Nonlinear partial differential equation

In mathematics and physics, a nonlinear partial differential equation is a partial differential equation with nonlinear terms. They describe many different physical systems, ranging from gravitation to fluid dynamics, and have been used in mathematics to solve problems such as the Poincaré conjecture and the Calabi conjecture. They are difficult to study: almost no general techniques exist that work for all such equations, and usually each individual equation has to be studied as a separate problem.

The distinction between a linear and a nonlinear partial differential equation is usually made in terms of the properties of the operator that defines the PDE itself.[1]

Methods for studying nonlinear partial differential equations

Existence and uniqueness of solutions

A fundamental question for any PDE is the existence and uniqueness of a solution for given boundary conditions. For nonlinear equations these questions are in general very hard: for example, the hardest part of Yau's solution of the Calabi conjecture was the proof of existence for a Monge–Ampere equation. The open problem of existence (and smoothness) of solutions to the Navier–Stokes equations is one of the seven Millennium Prize problems in mathematics.

Singularities

The basic questions about singularities (their formation, propagation, and removal, and regularity of solutions) are the same as for linear PDE, but as usual much harder to study. In the linear case one can just use spaces of distributions, but nonlinear PDEs are not usually defined on arbitrary distributions, so one replaces spaces of distributions by refinements such as Sobolev spaces.

An example of singularity formation is given by the Ricci flow: Richard S. Hamilton showed that while short time solutions exist, singularities will usually form after a finite time. Grigori Perelman's solution of the Poincaré conjecture depended on a deep study of these singularities, where he showed how to continue the solution past the singularities.

Linear approximation

The solutions in a neighborhood of a known solution can sometimes be studied by linearizing the PDE around the solution. This corresponds to studying the tangent space of a point of the moduli space of all solutions.

Moduli space of solutions

Ideally one would like to describe the (moduli) space of all solutions explicitly, and for some very special PDEs this is possible. (In general this is a hopeless problem: it is unlikely that there is any useful description of all solutions of the Navier–Stokes equation for example, as this would involve describing all possible fluid motions.) If the equation has a very large symmetry group, then one is usually only interested in the moduli space of solutions modulo the symmetry group, and this is sometimes a finite-dimensional compact manifold, possibly with singularities; for example, this happens in the case of the Seiberg–Witten equations. A slightly more complicated case is the self dual Yang–Mills equations, when the moduli space is finite-dimensional but not necessarily compact, though it can often be compactified explicitly. Another case when one can sometimes hope to describe all solutions is the case of completely integrable models, when solutions are sometimes a sort of superposition of solitons; this happens e.g. for the Korteweg–de Vries equation.

Exact solutions

It is often possible to write down some special solutions explicitly in terms of elementary functions (though it is rarely possible to describe all solutions like this). One way of finding such explicit solutions is to reduce the equations to equations of lower dimension, preferably ordinary differential equations, which can often be solved exactly. This can sometimes be done using separation of variables, or by looking for highly symmetric solutions.

Some equations have several different exact solutions.

Numerical solutions

Numerical solution on a computer is almost the only method that can be used for getting information about arbitrary systems of PDEs. There has been a lot of work done, but a lot of work still remains on solving certain systems numerically, especially for the Navier–Stokes and other equations related to weather prediction.

Lax pair

If a system of PDEs can be put into Lax pair form

then it usually has an infinite number of first integrals, which help to study it.

Euler–Lagrange equations

Systems of PDEs often arise as the Euler–Lagrange equations for a variational problem. Systems of this form can sometimes be solved by finding an extremum of the original variational problem.

Hamilton equations

Integrable systems

PDEs that arise from integrable systems are often the easiest to study, and can sometimes be completely solved. A well-known example is the Korteweg–de Vries equation.

