El problema de Cauchy asociado a una perturbación dispersiva de quinto orden de la ecuación de Benjamín
ilustraciones, diagramas
- Autores:
-
Correa Castañeda, Diego Fernando
- Tipo de recurso:
- Fecha de publicación:
- 2023
- Institución:
- Universidad Nacional de Colombia
- Repositorio:
- Universidad Nacional de Colombia
- Idioma:
- spa
- OAI Identifier:
- oai:repositorio.unal.edu.co:unal/84480
- Palabra clave:
- 510 - Matemáticas::515 - Análisis
ECUACIONES DIFERENCIALES PARCIALES
PROBLEMA DE CAUCHY
Differential equations, partial
Cauchy problem-8a. ed.
Buen planteamiento
Ecuaciones dispersivas
Regularización parabólica
Espacios de Bourgain
Well posedness
Dispersive equations
Parabolic regularization
Bourgain spaces
- Rights
- openAccess
- License
- Reconocimiento 4.0 Internacional
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| dc.title.spa.fl_str_mv |
El problema de Cauchy asociado a una perturbación dispersiva de quinto orden de la ecuación de Benjamín |
| dc.title.translated.eng.fl_str_mv |
The Cauchy problem associated to a fifth order perturbation of the Benjamin equation |
| title |
El problema de Cauchy asociado a una perturbación dispersiva de quinto orden de la ecuación de Benjamín |
| spellingShingle |
El problema de Cauchy asociado a una perturbación dispersiva de quinto orden de la ecuación de Benjamín 510 - Matemáticas::515 - Análisis ECUACIONES DIFERENCIALES PARCIALES PROBLEMA DE CAUCHY Differential equations, partial Cauchy problem-8a. ed. Buen planteamiento Ecuaciones dispersivas Regularización parabólica Espacios de Bourgain Well posedness Dispersive equations Parabolic regularization Bourgain spaces |
| title_short |
El problema de Cauchy asociado a una perturbación dispersiva de quinto orden de la ecuación de Benjamín |
| title_full |
El problema de Cauchy asociado a una perturbación dispersiva de quinto orden de la ecuación de Benjamín |
| title_fullStr |
El problema de Cauchy asociado a una perturbación dispersiva de quinto orden de la ecuación de Benjamín |
| title_full_unstemmed |
El problema de Cauchy asociado a una perturbación dispersiva de quinto orden de la ecuación de Benjamín |
| title_sort |
El problema de Cauchy asociado a una perturbación dispersiva de quinto orden de la ecuación de Benjamín |
| dc.creator.fl_str_mv |
Correa Castañeda, Diego Fernando |
| dc.contributor.advisor.none.fl_str_mv |
Pastrán Ramírez, Ricardo Ariel |
| dc.contributor.author.none.fl_str_mv |
Correa Castañeda, Diego Fernando |
| dc.subject.ddc.spa.fl_str_mv |
510 - Matemáticas::515 - Análisis |
| topic |
510 - Matemáticas::515 - Análisis ECUACIONES DIFERENCIALES PARCIALES PROBLEMA DE CAUCHY Differential equations, partial Cauchy problem-8a. ed. Buen planteamiento Ecuaciones dispersivas Regularización parabólica Espacios de Bourgain Well posedness Dispersive equations Parabolic regularization Bourgain spaces |
| dc.subject.lemb.spa.fl_str_mv |
ECUACIONES DIFERENCIALES PARCIALES PROBLEMA DE CAUCHY |
| dc.subject.lemb.eng.fl_str_mv |
Differential equations, partial Cauchy problem-8a. ed. |
| dc.subject.proposal.spa.fl_str_mv |
Buen planteamiento Ecuaciones dispersivas Regularización parabólica Espacios de Bourgain |
| dc.subject.proposal.eng.fl_str_mv |
Well posedness Dispersive equations Parabolic regularization Bourgain spaces |
| description |
ilustraciones, diagramas |
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2023 |
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2023-08-08T15:10:38Z |
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2023-08-08T15:10:38Z |
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2023-02-01 |
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Trabajo de grado - Maestría |
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info:eu-repo/semantics/masterThesis |
