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Abstract
In this paper we are interested in the large time behavior of linear kinetic equations with heavy-tailed local equilibria. Our main contribution concerns the kinetic Levy-Fokker-Planck equation, for which we adapt hypocoercivity techniques in order to show that solutions converge exponentially fast to the global equilibrium. Compared to the classical kinetic Fokker-Planck equation, the issues here concern the lack of symmetry of the non-local Levy-Fokker-Planck operator and the understanding of its regularization properties. As a complementary related result, we also treat the case of the heavy-tailed BGK equation.
Highlights
We consider a distribution function f ≡ f (t, x, v) depending on time t ≥ 0, position x ∈ Td = Rd /Zd and velocity v ∈ Rd which satisfies the fractional kinetic Fokker–Planck equation∂t f + v · ∇x f = ∇v · (v f ) − (−∆v )α/2 f . (1)Here we assume α ∈ (0, 2) and the fractional Laplacian −(−∆v )α/2 is such that for any Schwartz function g : Rd → R, one has F ((−∆v )α/2g )(ξ) = |ξ|αF (g )(ξ) where F ( · ) denotes the Fourier transform
Where P.V. stands for the principal value and the constant is given by Cd,α =
In the subsequent proofs, we denote by C a positive constant depending only on fixed numbers and its value may change from line to line
Summary
Note that fractional hypocoercivity has already been studied recently in [5] where a L2-hypocoercivity approach is developed In this sense, their framework is quite different, note that it is more general than ours (in terms of phase space and linear operators). In particular let us point out that we do not need fractional derivatives in our Lyapunov functionals and our proof does not rely on Fourier transform. In this sense our method differs completely from that of [11] and the recent [5] in which a mode by mode analysis is developed. In the subsequent proofs, we denote by C a positive constant depending only on fixed numbers (including d and α) and its value may change from line to line
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