Symmetry

Some systems of PDEs have large symmetry groups. For example, the Yang–Mills equations are invariant under an infinite-dimensional gauge group, and many systems of equations (such as the Einstein field equations) are invariant under diffeomorphisms of the underlying manifold. Any such symmetry groups can usually be used to help study the equations; in particular if one solution is known one can trivially generate more by acting with the symmetry group.

Sometimes equations are parabolic or hyperbolic "modulo the action of some group": for example, the Ricci flow equation is not quite parabolic, but is "parabolic modulo the action of the diffeomorphism group", which implies that it has most of the good properties of parabolic equations.

List of equations

See the extensive List of nonlinear partial differential equations.

See also

References

  1. ^ Logan, J. David (1994). An Introduction to Nonlinear Partial Differential Equations. New York: John Wiley & Sons. pp. 8–11. ISBN 0-471-59916-6.

Read other articles:

German police official and head of the Gestapo (1939–1945) Heinrich MüllerDirector of the GestapoIn office27 September 1939 – 1 May 1945Appointed byHeinrich HimmlerPreceded byReinhard HeydrichSucceeded bynone (Office abolished) Personal detailsBorn(1900-04-28)28 April 1900Munich, Kingdom of BavariaGerman EmpireDiedMay 1945 (aged 45)Berlin (assumed)Civilian awardsGolden Party BadgeNicknameGestapo MüllerMilitary serviceAllegiance German Empire Nazi GermanyServiceGerman ...

 

Ninoy Aquino DayBenigno Ninoy Aquino, Jr.Nama resmiMemperingati pembunuhan senator Benigno Ninoy Aquino, Jr.Dirayakan olehFilipinaTanggal21 AgustusFrekuensitahunan Hari Ninoy Aquino adalah hari libur nasional di Filipina dirayakan setahun sekali pada tanggal 21 Agustus, untuk memperingati pembunuhan mantan senator, Benigno Ninoy Aquino, Jr. Dia adalah suami dari Corazon Aquino, yang kemudian menjadi Presiden Filipina. Mereka berdua dianggap sebagai pahlawan demokrasi di Filipina. Pembunuhanny...

 

В Википедии есть статьи о других людях с фамилией Целиковская. Тамара Алексеевна Целиковскаяукр. Тамара Олексіївна Целіківська Основные сведения Страна  СССР→ Украина Дата рождения 17 ноября 1935(1935-11-17) Место рождения Елец, Липецкая область, РСФСР, СССР Дата ...

Boleto de las elecciones generales de Oklahoma del 2018. Las elecciones de medio mandato de Estados Unidos (mid-term election) son las elecciones generales que se efectúan el martes siguiente al primer lunes de noviembre cada dos años, en el punto medio de la legislatura de cuatro años de un presidente. En esos comicios se eligen los 435 escaños de la Cámara de Representantes de Estados Unidos, y 33 o 34 de los 100 del Senado de Estados Unidos. Además, en estas elecciones 34 de los 50 e...

 

Họ Nhím lông Cựu Thế giớiThời điểm hóa thạch: Tiền Miocen - gần đâyNhím lông cứng Cựu thế giớiPhân loại khoa họcGiới (regnum)AnimaliaNgành (phylum)ChordataLớp (class)MammaliaBộ (ordo)RodentiaPhân bộ (subordo)HystricomorphaPhân thứ bộ (infraordo)HystricognathiHọ (familia)HystricidaeFischer de Waldheim, 1817Các chi Atherurus Hystrix Trichys Họ Nhím lông Cựu thế giới, tại Việt Nam đơn giản gọi là họ Nhím[1] ...

 

602nd AAA Gun BattalionMemorial in FranceActive1942–1945CountryUSABranchArmyTypeAnti Aircraft ArtilleryEngagementsDefense of Paris, and Antwerp; Battle of the BulgeCommandersCommanding OfficerLt. Colonel Blair C. ForbesExecutive OfficerMajor George E. RogersMilitary unit Creating 602nd Anti-Aircraft Artillery Gun Battalion was an Anti-aircraft artillery battalion of the United States Army during World War II.[1] The unit began in 1942 as 1st Battalion of the 602d CA (AA) in Fort Bli...

