The Wigner–Poisson–Fokker–Planck system : global-in-time solution and dispersive effects

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1. [1] Arnold A., Self-consistent relaxation-time models in quantum mechanics, Comm. Partial Differential Equations21 (3&4) (1995) 473-506. Zbl0849.35113MR1387456
2. [2] Arnold A., Carrillo J.A., Dhamo E., On the periodic Wigner–Poisson–Fokker–Planck system, J. Math. Anal. Appl.275 (2002) 263-276. Zbl1014.35084
3. [3] Arnold A., López J.L., Markowich P.A., Soler J., An analysis of quantum Fokker–Planck models: A Wigner function approach, Rev. Mat. Iberoamericana20 (3) (2004) 771-814. Zbl1062.35097
4. [4] Arnold A., Ringhofer C., An operator splitting method for the Wigner–Poisson problem, SIAM J. Numer. Anal.33 (1996) 1622-1643. Zbl0860.65143
5. [5] Arnold A., Sparber C., Conservative quantum dynamical semigroups for mean-field quantum diffusion models, Comm. Math. Phys.251 (1) (2004) 179-207. Zbl1085.82004MR2096738
6. [6] Bouchut F., Existence and uniqueness of a global smooth solution for the Vlasov–Poisson–Fokker–Planck system in three dimensions, J. Funct. Anal.111 (1993) 239-258. Zbl0777.35059
7. [7] Bouchut F., Smoothing effect for the non-linear Vlasov–Poisson–Fokker–Planck system, J. Differential Equations122 (2) (1995) 225-238. Zbl0840.35053
8. [8] Brezzi F., Markowich P.A., The three-dimensional Wigner–Poisson problem: existence, uniqueness and approximation, Math. Methods Appl. Sci.14 (1) (1991) 35-61. Zbl0739.35080
9. [9] Caldeira A.O., Leggett A.J., Path integral approach to quantum Brownian motion, Physica A121 (1983) 587-616. Zbl0585.60082MR726154
10. [10] Cañizo J.A., López J.L., Nieto J., Global $L^1$-theory and regularity for the 3D non-linear Wigner–Poisson-Fokker–Planck system, J. Differential Equations198 (2004) 356-373. Zbl1039.35093
11. [11] Carpio A., Long time behavior for solutions of the Vlasov–Poisson–Fokker–Planck equation, Math. Meth. Appl. Sci.21 (1998) 985-1014. Zbl0911.35090
12. [12] Carrillo J.A., Soler J., Vázquez J.L., Asymptotic behaviour and selfsimilarity for the three dimensional Vlasov–Poisson–Fokker–Planck system, J. Funct. Anal.141 (1996) 99-132. Zbl0873.35066
13. [13] Castella F., $L^2$ solutions to the Schrödinger–Poisson system: existence, uniqueness, time behaviour, and smoothing effects, Math. Models Methods Appl. Sci.7 (8) (1997) 1051-1083. Zbl0892.35141
14. [14] Castella F., The Vlasov–Poisson–Fokker–Planck system with infinite kinetic energy, Indiana Univ. Math. J.47 (3) (1998) 939-964. Zbl0926.35004
15. [15] Castella F., Erdös L., Frommlet F., Markowich P.A., Fokker–Planck equations as scaling limits of reversible quantum systems, J. Stat. Phys.100 (3/4) (2000) 543-601. Zbl0988.82038
16. [16] Castella F., Perthame B., Strichartz estimates for kinetic transport equations, C. R. Acad. Sci. Paris, Ser. I332 (1) (1996) 535-540. Zbl0848.35095MR1383431
17. [17] Diósi L., On high-temperature Markovian equation for quantum Brownian motion, Europhys. Lett.22 (1993) 1-3. 
18. [18] Gruvin H.L., Govindan T.R., Kreskovsky J.P., Stroscio M.A., Transport via the Liouville equation and moments of quantum distribution functions, Solid State Electr.36 (1993) 1697-1709. 
19. [19] Helffer B., Nier F., Hypoelliptic Estimates and Spectral Theory for Fokker–Planck Operators and Witten Laplacians, Lecture Notes in Mathematics, vol. 1862, Springer-Verlag, Berlin, 2005. Zbl1072.35006
20. [20] Hérau F., Nier F., Isotropic hypoellipticity and trend to equilibrium for the Fokker–Planck equation with high degree potential, Arch. Rational Mech. Anal.171 (2) (2004). Zbl1139.82323
21. [21] Lions P.L., Perthame B., Propagation of moments and regularity for the 3-dimensional Vlasov–Poisson system, Invent. Math.105 (1991) 415-430. Zbl0741.35061
22. [22] Manzini C., The three dimensional Wigner–Poisson problem with inflow boundary conditions, J. Math. Anal. Appl.313 (1) (2006) 184-196. Zbl1160.35543
23. [23] Manzini C., Barletti L., An analysis of the Wigner–Poisson problem with time-dependent, inflow boundary conditions, Nonlin. Anal.60 (1) (2004) 77-100. Zbl1084.81051
24. [24] Markowich P.A., Ringhofer C., An analysis of the quantum Liouville equation, Z. Angew. Math. Mech.69 (1989) 121-127. Zbl0682.46047MR990011
25. [25] Markowich P.A., Ringhofer C., Schmeiser C., Semiconductor Equations, Springer, 1990. Zbl0765.35001MR1063852
26. [26] Pazy A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, 1983. Zbl0516.47023MR710486
27. [27] Perthame B., Time decay, propagation of low moments and dispersive effects for kinetic equations, Comm. Partial Differential Equations21 (1&2) (1996) 659-686. Zbl0852.35139MR1387464
28. [28] Reed M., Simon B., Methods of Modern Mathematical Physics I, Functional Analysis, Academic Press, 1980. Zbl0459.46001MR751959
29. [29] Risken H., The Fokker–Planck Equation, Springer, 1984. Zbl0546.60084
30. [30] Sparber C., Carrillo J.A., Dolbeault J., Markowich P.A., On the long time behavior of the quantum Fokker–Planck equation, Monatsh. Math.141 (3) (2004) 237-257. Zbl1061.35095
31. [31] Stein E.M., Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. Zbl0207.13501MR290095
32. [32] Steinrück H., The one-dimensional Wigner–Poisson problem and its relation to the Schrödinger–Poisson problem, SIAM J. Math. Anal.22 (4) (1992) 957-972. Zbl0738.35077
33. [33] Stroscio M.A., Moment-equation representation of the dissipative quantum Liouville equation, Superlattices and Microstructures2 (1986) 83-87. 
34. [34] Wigner E., On the quantum correction for thermodynamic equilibrium, Phys. Rev.40 (1932) 749-759. Zbl58.0948.07JFM58.0948.07

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