Communications in Mathematical Sciences
Volume 20 (2022)
Existence of smooth solutions to the Landau–Fermi–Dirac equation with Coulomb potential
Pages: 2315 – 2365
In this paper, we prove global-in-time existence and uniqueness of smooth solutions to the homogeneous Landau–Fermi–Dirac equation with Coulomb potential. The initial conditions are nonnegative, bounded and integrable. We also show that any weak solution converges towards the steady state given by the Fermi–Dirac statistics. Furthermore, the convergence is algebraic, provided that the initial datum is close to the steady state in a suitable weighted Lebesgue norm.
Landau–Fermi–Dirac equation, existence and uniqueness, regularity, coercivity, dissipation, long-time behavior, algebraic decay, H-theorem
2010 Mathematics Subject Classification
35Bxx, 35K55, 35K59, 35P15, 35Q84, 82C40, 82D10
WG is partially supported by the NSF-DMS grant 1840314.
MG is partially supported by the DMS-NSF 2019335 and would like to thank NCTS Mathematical Division of Taipei for their kind hospitality.
NZ acknowledges support from the Alexander von Humboldt Foundation (AvH) and from the Austrian Science Foundation (FWF), grants P30000, P33010.
Received 20 October 2021
Received revised 15 March 2022
Accepted 21 March 2022
Published 29 November 2022