Granja Estação Ferroviária de Granjaa estação da Granja, em 2010 Identificação: 39040 GJA (Granja)[1] Denominação: Estação de Granja Administração: Infraestruturas de Portugal (norte)[2] Classificação: E (estação)[1] Tipologia: C [3] Linha(s): Linha do Norte (PK 320+394) Altitude: 13 m (a.n.m) Coordenadas: 41°2′18.74″N × 8°38′49.55″W (=+41.03854;−8.6471) Localização na rede (mais mapas: 41° 02′ 18,74″ N, 8° 38′ 49,55″ O; IGeoE)...

 

University in Tallinn Estonian Academy of Music and TheatreEstonian Academy of Music and Theatre Campus BuildingFormer nameTallinn Higher Music SchoolTypePublicEstablished1919AddressRävala puiestee 16, 10141 Tallinn, Estonia, Tallinn, Estonia59°25′52″N 24°44′52″E / 59.4312°N 24.7479°E / 59.4312; 24.7479CampusUrbanWebsiteeamt.ee The Estonian Academy of Music and Theatre (Eesti Muusika- ja Teatriakadeemia) began as a mixed choir of the Estonia Society Musica...

 

この項目では、ファッション雑誌について説明しています。その他の用法については「アンアン」をご覧ください。 この記事は検証可能な参考文献や出典が全く示されていないか、不十分です。出典を追加して記事の信頼性向上にご協力ください。(このテンプレートの使い方)出典検索?: An・an – ニュース · 書籍 · スカラー · CiNii · J-STAGE&...

Japanese fantasy novel series and its franchise The Twelve KingdomsCover of the first book (first volume) of the Kodansha edition, featuring Yoko Nakajima十二国記(Jūni Kokuki)GenreEpic fantasy[1][2]Isekai[3] Novel seriesWritten byFuyumi OnoIllustrated byAkihiro YamadaPublished byKodanshaShinchoshaEnglish publisherNA: TokyopopImprintX Bunko White Heart(#1–7)Shinchō Bunko(#8–present; reprint)DemographicFemaleOriginal run1992 – presentVolumes9 (...

 

Сухопутні війська Республіки Корея 대한민국 육군 Daehanminguk Yuk-gun Прапор Сухопутних військ Республіки КореяЗасновано 5 вересня, 1948 рокуКраїна  Південна КореяНалежність Збройні сили Республіки КореяВид сухопутні військаТип арміяЧисельність 495,000 чол. (2014 р.)Штаб СВ м. Керен, пр...

 

Railway station in Hamamatsu, Japan Okaji Station岡地駅Okaji Station (then called Kigakōkōmae Station) in March 2006General informationLocationHosoe-chō Nakagawa 4672-3, Kita-ku, Hamamatsu-shi, Shizuoka-ken 431-1304JapanCoordinates34°48′44″N 137°39′56″E / 34.81222°N 137.66556°E / 34.81222; 137.66556Operated byTenryū Hamanako RailroadLine(s)■ Tenryū Hamanako LineDistance43.5 kilometers from KakegawaPlatforms1 side platformOther informationStatusUns...

SMA Unggul DelInformasiDidirikan10 April 2012JenisSwastaAkreditasiAkreditasi A (2020)[1]Kepala SekolahArini Desianti Parawi, S.PdModeratorDany Robson ManurungJumlah kelasKelas setiap tingkat : X PIIS : 1 Kelas, X PMIA : 4 Kelas, XI PMIA : 4 kelas, XI PIIS : 1 kelas, XII PMIA : 4 kelas, XII PIIS : 1 KelasJurusan atau peminatanIPA dan IPSRentang kelasX PMIA, X PIIS , XI PMIA , XI PIIS, XII PMIA, XII PIIS,KurikulumKurikulum Tingkat Satuan Pend...

