Avisos:
» Aviso importante: Las calificaciones
finales del curso se pueden encontrar en esta liga.
Sobre el curso:
El curso tiene dos objetivos principales: presentar al alumno
los resultados básicos sobre existencia de soluciones a
ecuaciones de onda nolineales, por un lado, así como dar una
introducción a la teoría de sistemas hiperbólicos de leyes de
conservación nolineales, por el otro. Asimismo, se presentará
una panorámica de la teoría de Kreiss-Lopatinski para sistemas
hiperbólicos con valores iniciales y de frontera, si el tiempo
lo permite.
Contenido:
- Temario del curso, calendario y bibliografía [PDF].
- Página del Posgrado en Ciencias Matemáticas.
Material auxiliar:
- El material de lectura para la contingencia sanitaria se puede descargar aquí.
- Material de lectura:
- E. Godlewski, P.-A. Raviart, Hyperbolic Systems of
Conservation Laws, Mathematiques & Applications
vol. 3/4, Ellipses, Paris, 1991.
- Los artículos para lectura y exposición al final del curso
son los siguientes (los artículos que ya han sido
seleccionados están tachados):
Ecuación de onda nolineal:
- M. Ikeda, M. Sobajima, K. Wakasa, J. Differ. Equ. 267 (2019), 5165-5201 [DOI]
- M. Kato, M. Sakuraba, Nonlinear Anal. 182 (2019), 209-225 [DOI]
- N. Lai, H. Takamura, Nonlinear Anal. 168 (2018), 222-237 [DOI]
- K. Nishihara, J. Math. Anal. Appl. 478 (2019), 458-465 [DOI]
- M. Sobajima, K. Wakasa, J. Math. Anal. Appl. 484 (2020), 123667 [DOI]
- L. Tebou, Z. Angew. Math. Phys. 71 (2020), 22 [DOI]
Ley escalar de conservación:
F. Ancona, O. Glass, K. T. Nguyen, SIAM J. Math. Anal. 51 (2019), 3020-3051[DOI]- S. Bianchini, E. Marconi, Arch. Ration. Mech. Anal. 226
(2017), 441-493 [DOI]
M.-J. Kang, A. Vasseur, Ann. I. H. Poincaré 34 (2017), 139-156[DOI]S. G. Krupa, A. Vasseur, J. Hyper. Differ. Equ. 16 (2019), 157-191[DOI]Y.-S. Kwon, Abstr. Appl. Anal. (2014) 690801, 1-7[DOI]
- L. Silvestre, Comm. Pure Appl. Math. 72 (2019), 1321-1348
[DOI]
- N. Yoshida, SIAM J. Math. Anal. 50 (2018), 891-932 [DOI]
Sistemas de leyes de conservación:
- C. M. Dafermos, Discr. Const. Dyn. Syst. 36 (2016), 4271-4285 [DOI]
C. M. Dafermos, Ricerche Mat. 67 (2018), 755-764[DOI]- C. Mascia, Acta Math. Sci. 35B (2015), 807-831 [DOI]
D. Serre, A. Vasseur, Contemp. Math. 658 (2016), 237-248[DOI]D. Serre, A. Vasseur, J. Ecole Polytech. Math. 1 (2014), 1-28[DOI]- A. Vasseur, L. Yao, Commun. Math. Sci. 14 (2016),
2215-2228 [DOI]
Bibliografía
Ecuación de onda nolineal:
- A. Bressan, Lecture Notes on Functional Analysis, vol. 143 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2013.
- T. Cazenave, A. Haraux, An introduction to semilinear evolution equations, vol. 13 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, New York, 1998.
- L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, Amer. Math. Soc., Providence, RI, 1998.
- L. Hörmander, Lectures on nonlinear hyperbolic differential equations, vol. 26 of Mathématiques & Applications (Berlin), Springer-Verlag, Berlin, 1997.
- F. John, Nonlinear wave equations, formation of singularities, vol. 2 of University Lecture Series, American Mathematical Society, Providence, RI, 1990.
- C. E. Kenig, Lectures on the energy critical nonlinear wave equation, vol. 122 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, 2015.
- J. Shatah, M. Struwe, Geometric wave equations, vol. 2 of Courant Lecture Notes in Mathematics, New York University, American Mathematical Society, Providence, RI, 1998.
- W. A. Strauss, Nonlinear wave equations, vol. 73 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, 1989.
- M. E. Taylor, Partial differential equations III.
Nonlinear equations, vol. 117 of Applied Mathematical
Sciences, Springer-Verlag, New York, 1997.
Sistemas hiperbólicos de leyes de conservación:
- S. Alinhac, Hyperbolic partial differential equations, Universitext, Springer Verlag, 2009.
- A. Bressan, Hyperbolic systems of conservation laws: The one-dimensional Cauchy problem, vol. 20 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2000.
- S. Bianchini, A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. of Math. (2) 161 (2005), no. 1, 223-342.
- C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, vol. 325, Springer-Verlag, fourth ed., 2016.
- E. Godlewski, P. A. Raviart, Hyperbolic Systems of
Conservation Laws, Mathematiques & Applications
vol. 3/4, Ellipses, Paris, 1991.
- E. Godlewski, P. A. Raviart, Numerical approximation
of hyperbolic systems of conservation laws, vol. 118
of Applied Mathematical Sciences, Springer-Verlag, New York,
1996.
- P. D. Lax, Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM, Philadelphia, PA, 1973.
- T.-P. Liu, Viscous and Hyperbolic Conservation Laws, SIAM, Philadelphia, PA, 2000.
- A. Majda, Compressible fluid flow and systems of
conservation laws in several space variables, vol. 53
of Applied Mathematical Sciences, Springer-Verlag, New York,
1984.
- D. Serre, Systems of Conservation Laws 1. Hyperbolicity, entropies, shock waves, Cambridge University Press, 1999.
- D. Serre, Systems of Conservation Laws 2. Geometric structures, oscillations and initial-boundary value problems, Cambridge University Press, 2000.
- J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, second ed., 1994.
- M. Renardy, R. C. Rogers, An introduction to partial differential equations, second ed., vol. 13 of Texts in Applied Mathematics, Springer-Verlag, New York, 2004.
Teoría de Kreiss-Lopatinski:
- S. Benzoni-Gavage, D. Serre, Multidimensional hyperbolic partial differential equations, first order systems and applications, Oxford University Press, 2007.
- R. L. Higdon, Initial-boundary value problems for linear hyperbolic systems, SIAM Review 28 (1986), no.2, 177-217.
- H.-O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math. 23 (1970), 277-298.
- A. Majda, The existence of multi-dimensional shock fronts, Mem. Amer. Math. Soc. 43 (1983), no. 281.
- A. Majda, The stability of multi-dimensional shock fronts, Mem. Amer. Math. Soc. 41 (1983), no. 275.
- A. Majda, Compressible fluid flow and systems of conservation laws in several space variables, vol. 53 of Applied Mathematical Sciences, Springer-Verlag, New York, 1984.
- G. Métivier, Stability of multidimensional shocks, in Advances in the theory of shock waves, vol. 47 of Progress in Nonlinear Differential Equations, Birkhäuser Boston, Boston, MA, 2001, pp. 25-103.
- G. Métivier, Stability of multi-dimensional weak shocks, Comm. Partial Differ. Equ. 15 (1990), no. 7, 983-1028.
- D. Serre, Systems of Conservation Laws 2. Geometric structures, oscillations and initial-boundary value problems, Cambridge University Press, 2000.