I_8 \leq \epsilon C\|w_\epsilon\|_{H^1(\Omega_l^\epsilon)}$ and $I_9\leq C\epsilon \|u_0\|_{H^2(\Omega_h^\epsilon)}(\|v_\epsilon\|_{H^2(\Omega_l^\epsilon)}+\|v_0\|_{H^2(\Omega;H^2(B(x)))})$.
For $I_{10}$ and $I_{11}$ we use the assumptions on $q$ and $q^\epsilon$ (as stated in (A2) and (B2)) to get $I_{10}+I_{11}\leq \sqrt{\epsilon}C\|z_\epsilon\|_{H^1(\Omega_h^\epsilon)}$.

Now we obtain
\begin{align*}
&\frac{1}{2}\partial_t\|z_\epsilon\|^2_{L^2(\Omega_h^\epsilon)}+\frac{1}{2}\partial_t\|w_\epsilon\|^2_{L^2(\Omega_l^\epsilon)}
+D_h\|\nabla z_\epsilon\|^2_{L^2(\Omega_h^\epsilon)}+\epsilon^2
D_l\|\nabla w_\epsilon\|^2_{L^2(\Omega_l^\epsilon)}\leq \|z_\epsilon\|_{L^2(\Omega_h^\epsilon)}+\\
&\epsilon C_1\|u_0\|_{H^2(\Omega_h^\epsilon)}(\|v_\epsilon\|_{H^2(\Omega_l^\epsilon)}+\|v_0\|_{H^2(\Omega;H^2(B(x)))})+\sqrt{\epsilon}C_2\|z_\epsilon\|_{H^1(\Omega_h^\epsilon)}+\epsilon C_3 \|w_\epsilon\|_{H^1(\Omega_l^\epsilon)}
\end{align*}
Using the energy bounds (\ref{EB1}) and  (\ref{EB2}) together with
a Gronwall-type argument lead to
\begin{align}
\|z_\epsilon\|^2_{L^\infty(I,L^2(\Omega_h^\epsilon))}+&
\|w_\epsilon\|^2_{L^\infty(I,L^2(\Omega_l^\epsilon))}+
\|z_\epsilon\|^2_{L^2(I,H^1(\Omega_h^\epsilon))}+\nonumber\\
&+\epsilon\|w_\epsilon\|^2_{L^2(I,H^1(\Omega_l^\epsilon))}\label{inest}\leq \epsilon \tilde{C}_1 +\sqrt{\epsilon}\tilde{C}_2 \|z_\epsilon\|_{L^2(I,H^1(\Omega_h^\epsilon))},
\end{align}
which, in particular, implies
\begin{align*}
\|z_\epsilon\|^2_{L^2(I,H^1(\Omega_h^\epsilon))}
\leq \epsilon \tilde{C}_1 +\sqrt{\epsilon}\tilde{C}_2 \|z_\epsilon\|_{L^2(I,H^1(\Omega_h^\epsilon))}.
\end{align*}
and thus
\begin{align*}
\|z_\epsilon\|_{L^2(I,H^1(\Omega_h^\epsilon))}
\leq \sqrt{\epsilon}\frac{1}{2}( \tilde{C}^2_2+\sqrt{\tilde{C}_2+4\tilde{C}_1}).
\end{align*}
Combining this with \eqref{inest} gives the result
\begin{align*}
\|z_\epsilon\|_{L^\infty(I,L^2(\Omega_h^\epsilon))}+
\|w_\epsilon\|_{L^\infty(I,L^2(\Omega_l^\epsilon))}+
\|z_\epsilon\|_{L^2(I,H^1(\Omega_h^\epsilon))}+&
\epsilon\|w_\epsilon\|_{L^2(I,H^1(\Omega_l^\epsilon))}\leq
c\sqrt{\epsilon},
\end{align*}
where the constant $c$ is independent of $\epsilon$.
The last step uses the evident estimate $\|\epsilon u_1(1-\chi_\epsilon)\|_{H^1(\Omega_h^\epsilon)}\leq C\sqrt{\epsilon}$, and the theorem is proven.
\end{proof}

\section*{Acknowledgments}
We acknowledge fruitful  discussions with Gregory Chechkin regarding homogenization techniques for non-periodic media.
We also thank Eduard Marusic-Paloka for an interesting correspondence on the best corrector estimates (upper bounds on convergence rates) existing
for the stationary Stokes and Navier-Stokes problems.

