Avisos:
» Ya está disponible la lista de artículos
para la exposición final (ver abajo).
Sobre el curso:
El objetivo principal del curso es introducir al estudiante a la teoría de semigrupos y sus aplicaciones a ecuaciones diferenciales parciales de evolución. Se discutirán: integral de Bochner y espacios dependientes del tiempo, teoría de generación de semigrupos (teoremas de tipo Hille-Yosida y de Lumer-Philips), regularidad, estabilidad de semigrupos, el problema abstracto de Cauchy y semigrupos de evolución. Asimismo, se prestará especial atención a las aplicaciones a ecuaciones diferenciales parciales de tipo parabólico y de tipo hiperbólico.
Contenido:
- Calendario UNAM 2026-1 (plan semestral): [PDF]
- Temario del curso, calendario y bibliografía [PDF]
- Página del Posgrado en Ciencias Matemáticas.
Material auxiliar:
- Ya está disponible la lista de artículos para la exposición final:
L. Arlotti, A perturbation theorem for positive contraction semigroups onL1-spaces with applications to transport equations and Kolmogorov’s differential equations,Acta Appl. Math.23 (1991), no. 2, 129–144.
- J. M. Ball, Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proc. Amer. Math. Soc. 63 (1977), no. 2, 370-373.
- D. Cramer, Y. Latushkin, Gearhart-Prüss theorem in stability for wave equations: a survey, in Evolution equations, G. Goldstein, R. Nagel, and S. Romanelli, eds., vol. 234 of Lecture Notes in Pure and Appl. Math., Dekker, New York, 2003, pp. 105-119.
K. Igari, Degenerate parabolic differential equations,Publ. Res. Inst. Math. Sci.9 (1974), 493–504.K. Igari, Well-posedness of the Cauchy problem for some evolution equations,Publ. Res. Inst. Math. Sci.9 (1973/74), 613–629.
- T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal. 58 (1975), no. 3, 181-205.
- T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in Spectral theory and differential equations (Proc. Sympos., Dundee, 1974), W. N. Everitt, ed., no. 448 in Lecture Notes in Mathematics, Springer, Berlin, 1975, pp. 25-70.
Bibliografía
- J. M. Ball, Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proc. Amer. Math. Soc. 63 (1977), no. 2, 370-373.
- C. Chicone, Y. Latushkin, Evolution semigroups in
dynamical systems and differential equations, vol. 70
of Mathematical Surveys and Monographs, American
Mathematical Society, Providence, RI, 1999.
- D. Cramer, Y. Latushkin, Gearhart-Prüss theorem in stability for wave equations: a survey, in Evolution equations, G. Goldstein, R. Nagel, and S. Romanelli, eds., vol. 234 of Lecture Notes in Pure and Appl. Math., Dekker, New York, 2003, pp. 105-119.
- K. J. Engel, R. Nagel, One-parameter semigroups for linear evolution equations, vol. 194 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2000.
- K. J. Engel, R. Nagel, A short course on operator semigroups, Universitext, Springer-Verlag, New York, 2006.
- L. C. Evans, Application of nonlinear semigroup theory to certain partial differential equations, in Nonlinear evolution equations (Proc. Sympos., Univ. Wisconsin, Madison, Wis., 1977), M. G. Crandall, ed., vol. 40 of Publ. Math. Res. Center Univ. Wisconsin, Academic Press, New York-London, 1978, pp. 163-188.
- L. C. Evans, Partial differential equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, second ed., 2010.
- T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal. 58 (1975), no. 3, 181-205.
- T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in Spectral theory and differential equations (Proc. Sympos., Dundee, 1974), W. N. Everitt, ed., no. 448 in Lecture Notes in Mathematics, Springer, Berlin, 1975, pp. 25-70.
- A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems, vol. 16 of Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Verlag, Basel, 1995.
- A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983.
- M. Renardy, R. C. Rogers, An introduction to partial differential equations, vol. 13 of Texts in Applied Mathematics, Springer-Verlag, New York, second ed., 2004.
- I. I. Vrabie, C0-semigroups and applications, vol. 191 of North-Holland Mathematics Studies, North-Holland Publishing Co., Amsterdam, 2003.
- K. Yosida, Functional analysis, Classics in
Mathematics, Springer-Verlag, Berlin, sixth ed., 1980.