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

Abstract

We study the degenerate Kolmogorov equations (also known as kinetic Fokker-Planck equations) in nondivergence form. The leading coefficients a ij are merely measurable in t and satisfy the vanishing mean oscillation (VMO) condition in x, v with respect to some quasi-metric. We also assume boundedness and uniform nondegeneracy of a ij with respect to v. We prove global a priori estimates in weighted mixed-norm Lebesgue spaces and solvability results. We also show an application of the main result to the Landau equation. Our proof does not rely on any kernel estimates.

Kinetic Fokker-Planck and Landau Equations with Specular Reflection Boundary Condition

“…By using a perturbation argument, we extend the aforementioned unique solvability result to the viscous linearized Landau equation, which contains the non-local term K g f defined in (1.7). We then prove uniform in ν bounds by combining the standard energy estimate (see Lemma 5.7) with the S p estimates of [5]. Finally, by using the weak* compactness argument, we prove the existence of the finite energy strong solution (see Definition 1.3) to the linearized Landau equation (1.5).…”

“…T (see Section 4) to the extended equation (2.17) and conclude that a finite energy weak solution with θ ≥ 2 is of class S 2 (Σ T ). We then use the S p regularity results of [5] (see Appendix D) to conclude that f ∈ S p (Σ T ). Unfortunately, the mirror extension argument works only for very particular diffusion operators in the velocity variable, for example, ∆ v and ∇ v • (σ G ∇ v ), where σ G is defined in (1.6).…”

“…By the existence result for the generalized kinetic Fokker-Planck equation (see Theorem 1.4), we prove the existence of a finite energy weak solution in the sense of Definition 1.1. To prove the unique solvability in the class of the finite energy strong solutions, we use the mirror extension method combined with the S 2 estimate of [5] (see Appendix D).…”

“…Then, repeating the above argument we used to estimate K 1 u, we get In this section, we present the main results of [5]. Denote The following theorem is a simplified version of Theorem 2.4 of [5].…”

“…Away from the boundary ∂Ω, by the partition of unity argument, we may reduce these equations to the ones on the whole space R 7 T . To show that f ∈ S 2 away from the boundary, we use the unique solvability result of [5] (see Theorem D.5). As discussed in the Appendix D (see Remark D.2), this theorem is applicable if the leading coefficients are of class L ∞ ((0, T ),…”