Well-posedness of a non-local abstract Cauchy problem with a singular integral
Haiyan JIANG1, Tiao LU2,3(), Xiangjiang ZHU2
1. School of Mathematical Sciences, Beijing Institute of Technology, Beijing 100081, China 2. School of Mathematical Sciences, Peking University, Beijing 100871, China 3. CAPT, HEDPS, LMAM, IFSA Collaborative Innovation Center of MoE, Peking University, Beijing 100871, China
A non-local abstract Cauchy problem with a singular integral is studied, which is a closed system of two evolution equations for a real-valued function and a function-valued function. By proposing an appropriate Banach space, the well-posedness of the evolution system is proved under some bounded-ness and smoothness conditions on the coeffcient functions. Furthermore, an isomorphism is established to extend the result to a partial integro-differential equation with a singular convolution kernel, which is a generalized form of the stationary Wigner equation. Our investigation considerably improves the understanding of the open problem concerning the well-posedness of the stationary Wigner equation with inflow boundary conditions.
. [J]. Frontiers of Mathematics in China, 2019, 14(1): 77-93.
Haiyan JIANG, Tiao LU, Xiangjiang ZHU. Well-posedness of a non-local abstract Cauchy problem with a singular integral. Front. Math. China, 2019, 14(1): 77-93.
AArnold, HLange, P FZweifel. A discrete-velocity, stationary Wigner equation. J Math Phys, 2000, 41(11): 7167–7180
2
J,Banasiak LBarletti. On the existence of propagators in stationary Wigner equation without velocity cut-off. Transport Theor Stat, 2001, 30(7): 659–672
3
LBarletti, P FZweifel. Parity-decomposition method for the stationary Wigner equation with inflow boundary conditions. Transport Theor Stat, 2001, 30(4-6): 507–520
4
H OFattorini. The Cauchy Problem. Encyclopedia Math Appl, Vol 18. Reading: Addison-Wesley, 1983
5
W RFrensley. Wigner function model of a resonant-tunneling semiconductor device. Phys Rev B, 1987, 36: 1570–1580
6
W RFrensley. Boundary conditions for open quantum systems driven far from equilibrium. Rev Modern Phys, 1990, 62(3): 745–791
7
XHu, STang, MLeroux. Stationary and transient simulations for a one-dimensional resonant tunneling diode. Commun Comput Phys, 2008, 4(5): 1034–1050
8
RLi, TLu, Z-PSun. Stationary Wigner equation with inflow boundary conditions: Will a symmetric potential yield a symmetric solution? SIAM J Appl Math, 2014, 70(3): 885–897
9
RLi, TLu, Z-P.SunParity-decomposition and moment analysis for stationary Wigner equation with inflow boundary conditions. Front Math China, 2017, 12(4): 907–919
