Abstract
In this paper, we present new estimates for the entropy dissipation of the Landau–Fermi–Dirac equation (with hard or moderately soft potentials) in terms of a weighted relative Fisher information adapted to this equation. Such estimates are used for studying the large time behaviour of the equation, as well as for providing new a priori estimates (in the soft potential case). An important feature of such estimates is that they are uniform with respect to the quantum parameter. Consequently, the same estimations are recovered for the classical limit, that is the Landau equation.
Highlights
Entropy production estimates have been significantly used in recent years for the study of partial differential equations
Some non optimal functional inequalities linking the entropy production to the associated entropy have been obtained therein which, when combined with a careful spectral analysis, yield an optimal rate of convergence to equilibrium. It is the purpose of this paper to provide a systematic study of the entropy and entropy production functional for the spatially homogeneous Landau–Fermi–Dirac equation, extending the results obtained in [5] and complementing them with a study of the soft potentials case
We aim to provide a systematic study of the entropy production which in particular applies to the study of soft potentials
Summary
Entropy production estimates have been significantly used in recent years for the study of partial differential equations. This latter equation arises in the modelling of plasma and can be derived from the Boltzmann equation in the so-called grazing collision limit For this model, the equivalent of Cercignani’s conjecture was proven to hold first in the special case of the so-called Maxwell molecules in [23], together with weaker versions of the functional inequality linking the entropy to its entropy production for hard potentials. Some non optimal functional inequalities linking the entropy production to the associated entropy have been obtained therein which, when combined with a careful spectral analysis, yield an optimal (exponential) rate of convergence to equilibrium It is the purpose of this paper to provide a systematic study of the entropy and entropy production functional for the spatially homogeneous Landau–Fermi–Dirac equation, extending the results obtained in [5] and complementing them with a study of the soft potentials case. The results of the present contribution in that context will be applied to the long-time behaviour of solutions to the Landau–Fermi–Dirac equation with moderately soft potentials in the forthcoming work [6]
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