Fractional Diffusion Limit for a Fractional Vlasov--Fokker--Planck Equation

Abstract

This paper is devoted to the rigorous derivation of the macroscopic limit of a Vlasov-Fokker-Planck equation in which the Laplacian is replaced by a fractional Laplacian. The evolution of the density is governed by a fractional heat equation with the addition of a convective term coming from the external force. The analysis is performed by a modified test function method and by obtaining a priori estimates from quadratic entropy bounds. In addition, we give the proof of existence and uniqueness of solutions to the Vlasov-fractional-Fokker-Planck equation.

Anomalous Diffusion Limit of Kinetic Equations in Spatially Bounded Domains

“…For a more detailed presentation of the equilibrium of L s we refer the reader to [1] and references within.…”

“…The existence and uniqueness of such weak solutions can be established by adapting the method of Carrillo in [8] or Mellet and Vasseur in [30] in order to handle the non-local property of the diffusion operator. In the whole space, this was done my the author and Aceves-Sanchez in [1]. We do not dwell on this issue for it is not the focus of this paper.…”

“…Finally, the second inequality in (39) comes from the modified logarithmic Sobolev inequality of Gentil-Imbert (Theorem 3 in [19]) which we can use here because F (v) is the infinitely divisible law associated with the Lévy measure 1/|v| d+2s . We refer the interested reader to [1] for a proof of this functional inequality in the fractional Laplacian case.…”

Boundedness and homogeneous asymptotics for a fractional logistic Keller-Segel equations

“…Interestingly, the (fractional) Keller-Segel system can be recovered as limit cases of other equations. In this regards, Lattanzio & Tzavaras [46] considered the Keller-Segel system as high friction limits of the Euler-Poisson system with attractive potentials (note that the case with fractional diffusion corresponds to the nonlocal pressure law p(u) = Λ α−2 u(x)) while Bellouquid, Nieto & Urrutia [7] obtained the fractional Keller-Segel system as a hydrodynamic limit of a kinetic equation (see also Chalub, Markowich, Perthame & Schmeiser [21], Mellet, Mischler & Mouhot [62], Aceves-Sanchez & Mellet [2] and Aceves-Sanchez & Cesbron [1]).…”

“…We will restrict ourselves to the (relevant biologically) case of u ≥ 0, ensured by u 0 ≥ 0. The parameter χ > 0 quantifies the sensitivity of organisms to the attracting chemical signal and ν ≥ 0 models its decay 1 .…”