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info:eu-repo/semantics/acceptedVersion |
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Text |
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https://repositorio.unal.edu.co/handle/unal/84480 |
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Universidad Nacional de Colombia |
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Repositorio Institucional Universidad Nacional de Colombia |
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https://repositorio.unal.edu.co/ |
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Universidad Nacional de Colombia Repositorio Institucional Universidad Nacional de Colombia |
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spa |
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Iorio, Jr R. J. y Valéria de Magalhães Iorio, Fourier Analysis and Partial Differential Equations, Cambridge University Press, 2001. Kenig, Ponce y Vega, A bilinear estimate with applications to the KdV equation, Journal of the american mathematical society, Volume 9, Number 2, April 1996. Stein Elias y Shakarchi Rami, Fourier Analysis: An Introduction, Princeton University Press, (2003). Linares, F. y Ponce, G., Introduction to Nonlinear Dispersive Equations, Springer, 2009. Gleeson H. , Hammerton P., Papageorgiou D. T. , Vanden-Broeck J.-M. , A new application of the Korteweg–de Vries Benjamin-Ono equation in interfacial electrohydrodynamics, American Institute of Physics, 2007. Linares Felipe, $L^2$ Global Well-Posedness of the Initial Value Problem Associated to the Benjamin Equation, IMPA, 1998. Kenig, Ponce y Vega, The Cauchy problem for the Korteweg de Vries equation in Sobolev spaces of negative indices, Duke Math. J. 71 (1993), 1 - 21. MR 94g:35196. Bourgain, Fourier Transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Geometric and Functional Analysis, Volume 3, Number 3, 1993. Chen, Guo y Xiao, Sharp well-posedness for the Benjamin equation, Journal of the american mathematical society, Nonlinear Analysis 74 (2011) 6209–6230. Junfeng Li, Xia Li, Well-posedness for the fifth order KP-II initial data problem in $H^{s,0}(\R \times \mathbb{T})$, J. Differential Equations 262 (2017) 2196–2230. Korteweg D.J. y de Vries F., On the Change of Form of Long Waves Advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves, Philosophical Magazine, 39, 422—443, 1895. Kato T., On the Cauchy Problem for the (generalized) Korteweg - de Vries Equation, Advances in Mathematics Supplementary Studies, vol. 8, M.G. Crandall, ed., Academic Press (1983) 93-128 Kruzkhov S., Faminskii A., Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation, Math. USSR Sbornik 48 (1984) 391-421. Lax P.D., A Hamiltonian approach to KdV and other equations in Nonlinear Evolution Equations, Math. M.G. Crandall, ed., Academic Press (1985) 207-224. Newell A. C., Solitons in Mathematics and Physics, Regional Conference Series in Applied Mathematics, SIAM (1985). Novikov S. , Manakov S.V., Pitaeviskii L.P. y Zakharov V.E., Theory of Solitons - The Inverse Scattering Method, Contemporary Soviet Mathematics, Consultants Bureau, New York (1984). Whitham G.B., Linear and Nonlinear Waves, Wiley (1974). Benjamin, T.B., Internal Waves of Permanent Form in Fluids of Great Depth, J. Fluid Mech. 29, (1967), 559-592. Dix D., Temporal asymptotic behavior of solutions of the Benjamin-Ono-Burgers equation, J. Differ. Equations 90 (2) (1991) 238-287. Dan Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag Berlin Heidelberg New York 1981. Iorio, Jr. R. J., KdV, BO and friends in weighted Sobolev spaces, in Function Analytic Methods for Partial Differential Equations, Lecture Notes in Mathematics, vol. 1450, Springer (1990) 