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

Arnold, Anton; Dhamo, Elidon; Manzini, Chiara

Annales de l'I.H.P. Analyse non linéaire, Tome 24 (2007) no. 4, pp. 645-676

[1] Arnold A., Self-consistent relaxation-time models in quantum mechanics, Comm. Partial Differential Equations 21 (3&4) (1995) 473-506. | Zbl | MR

[2] Arnold A., Carrillo J.A., Dhamo E., On the periodic Wigner-Poisson-Fokker-Planck system, J. Math. Anal. Appl. 275 (2002) 263-276. | Zbl

[3] Arnold A., López J.L., Markowich P.A., Soler J., An analysis of quantum Fokker-Planck models: A Wigner function approach, Rev. Mat. Iberoamericana 20 (3) (2004) 771-814. | Zbl

[4] Arnold A., Ringhofer C., An operator splitting method for the Wigner-Poisson problem, SIAM J. Numer. Anal. 33 (1996) 1622-1643. | Zbl

[5] Arnold A., Sparber C., Conservative quantum dynamical semigroups for mean-field quantum diffusion models, Comm. Math. Phys. 251 (1) (2004) 179-207. | Zbl | MR

[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. | Zbl

[7] Bouchut F., Smoothing effect for the non-linear Vlasov-Poisson-Fokker-Planck system, J. Differential Equations 122 (2) (1995) 225-238. | Zbl

[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. | Zbl

[9] Caldeira A.O., Leggett A.J., Path integral approach to quantum Brownian motion, Physica A 121 (1983) 587-616. | Zbl | MR

[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 Equations 198 (2004) 356-373. | Zbl

[11] Carpio A., Long time behavior for solutions of the Vlasov-Poisson-Fokker-Planck equation, Math. Meth. Appl. Sci. 21 (1998) 985-1014. | Zbl

[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. | Zbl

[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. | Zbl

[14] Castella F., The Vlasov-Poisson-Fokker-Planck system with infinite kinetic energy, Indiana Univ. Math. J. 47 (3) (1998) 939-964. | Zbl

[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. | Zbl

[16] Castella F., Perthame B., Strichartz estimates for kinetic transport equations, C. R. Acad. Sci. Paris, Ser. I 332 (1) (1996) 535-540. | Zbl | MR

[17] Diósi L., On high-temperature Markovian equation for quantum Brownian motion, Europhys. Lett. 22 (1993) 1-3.

[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] 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. | Zbl

[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).

[21] Lions P.L., Perthame B., Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Invent. Math. 105 (1991) 415-430. | Zbl

[22] Manzini C., The three dimensional Wigner-Poisson problem with inflow boundary conditions, J. Math. Anal. Appl. 313 (1) (2006) 184-196.

[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. | Zbl

[24] Markowich P.A., Ringhofer C., An analysis of the quantum Liouville equation, Z. Angew. Math. Mech. 69 (1989) 121-127. | Zbl | MR

[25] Markowich P.A., Ringhofer C., Schmeiser C., Semiconductor Equations, Springer, 1990. | Zbl | MR

[26] Pazy A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, 1983. | Zbl | MR

[27] Perthame B., Time decay, propagation of low moments and dispersive effects for kinetic equations, Comm. Partial Differential Equations 21 (1&2) (1996) 659-686. | Zbl | MR

[28] Reed M., Simon B., Methods of Modern Mathematical Physics I, Functional Analysis, Academic Press, 1980. | Zbl | MR

[29] Risken H., The Fokker-Planck Equation, Springer, 1984. | Zbl

[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. | Zbl

[31] Stein E.M., Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. | Zbl | MR

[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. | Zbl

[33] Stroscio M.A., Moment-equation representation of the dissipative quantum Liouville equation, Superlattices and Microstructures 2 (1986) 83-87.

[34] Wigner E., On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40 (1932) 749-759. | JFM