\bibliographystyle{siam}

redpajama_set_name: RedPajamaArXiv


text:
Obama's dangerous claim to executive power
Betsy McCaughey Author, "Beating Obamacare"
October 09, 2013 10:58 AM ET
Negotiating with Republicans has never been in Obama's playbook. Three days after his inauguration in 2009, President Barack Obama silenced Republican lawmakers who voiced concerned about the enormity of spending in his stimulus bill by uttering two brash words, "I won."
That was his governing philosophy, as he rammed through the American Reinvestment and Recovery Act.
But in the fall of 2010. Republicans swept into control of the House of Representatives, and since then Obama's agenda has been stalled. With little prospect of gaining control of the House in 2014, Obama is resorting to discrediting the Constitution's limits on presidential power rather than bargain with Congressional Republicans.
Here are the President's own words, explaining why he refuses to negotiate to end the government shutdown and resolve the fast approaching debt ceiling problem. His views on presidential power ought to alarm all Americans.
"I will not pay ransom" said Obama, for a stopgap spending bill to open the government.
Historically presidents have had to make concessions to Congress to secure funding. President Reagan endured eight shutdowns. Each time he negotiated with the Democratic Speaker of the House, Tip O'Neill, conceding on issues from mid-range missile defense to support for Nicaraguan contras, to quickly end the shutdown.
Fast forward to the current showdown. The House Republicans' fourth (and latest) offer asks for only two changes in Obamacare: First, eliminate the subsidy for members of Congress, which has outraged the public. And second, delay for a year making insurance mandatory for individuals. Let anyone enroll in Obamacare who wants to. But don't penalize individuals for being uninsured in 2014 when the president has already postponed the penalty on big companies for not insuring workers.
"The Affordable Care Act is a law that passed the House, it passed the Senate, the Supreme Court ruled it constitutional … and it is settled," said Obama.
Not so. The health program the president is imposing on us is not the Affordable Care Act. The president has dismembered and mangled it. Gone is the employer mandate, the cap on out of pocket expenses, income verification and over half the deadlines specified in the law. The President delayed or did away with these features, without asking Congress. Illegally. Then he added 1,472 waivers and connived a subsidy for members of Congress that no one else in American earning $174,000 a year could get. The Supreme Court has ruled twice that presidents cannot delay, amend, and repeal parts of laws.
"It has not been done in the past and we're not going to start doing it now," said Obama, explaining why he will not negotiate with House Republicans over raising the debt ceiling.
The truth is, presidents have often had to make concessions to get the debt ceiling raised. In 1995 Speaker of the House Newt Gingrich tried to extract major spending reforms from President Bill Clinton. The two sides sparred for more than a week past the "deadline" for a hike. That is what the framers intended.
"That's not how our constitutional system is designed," the president said disingenuously. James Madison, a main designer of that constitutional system explained in Federalist 58 that the president must come to Congress as a supplicant to borrow, tax, or spend. That dependence on Congress is "the most effectual tool to remedy" any grievance the people could have against their president.
That brings us to the nefarious proposal taking shape on Capitol Hill. Senate Democrats reportedly will offer a bill shifting to the president the discretion to raise the debt ceiling any time through 2014, except if two-thirds of each house of Congress vote to disapprove. This device, first tried in the Budget Control Act of 2011, weasels around the U. S. Constitution's system of checks and balances and substantially enlarges executive power.
Who is Obama, the 44th president, to demand more borrowing latitude than his 43 precedessors?.
As Madison warned in Federalist 62, "an elective despot is not what we fought for."
Tags : barack obama betsy mccaughey government shutdown james madison