Semigroups for general transport equations with abstract boundary conditions

Luisa Arlotti (1) , Jacek Banasiak (2) , Bertrand Lods (3)

1 Università degli Studi di Udine - University of Udine [Italie]
2 School of Mathematical Sciences, University of KwaZulu-Natal
3 LMBP - Laboratoire de Mathématiques Blaise Pascal

We investigate C_0-semigroup generation properties of the Vlasov equation with general boundary conditions modeled by an abstract boundary operator H. For conservative boundary conditions we show that there is an extension A of the free streaming operator T_H which generates a C_0-semigroup $(V(t))_{t \geq 0}$ in $L^1$. Furthermore, we precisely describe its domain and provide necessary and sufficient conditions ensuring that $(V(t))_{t \geq 0}$ is stochastic.

Mots clés: Characteristic curves; Transport equation; Boundary conditions; $C_0$-semigroups; Characteristic curves.

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

Luisa Arlotti, Jacek Banasiak, Bertrand Lods. Semigroups for general transport equations with abstract boundary conditions. 2006. ⟨hal-00110239v1⟩