105-121. H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan 39 (1975) 1082-1091. Iorio Jr. R. J. , On The Cauchy Problem For The Benjamin-Ono Equation, Communications in partial differential equations. v. 11, n. 10, p. 1031-1081, 1986. Ponce G., On the global well-posedness of the Benjamin-Ono equation, Differ. Integral Equ. 4 (3) (1991) 527-542. Tao T., Global well-posedness of the Benjamin-Ono equation in $H^1(\mathbb{R})$, Volume 1, J. Hyperbolic Differ. Equ. (2003). T. B. Benjamin, A new kind of solitary waves, J. Fluid Mech. 245 (1992), 401-411. Ben-Artzi M. y Devinatz A. , The limiting absorption principle for partial differential operators, Mem. Amer. Math. Soc. 66 (Marzo 1987) no. 364. Brandt S. y Dahmen H.D. , The picture Book of Quantum Mechanics, 2nd ed., Springer (1995). Cycon H.L. , Froese R.G., Kirsh W. y Simon B., Schödinger Operators, Springer (1987). Drezinski J. y Gérard C., Scattering Theory of Classical and Quantum N-Particle Systems, Springer (1997). Gottfried K., Quantum Mechanics vol 1: Fundamentals, W.A. Benjamin (1996). Jensen A. , Scattering theory for Stark Hamiltonians, Proc. Indian Acad. Sci. (Math.Sci.) 104 (1994) 599-651. Merzbacher E., Quantum Mechanics, 2nd ed., Wiley (1970). Schecter M., Operator Methods in Quantum Mechanics, Elsevier-North-Holland (1981). Iorio , Jr R. J. , Tópicos na teoria da equação de Schrödinger, IMPA. Fonseca G., Pastrán R., Rodríguez G., The IVP for a nonlocal perturbation of the Benjamin-Ono equation in classical and weighted Sobolev spaces, Journal of Mathematical Analysis and Applications (2019). Pastrán R., Riaño 0., On the well-posedness for the Chen-Lee equation in periodic Sobolev spaces, Revista Colombiana de Matematicas (2006). Pastrán R., Riaño 0., Sharp Well-posedness for the Chen-Lee equation, Communications on Pure and Applied Analysis (2016). Tao T. , Multilinear weighted convolution of L2 functions, and applications to nonlinear dispersive equations, Amer. J. Math. 123(5) (2001) 839–908. Molinet, L., Saut, J. C., and Tzvetkov, N., Ill-posedness issues fo the Benjamin- Ono and related equations. SIAM J. Math. Anal. 33, 4 (2001), 982–988. Coddington y Levinson, Theory of Ordinary Differential Equations. McGraw-Hill (1963). Yosida K., Functional Analysis, 2nd ed. Springer (1968). |
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94 páginas |
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Universidad Nacional de Colombia |
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Bogotá - Ciencias - Maestría en Ciencias - Matemáticas |
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Facultad de Ciencias |
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Bogotá, Colombia |
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Universidad Nacional de Colombia - Sede Bogotá |
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Reconocimiento 4.0 Internacionalhttp://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Pastrán Ramírez, Ricardo Ariel891469efcca94bc8d850adfe63d31d59Correa Castañeda, Diego Fernando424ae976d77805f80c541ac8e378da562023-08-08T15:10:38Z2023-08-08T15:10:38Z2023-02-01https://repositorio.unal.edu.co/handle/unal/84480Universidad Nacional de ColombiaRepositorio Institucional Universidad Nacional de Colombiahttps://repositorio.unal.edu.co/ilustraciones, diagramasEn el contexto de la electrodinámica de fluidos, se dedujo la siguiente ecuación: $u_t + u_{xxxxx} - u_{xxx} + \sigma\, u_{xx}+uu_x=0$, donde $\sigma$ es la llamada "transformada de Hilbert". En este trabajo se estudió el problema de Cauchy asociado a esta ecuación, obteniendo resultados de bien planteado local en los siguientes casos: primero, tomando un dato inicial real arbitrario en el espacio periódico de Sobolev $H^s (T)$, cuando $s>3/2$, y segundo, cuando el dato