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

Abstract

We present a priori estimates and unique solvability results in the mixed-norm Lebesgue spaces for kinetic Kolmogorov-Fokker-Planck (KFP) equation in divergence form. The leading coefficients are bounded uniformly nondegenerate with respect to the velocity variable v and satisfy a vanishing mean oscillation (VMO) type condition. 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. Introduction and the main results 1.1. Notation and assumptions 1.2. Function spaces. 1.3. Main results 1.4. L 2 theory for the kinetic equations with bounded measurable coefficients 1.5. Motivation 1.6. Related works 1.7. Organization of the paper 2. S p -estimate for the model equation 3. Mixed-norm estimate for the model equation 3.1. Proof of Proposition 3.3 3.2. Proof of Theorem 3.1 4. Proof of Theorem 1.12 4.1. Proof of the assertion (i) 4.2. Proof of Theorem 1.12 (ii) and (iii).

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

“…The following 'nonlocal' lemma is similar to Lemma 5.5 of [17] and Lemma 3.8 in [18]. In the present authors' opinion, such 'nonlocal' lemmas are the technical novelties of the papers [17] - [18], and the current article.…”

“…It is a fundamental problem to establish the maximal regularity for the KFP equation in various functional spaces such as Hölder spaces (see [3], [9], [11], [24], [25], [27], [34], [35], [38]) and L p spaces (see [6], [10], [17] - [18] and the references therein) that is analogous to the theory developed for nondegenerate equations (see, for example, [22], [29], [31]). Such results play a crucial role in the studies of the conditional regularity of the LE (see [24]) and boundary value problem for the LE with the specular reflection boundary condition (see [12] - [13]).…”

“…See the details in Section 1.5. This article is a part of the present authors' program to develop the maximal regularity results for the KFP equation with rough leading coefficients via kernel free approach (see [17] - [18]).…”