Communications in Mathematical Sciences

Volume 20 (2022)

Number 8

Existence of smooth solutions to the Landau–Fermi–Dirac equation with Coulomb potential

Pages: 2315 – 2365



William Golding (Department of Mathematics, University of Texas, Austin, Tx., U.S.A.)

Maria Pia Gualdani (Department of Mathematics, University of Texas, Austin, Tx., U.S.A.)

Nicola Zamponi (Institute for Analysis and Scientific Computing, Vienna University of Technology, Vienna, Austria)


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

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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