 

Artikel ini sebatang kara, artinya tidak ada artikel lain yang memiliki pranala balik ke halaman ini.Bantulah menambah pranala ke artikel ini dari artikel yang berhubungan atau coba peralatan pencari pranala.Tag ini diberikan pada Februari 2023. SMA SWASTA AL HUSNA TANGERANGInformasiDidirikan1979JenisSekolah SwastaAkreditasiBNomor Statistik Sekolah302020411058Nomor Pokok Sekolah Nasional20606551Kepala SekolahDrs. AMSARI, S.Pd., M.Pd.I.Jurusan atau peminatanIPA dan IPSRentang kelasX, XI I...

 

Casa de Salomón Cohen Vista del edificio (2017)LocalizaciónPaís EspañaUbicación Calle Cardenal Cisneros - Calle Sor Alegría, MelillaEspaña EspañaCoordenadas 35°17′39″N 2°56′36″O / 35.294303333333, -2.9434066666667Información generalEstilo modernismoDiseño y construcciónArquitecto Emilio Alzugaray[editar datos en Wikidata] La Casa de Salomón Cohen es un edificio modernista, una de las mejores obras de Emilio Alzugaray.[1]​ Está situad...

رجاء بلمليح معلومات شخصية الميلاد 22 أبريل 1962  الدار البيضاء  الوفاة 2 سبتمبر 2007 (45 سنة) [1]  الدار البيضاء  سبب الوفاة سرطان  مواطنة المغرب الإمارات العربية المتحدة  الحياة الفنية النوع موسيقى عربية الآلات الموسيقية صوت بشري  المهنة مغنية  اللغة الأم ...

 

Demographics of IslamabadPopulation pyramid of Islamabad in 2017Population1,014,825 (2017) Islamabad had an estimated population of 1,014,825 according to the 2017 Census.[1] Urdu, the national and first official language of the country, is predominantly spoken within the city due to the ethnic mix of populations. English, the second official language, is also commonly understood. Other languages include Punjabi and Pashto. The mother tongue of the majority of the population is Punjab...

 

Argentine actress Blanca Lagrotta'Born(1921-02-14)February 14, 1921October 22, 1978(1978-10-22) (aged 57)OccupationActress Blanca Lagrotta (14 February 1921 – 22 October 1978) was an Argentine actress. She starred in films such as El nieto de Congreve (1949), Deshonra (1952), Mercado de abasto (1955), Setenta veces siete (1962), Amor libre (1969), Operación Masacre (1972), Un mundo de amor (1975) and El fantástico mundo de María Montiel (1978).[1] In Leopoldo Torre Nilsson's...

Zoe Benjamin, lecturer in psychology and principles of education, c. 1925 Sophia Zoe Benjamin (24 December 1882 – 13 April 1962) was a pioneer of early childhood education in Australia. History Zoe was born in Adelaide, South Australia to Philip Benjamin (1848–1924) and his wife Miriam Minnie Benjamin, née Cohen (1852–1918), Orthodox Jews. Philip was a nephew of Judah Moss Solomon (1818–1880) and closely related to Vaiben Louis Solomon (1853–1908), Elias Solomon MLA, MHR (1839–18...

 

British comedic culinary podcast PodcastOff Menu with Ed Gamble and James AcasterPresentationHosted byEd Gamble, James AcasterGenreComedy, food, interviewLanguageEnglishLength45–120 minutesProductionNo. of episodes200PublicationOriginal release5 December 2018 (5 December 2018) Off Menu with Ed Gamble and James Acaster is a food and comedy podcast featuring Ed Gamble and James Acaster, in which guests are invited to select their dream menu by both comedians. Off Menu was launched in Dec...

 

Strategi Solo vs Squad di Free Fire: Cara Menang Mudah!