Hölder Continuity for a Family of Nonlocal Hypoelliptic Kinetic Equations

Abstract

In this work, Holder continuity is obtained for solutions to the nonlocal kinetic Fokker-Planck Equation, and to a family of related equations with general integro-differential operators. These equations can be seen as a generalization of the Fokker-Planck Equation, or as a linearization of non-cutoff Boltzmann. Difficulties arise because our equations are hypoelliptic, so we utilize the theory of averaging lemmas. Regularity is obtained using De Giorgi's method, so it does not depend on the regularity of initial conditions or coefficients. This work assumes stronger constraints on the nonlocal operator than in the work of Imbert and Silvestre [22], but allows unbounded source terms.2 STOKOLS space with their velocities changing in a stochastic manner. If the velocity of a given particle varied according to the Weiner process, then f would obey a (local) kinetic Fokker-Planck Equation.However, when the velocity of each particle varies according to a Levy process (without drift), the density function obeys (1). A Levy process, unlike the Weiner process, allows individual particles to change velocity suddenly and discontinuously, which better approximates the effect of elastic collisions.Another important model from the statistical mechanics of particles is the Boltzmann Equation). In the non-cutoff case, the Boltzmann Equation sometimes enjoys a regularization effect similar the fractional Fokker-Planck equation (Alexandre, Morimoto, Ukai, Xu, and Yang [2]). Our equation (1) is closely related to the linear approximation of the bilinear collision operator Q(⋅, ⋅). If the mass, energy, and entropy of a solution are assumed to be uniformly bounded, then regularization due to hypoellipticity is observed for the Boltzmann Equation (Imbert and Silvestre [22]), and also for the closely related Landau Equation (Henderson and Snelson [18], Cameron, Silvestre, and Snelson [8]). Note that [22] rewrites the Boltzmann equation in the form (1), but with kernel satisfying weaker constraints than (2). Their regularity results are discussed below. The most important assumption these papers require is that the mass is bounded away from the vacuum, which is connected to the coercivity of the collision operator. In [19], Henderson, Snelson, and Tarfulea show that this assumption really does hold for the Landau Equation. See Mouhot [30] for a thorough review of the current state of research on this front.Equation (1) is a typical hypoelliptic equation. Although regularization of the integral operator happens only in v, we will gain regularity in t, x thanks to the mixing property of the transport operator. This is reminiscent of the hypoelliptic theory based on C ∞ of Hörmander [20] and Kolmogorov. Averaging lemmas such as [14] (Golse, Lions, Perthame, Sentis) can be seen as an H s theory of hypoellipticity.This H s theory has already been applied specifically to the nonlocal kinetic Fokker-Planck Equation. Lerner, Morimoto, showed that solutions to certain fractional kinetic equations are in a Sobolev space H σ in all three variable...

Pencil-Beam Approximation of Stationary Fokker--Planck

“…Several recent results related to the more general and/or closely related non-local kinetic models, such as the Boltzmann equation without cut-off, may be found in [21,22,23,30]. It is expected that results along those lines could lead to quantitative approximation estimates in the 1-Wasserstein sense between the (stationary) radiative transfer solution, in the narrow beam regime, and the fractional Fermi pencil-beam solution.…”

Pencil-beam approximation of fractional Fokker-Planck

“…Several recent results related to the more general and/or closely related non-local kinetic models, such as the Boltzmann equation without cut-off, may be found in [22,23,24,31]. It is expected that results along those lines could lead to quantitative approximation estimates in the 1-Wasserstein sense between the (stationary) radiative transfer solution, in the narrow beam regime, and the fractional Fermi pencilbeam solution.…”

“…We proved above that the radiative transfer solution converged in the highly forward-peaked limit, in a weak sense (and with no known estimates), to the fractional Fokker-Planck solution (Theorem 2.6), and furthermore, the latter solution can be subsequently approximated by a pencil-beam in the narrow beam regime (Theorem 1.1, above). The main obstacle that prevents the application of the regularity results in [22,23,24,31] resides in the a-priori regularity assumed on the solutions. The Hölder estimates proved in [22,31] hold for weak solutions and thus are appropriate for our work.…”

“…The main obstacle that prevents the application of the regularity results in [22,23,24,31] resides in the a-priori regularity assumed on the solutions. The Hölder estimates proved in [22,31] hold for weak solutions and thus are appropriate for our work. In contrast, the local Schauder estimates and their global extension obtained respectively in [23] and [24], assume solutions to be at least classical.…”

Hölder Regularity up to the Boundary for Critical SQG on Bounded Domains

“…The method is powerful for showing C α regularity of elliptic-and parabolic-type equations. It has been applied in a variety of situations for non-local problems, such as the fractional heat equation in [CCV11], the time-fractional case in [ACV16], the 3D Quasigeostrophic problem in [NV18a], or the kinetic setting by Imbert and Silvestre [IS16] or in [Sto18]. The method has also been applied in more exotic, non-elliptic situations such as Hamilton-Jacobi equations (see [CV17], [SV18]).…”