In this paper I investigate the homogenizability of linear transport equations with periodic data. Some results on homogenizability and on the form of the limit are known in literature. In particular, in [9], I proved the homogenizability in the two-dimensional case for nonvanishing functions, and, on the other hand I gave an example of a nonhomogenizable equation in the three-dimensional case. In this paper, I describe an example of a nonhomogenizable equation in two dimensions. As in [9], I study the problem using an equivalent formulation in terms of dynamical system properties of the associated ODEs.
@article{ASNSP_2009_5_8_1_175_0,
author = {Peirone, Roberto},
title = {A nonhomogenizable linear transport equation in $\mathbb{R}^2$},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {175--206},
year = {2009},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 8},
number = {1},
mrnumber = {2512205},
zbl = {1184.35038},
language = {en},
url = {https://www.numdam.org/item/ASNSP_2009_5_8_1_175_0/}
}
TY - JOUR
AU - Peirone, Roberto
TI - A nonhomogenizable linear transport equation in $\mathbb{R}^2$
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2009
SP - 175
EP - 206
VL - 8
IS - 1
PB - Scuola Normale Superiore, Pisa
UR - https://www.numdam.org/item/ASNSP_2009_5_8_1_175_0/
LA - en
ID - ASNSP_2009_5_8_1_175_0
ER -
%0 Journal Article
%A Peirone, Roberto
%T A nonhomogenizable linear transport equation in $\mathbb{R}^2$
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2009
%P 175-206
%V 8
%N 1
%I Scuola Normale Superiore, Pisa
%U https://www.numdam.org/item/ASNSP_2009_5_8_1_175_0/
%G en
%F ASNSP_2009_5_8_1_175_0
Peirone, Roberto. A nonhomogenizable linear transport equation in $\mathbb{R}^2$. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 1, pp. 175-206. https://www.numdam.org/item/ASNSP_2009_5_8_1_175_0/
[1] , Remarks on some linear hyperbolic equations with oscillatory coefficients, In: “Third International Conference on Hyperbolic Problems" (Uppsala 1990), Studentlitteratur, Lund, 1991, 119–130. | MR | Zbl
[2] , and , “Ergodic Theory," Springer-Verlag, New York, 1982. | MR
[3] , On the convergence of solutions of some evolution differential equations, Set-Valued Anal. 2 (1994), 175–182. | MR | Zbl
[4] W. E., Homogenization of linear and nonlinear transport equations, Comm. Pure Appl. Math. 45 (1992), 301–326. | MR
[5] and , Homogenization of linear transport equations with oscillatory vector fields, SIAM J. Appl. Math. 52 (1992), 34–45. | MR | Zbl
[6] and , “Introduction to the Modern Theory of Dynamical Systems", Cambridge University Press, Cambridge, 1995. | MR | Zbl
[7] and , Rotation sets for Maps of Tori, J. London Math. Soc. (2) 40 (1989), 490–506. | MR | Zbl
[8] , Homogenization of ordinary and linear transport equations, Differential Integral Equations 9 (1996), 323–334. | MR | Zbl
[9] , Convergence of solutions of linear transport equations, Ergodic Theory Dynam. Systems 23 (2003), 919–933. | MR | Zbl
[10] , Homogenization of two-dimensional linear flows with integral invariance, SIAM J. Appl. Math. 57 (1997), 1390–1405. | MR | Zbl
