Smoothing effects for the classical solutions to the Landau-Fermi-Dirac equation

Abstract

The smoothness of the solutions to the full Landau equation for Fermi-Dirac particles is investigated. It is shown that the classical solutions near equilibrium to the Landau-Fermi-Dirac equation have a regularizing effects in all variables (time, space and velocity), that is, they become immediately smooth with respect to all variables.

Keywords

Landau-Fermi-Dirac equation / Classical solutions / Smoothing effect

References

[1] Alexandre R., Ukai S., Morimoto Y. Uncertainty principle and kinetic equations. J. Funct. Anal., 2008, 255: 2013-2066

[2] Arsen’ev A. A., Buryak O. E.. On a connection between the solution of the Boltzmann equation and the solution of the Landau-Fokker-Planck equation. Math. USSR Sb., 1991, 69: 465-478

[3] Bagland V.. Well-posedness for the spatially homogeneous Landau-Fermi-Dirac equation for hard potentials. Proc. R. Soc. Edinb., Sect. A, 2004, 134: 415-447

[4] Bagland V., Lemou M.. Equilibrium states for the Landau-Fermi-Dirac equation. Banach Cent. Publ., 2004, 66: 29-37

[5] Bouchut F., Desvillettes L.. Averaging lemmas without time Fourier transform and application to discretized kinetic equations. Proc. Roy. Soc. Edinburgh, 1999, 129A: 19-36

[6] Chapman S., Cowling T. G.. The Mathematical Theory of Non-uniform Cases, 1970, Cambridge: Cambridge University Press

[7] Chen H., Li W. X., Xu C. J.. Analytic smoothness effect of solutions of spatially homogeneous Landau equation. J. Diff. Eq., 2010, 24(1): 77-94

[8] Chen Y. M.. Smoothing effects for weak solutions of the spatially homogeneous Landau-Fermi-Dirac equation for hard potentials. Acta Appl. Math., 2011, 113(1): 101-116

[9] Chen Y. M., Desvillettes L., He L. B.. Smoothing effects for classical solutions of the full Landau equation. Arch. Ration Mech. Anal., 2009, 193: 21-55

[10] Danielewicz P.. Nonrelativistic and relativistic Landau/Fokker-Planck equation for arbitrary statistics. Physica A, 1980, 100: 167-182

[11] Degond P., Lemou M.. Dispersion relations for the linearized Fokker-Planck equation. Arch. Ration. Mech. Anal., 1997, 138: 137-167

[12] Desvillettes L.. On asymptotics of the Boltzmann equation when the collisions become grazing. Transport Theory Statist. Phys., 1992, 21: 259-276

[13] Desvillettes L., Villani C.. On the spatially homogeneous Landau equation for hard potentials (I-II). Comm. Part. Diff. Eq., 2002, 25: 179-298

[14] Dolbeault J.. Kinetic models and quantum effects: a modified Boltzmann equation for Fermi-Dirac particles. Arch. Ration. Mech. Anal., 1994, 127: 101-131

[15] El Safadi M.. Smoothness of weak solutions of the spatially homogeneous Landau equation. Anal. Appl., 2007, 5: 29-49

[16] Guo Y.. The Landau equation in a periodic box. Comm. Math. Phys., 2002, 231: 391-434

[17] Lemou M.. Linearized quantum and relativistic Fokker-Planck-Landau equations. Math. Models Method Appl. Sci., 2000, 23: 1093-1119

[18] Liu S. Q., Ma X.. Global classical solutions of the Landau-Fermi-Dirac equation on torus. Chin. Ann. Math., 2012, 33A(1): 39-64

[19] Lu X.. On the Boltzmann equation for Fermi-Dirac particles with very soft potentials: global existence of weak solutions. J. Diff. Eq., 2008, 245(7): 1705-1761

[20] Villani C.. A review of mathematical topics in collisional kinetic theory. Handb. Math. Fluid. Dyn., 2002, 1: 71-305