inicial pertenece a $L^2 (R)$. (Texto tomado de la fuente)In the context of the Fluid electrodynamics, the next equation was deduced: $u_t + u_{xxxxx} - u_{xxx} + \sigma\, u_{xx}+uu_x=0$, where $\sigma$ is the so called "Hilbert transform". In this work, the Cauchy problem associated to this equation was studied, obtaining results of local well - posedness in the next cases: first, taking an arbitrary real initial data in the periodic Sobolev space $H^s (T)$, when $s>3/2$, and second, when the initial data belongs to $L^2 (R)$.MaestríaMagíster en Ciencias - MatemáticasEcuaciones diferenciales parciales de tipo dispersivoPara obtener el primer resultado se usó la llamada "Regularización parabólica", y para el segundo, se usaron los llamados "Espacios de Bourgain".94 páginasapplication/pdfspaUniversidad Nacional de ColombiaBogotá - Ciencias - Maestría en Ciencias - MatemáticasFacultad de CienciasBogotá, ColombiaUniversidad Nacional de Colombia - Sede Bogotá510 - Matemáticas::515 - AnálisisECUACIONES DIFERENCIALES PARCIALESPROBLEMA DE CAUCHYDifferential equations, partialCauchy problem-8a. ed.Buen planteamientoEcuaciones dispersivasRegularización parabólicaEspacios de BourgainWell posednessDispersive equationsParabolic regularizationBourgain spacesEl problema de Cauchy asociado a una perturbación dispersiva de quinto orden de la ecuación de BenjamínThe Cauchy problem associated to a fifth order perturbation of the Benjamin equationTrabajo de grado - Maestríainfo:eu-repo/semantics/masterThesisinfo:eu-repo/semantics/acceptedVersionTexthttp://purl.org/redcol/resource_type/TMIorio, Jr R. J. y Valéria de Magalhães Iorio, Fourier Analysis and Partial Differential Equations, Cambridge University Press, 2001.Kenig, Ponce y Vega, A bilinear estimate with applications to the KdV equation, Journal of the american mathematical society, Volume 9, Number 2, April 1996.Stein Elias y Shakarchi Rami, Fourier Analysis: An Introduction, Princeton University Press, (2003).Linares, F. y Ponce, G., Introduction to Nonlinear Dispersive Equations, Springer, 2009.Gleeson H. , Hammerton P., Papageorgiou D. T. , Vanden-Broeck J.-M. , A new application of the Korteweg–de Vries Benjamin-Ono equation in interfacial electrohydrodynamics, American Institute of Physics, 2007.Linares Felipe, $L^2$ Global Well-Posedness of the Initial Value Problem Associated to the Benjamin Equation, IMPA, 1998.Kenig, Ponce y Vega, The Cauchy problem for the Korteweg de Vries equation in Sobolev spaces of negative indices, Duke Math. J. 71 (1993), 1 - 21. MR 94g:35196.Bourgain, Fourier Transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Geometric and Functional Analysis, Volume 3, Number 3, 1993.Chen, Guo y Xiao, Sharp well-posedness for the Benjamin equation, Journal of the american mathematical society, Nonlinear Analysis 74 (2011) 6209–6230.Junfeng Li, Xia Li, Well-posedness for the fifth order KP-II initial data problem in $H^{s,0}(\R \times \mathbb{T})$, J. Differential Equations 262 (2017) 2196–2230.Korteweg D.J. y de Vries F., On the Change of Form of Long Waves Advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves, Philosophical Magazine, 39, 422—443, 1895.Kato T., On the Cauchy Problem for the (generalized) Korteweg - de Vries Equation, Advances in Mathematics Supplementary Studies, vol. 8, M.G. Crandall, ed., Academic Press (1983) 93-128Kruzkhov S., Faminskii A., Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation, Math. USSR Sbornik 48 (1984) 391-421.Lax P.D., A Hamiltonian approach to KdV and other equations in Nonlinear Evolution Equations, Math. M.G. Crandall, ed., Academic Press (1985) 207-224.Newell A. C., Solitons in Mathematics and Physics, Regional Conference Series in Applied Mathematics, SIAM (1985).Novikov S. , Manakov S.V., Pitaeviskii