A nonhomogenizable linear transport equation in 2
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 1, pp. 175-206.

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.

Classification : 35B27, 37E45
Peirone, Roberto 1

1 Università di Roma “Tor Vergata”, Dipartimento di Matematica, Via della Ricerca Scientifica, 00133, Roma, Italia
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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. http://www.numdam.org/item/ASNSP_2009_5_8_1_175_0/

[1] Y. Brenier, 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] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, “Ergodic Theory," Springer-Verlag, New York, 1982. | MR

[3] E. De Giorgi, 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] T. Y. Hou and X. Xin, Homogenization of linear transport equations with oscillatory vector fields, SIAM J. Appl. Math. 52 (1992), 34–45. | MR | Zbl

[6] A. Katok and B. Hasselblatt, “Introduction to the Modern Theory of Dynamical Systems", Cambridge University Press, Cambridge, 1995. | MR | Zbl

[7] M. Misiurewicz and K. Ziemian, Rotation sets for Maps of Tori, J. London Math. Soc. (2) 40 (1989), 490–506. | MR | Zbl

[8] R. Peirone, Homogenization of ordinary and linear transport equations, Differential Integral Equations 9 (1996), 323–334. | MR | Zbl

[9] R. Peirone, Convergence of solutions of linear transport equations, Ergodic Theory Dynam. Systems 23 (2003), 919–933. | MR | Zbl

[10] T. Tassa, Homogenization of two-dimensional linear flows with integral invariance, SIAM J. Appl. Math. 57 (1997), 1390–1405. | MR | Zbl