Frontiers of Mathematics in China >

2019 , Vol. 14 >Issue 1: 77 - 93

DOI: https://doi.org/10.1007/s11464-019-0750-3

RESEARCH ARTICLE

Well-posedness of a non-local abstract Cauchy problem with a singular integral

  • Haiyan JIANG 1 ,
  • Tiao LU , 2,3 ,
  • Xiangjiang ZHU 2
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  • 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

Received date: 13 Jan 2018

Accepted date: 05 Jan 2019

Published date: 22 Mar 2019

Copyright

2019 Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature
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Abstract

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.

Key words： Partial integro-differential equation (PIDE); singular integral; well-posedness; Wigner equation

Cite this article

Haiyan JIANG , Tiao LU , Xiangjiang ZHU . Well-posedness of a non-local abstract Cauchy problem with a singular integral[J]. Frontiers of Mathematics in China, 2019 , 14(1) : 77 -93 . DOI: 10.1007/s11464-019-0750-3

References

Publishing order | Descend order by publishing year | Descend order by cited within
1
Arnold A, Lange H, Zweifel P F. A discrete-velocity, stationary Wigner equation. J Math Phys, 2000, 41(11): 7167–7180

2
Banasiak J,Barletti L. On the existence of propagators in stationary Wigner equation without velocity cut-off. Transport Theor Stat, 2001, 30(7): 659–672

3
Barletti L, Zweifel P F. Parity-decomposition method for the stationary Wigner equation with inflow boundary conditions. Transport Theor Stat, 2001, 30(4-6): 507–520

4
Fattorini H O. The Cauchy Problem. Encyclopedia Math Appl, Vol 18. Reading: Addison-Wesley, 1983

5
Frensley W R. Wigner function model of a resonant-tunneling semiconductor device. Phys Rev B, 1987, 36: 1570–1580

6
Frensley W R. Boundary conditions for open quantum systems driven far from equilibrium. Rev Modern Phys, 1990, 62(3): 745–791

7
Hu X, Tang S, Leroux M. Stationary and transient simulations for a one-dimensional resonant tunneling diode. Commun Comput Phys, 2008, 4(5): 1034–1050

8
Li R, Lu T, Sun Z-P. 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
Li R, Lu T, Sun Z-P.Parity-decomposition and moment analysis for stationary Wigner equation with inflow boundary conditions. Front Math China, 2017, 12(4): 907–919

10
Markowich P A, Ringhofer C A, Schmeiser C. Semiconductor Equations. Vienna: Springer, 1990

DOI

11
Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. New York: Springer, 1983

DOI

12
Wigner E. On the quantum corrections for thermodynamic equilibrium. Phys Rev, 1932, 40: 749–759

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