Skip to main content

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

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.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Arnold A, Lange H, Zweifel P F. A discrete-velocity, stationary Wigner equation. J Math Phys, 2000, 41(11): 7167–7180

    MathSciNet  Article  MATH  Google Scholar 

  2. 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

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    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

    Article  MATH  Google Scholar 

  4. 4.

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

    Google Scholar 

  5. 5.

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

    Article  Google Scholar 

  6. 6.

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

    Article  Google Scholar 

  7. 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

    Google Scholar 

  8. 8.

    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

    Article  MATH  Google Scholar 

  9. 9.

    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

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

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

    Book  MATH  Google Scholar 

  11. 11.

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

    Book  MATH  Google Scholar 

  12. 12.

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

    Article  MATH  Google Scholar 

Download references

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).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Tiao Lu.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

Keywords

  • Partial integro-differential equation (PIDE)
  • singular integral
  • well-posedness
  • Wigner equation

MSC

  • 35S10
  • 81S30
  • 47D03