L.P. y Zakharov V.E., Theory of Solitons - The Inverse Scattering Method, Contemporary Soviet Mathematics, Consultants Bureau, New York (1984).Whitham G.B., Linear and Nonlinear Waves, Wiley (1974).Benjamin, T.B., Internal Waves of Permanent Form in Fluids of Great Depth, J. Fluid Mech. 29, (1967), 559-592.Dix D., Temporal asymptotic behavior of solutions of the Benjamin-Ono-Burgers equation, J. Differ. Equations 90 (2) (1991) 238-287.Dan Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag Berlin Heidelberg New York 1981.Iorio, Jr. R. J., KdV, BO and friends in weighted Sobolev spaces, in Function Analytic Methods for Partial Differential Equations, Lecture Notes in Mathematics, vol. 1450, Springer (1990) 105-121.H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan 39 (1975) 1082-1091.Iorio Jr. R. J. , On The Cauchy Problem For The Benjamin-Ono Equation, Communications in partial differential equations. v. 11, n. 10, p. 1031-1081, 1986.Ponce G., On the global well-posedness of the Benjamin-Ono equation, Differ. Integral Equ. 4 (3) (1991) 527-542.Tao T., Global well-posedness of the Benjamin-Ono equation in $H^1(\mathbb{R})$, Volume 1, J. Hyperbolic Differ. Equ. (2003).T. B. Benjamin, A new kind of solitary waves, J. Fluid Mech. 245 (1992), 401-411.Ben-Artzi M. y Devinatz A. , The limiting absorption principle for partial differential operators, Mem. Amer. Math. Soc. 66 (Marzo 1987) no. 364.Brandt S. y Dahmen H.D. , The picture Book of Quantum Mechanics, 2nd ed., Springer (1995).Cycon H.L. , Froese R.G., Kirsh W. y Simon B., Schödinger Operators, Springer (1987).Drezinski J. y Gérard C., Scattering Theory of Classical and Quantum N-Particle Systems, Springer (1997).Gottfried K., Quantum Mechanics vol 1: Fundamentals, W.A. Benjamin (1996).Jensen A. , Scattering theory for Stark Hamiltonians, Proc. Indian Acad. Sci. (Math.Sci.) 104 (1994) 599-651.Merzbacher E., Quantum Mechanics, 2nd ed., Wiley (1970).Schecter M., Operator Methods in Quantum Mechanics, Elsevier-North-Holland (1981).Iorio , Jr R. J. , Tópicos na teoria da equação de Schrödinger, IMPA.Fonseca G., Pastrán R., Rodríguez G., The IVP for a nonlocal perturbation of the Benjamin-Ono equation in classical and weighted Sobolev spaces, Journal of Mathematical Analysis and Applications (2019).Pastrán R., Riaño 0., On the well-posedness for the Chen-Lee equation in periodic Sobolev spaces, Revista Colombiana de Matematicas (2006).Pastrán R., Riaño 0., Sharp Well-posedness for the Chen-Lee equation, Communications on Pure and Applied Analysis (2016).Tao T. , Multilinear weighted convolution of L2 functions, and applications to nonlinear dispersive equations, Amer. J. Math. 123(5) (2001) 839–908.Molinet, L., Saut, J. C., and Tzvetkov, N., Ill-posedness issues fo the Benjamin- Ono and related equations. SIAM J. Math. Anal. 33, 4 (2001), 982–988.Coddington y Levinson, Theory of Ordinary Differential Equations. McGraw-Hill (1963).Yosida K., Functional Analysis, 2nd ed. Springer (1968).EstudiantesPúblico generalORIGINAL1010234652.2023.pdf1010234652.2023.pdfTesis de Maestría en Ciencias - Matemáticasapplication/pdf851290https://repositorio.unal.edu.co/bitstream/unal/84480/6/1010234652.2023.pdfca2df534bdc88ef0bdd3d6c4370edbe1MD56LICENSElicense.txtlicense.txttext/plain; charset=utf-85879https://repositorio.unal.edu.co/bitstream/unal/84480/5/license.txteb34b1cf90b7e1103fc9dfd26be24b4aMD55THUMBNAIL1010234652.2023.pdf.jpg1010234652.2023.pdf.jpgGenerated Thumbnailimage/jpeg4145https://repositorio.unal.edu.co/bitstream/unal/84480/7/1010234652.2023.pdf.jpg7cfcda30458dc8a736206ab9d099e8ccMD57unal/84480oai:repositorio.unal.edu.co:unal/844802024-08-18 23:13:09.861Repositorio Institucional Universidad Nacional de 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