Global $${L}_{p}$$ Estimates for Kinetic Kolmogorov–Fokker–Planck Equations in Nondivergence Form

Abstract

We present global Schauder type estimates in all variables and unique solvability results in kinetic Hölder spaces for kinetic Kolmogorov-Fokker-Planck (KFP) equations. The leading coefficients are Hölder continuous in the x, v variables and are merely measurable in the temporal variable. Our proof is inspired by Campanato's approach to Schauder estimates and does not rely on the estimates of the fundamental solution of the KFP operator.

Global $L_p$ estimates for kinetic Kolmogorov-Fokker-Planck equations in divergence form

“…To prove Theorem 1.12, we use the results and techniques of [8], which are based on N.V. Krylov's kernel free approach to nondegenerate parabolic equations (see [16], Chapters 4 -7). The main part of the argument is the mean oscillation estimates of (−∆ x ) 1/6 u and D v u in the case when the coefficients a ij are independent of the x and v variables, and b ≡ b = 0, c = 0, λ = 0.…”

“…Nondivergence form equations For a thorough review of the classical theory for the generalized KFP equations, we refer the reader to [2]. An overview of the literature on the Sobolev theory for KFP equations in nondivergence form can be found in a recent paper [8]. We also mention briefly the following papers:…”

“…• [3] on the interior S p estimate with leading coefficients of class V M O with respect to ρ; • the articles [4], [6], [5], [21] where the global S p estimate is proved under the following assumptions: either the leading coefficients are constant or independent of x, v, or they are continuous with respect to ρ; • [8], where the present authors proved the global S p estimate and unique solvability results with the coefficients satisfying Assumption 1.2.…”

“…The goal of this section is to prove Theorem 1.12 with L p in place of the weighted mixed-norm Lebesgue space, P = P 0 , and without the lower-order terms (see Theorem 2.1). We do this by using the duality argument and the S p -estimate taken from [8], which we state below (see Theorem 2.2).…”

“…We consider the L 2 case separately and treat more general equations which include the relativistic KFP equation. This work is a continuation of [8], where the present authors studied the KFP equations in nondivergence form. Contents 1.…”