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 boundedness and smoothness conditions on the coefficient 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 in ow boundary conditions.
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References
Arnold A, Lange H, Zweifel P F. A discrete-velocity, stationary Wigner equation. J Math Phys, 2000, 41(11): 7167–7180
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
Barletti L, Zweifel P F. Parity-decomposition method for the stationary Wigner equation with in flow boundary conditions. Transport Theor Stat, 2001, 30(4-6): 507–520
Fattorini H O. The Cauchy Problem. Encyclopedia Math Appl, Vol 18. Reading: Addison-Wesley, 1983
Frensley W R. Wigner function model of a resonant-tunneling semiconductor device. Phys Rev B, 1987, 36: 1570–1580
Frensley W R. Boundary conditions for open quantum systems driven far from equilibrium. Rev Modern Phys, 1990, 62(3): 745–791
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
Li R, Lu T, Sun Z-P. Stationary Wigner equation with in flow boundary conditions: Will a symmetric potential yield a symmetric solution? SIAM J Appl Math, 2014, 70(3): 885–897
Li R, Lu T, Sun Z-P. Parity-decomposition and moment analysis for stationary Wigner equation with in flow boundary conditions. Front Math China, 2017, 12(4): 907–919
Markowich P A, Ringhofer C A, Schmeiser C. Semiconductor Equations. Vienna: Springer, 1990
Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. New York: Springer, 1983
Wigner E. On the quantum corrections for thermodynamic equilibrium. Phys Rev, 1932, 40: 749–759
Acknowledgements
The authors are grateful to Prof. Ruo Li and Dr. Zhangpeng Sun for their helpful discussion. This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11671038, 91630130, 91434201, 11421101).
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Jiang, H., Lu, T. & Zhu, X. Well-posedness of a non-local abstract Cauchy problem with a singular integral. Front. Math. China 14, 77–93 (2019). https://doi.org/10.1007/s11464-019-0750-3
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DOI: https://doi.org/10.1007/s11464-019-0750-3