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 condition in x, v with respect to some quasi-metric. We also assume the 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.

Appendix A

Lemma A.1

Let $\sigma > 0$, $R > 0$, $p \geqq 1$ be numbers, and $f \in L_{p, \text {loc} } ({\mathbb {R}}^d)$. Denote

$$\begin{aligned} g ( x) = \int _{|y| > R^3 } f ( x+y) |y|^{- (d + \sigma )} \, \mathrm{d}y. \end{aligned}$$

Then,

$$\begin{aligned} (|g|^p)^{1/p}_{ B_{R^3 } }\leqq N (d , \sigma ) R^{- 3 \sigma } \sum _{k = 0}^{\infty } 2^{- 3 k\sigma } (|f|^p)^{1/p}_{ B_{ (2^kR)^3 } }. \end{aligned}$$

Proof

By Hölder’s inequality for any $x \in B_{R^3 }$, we have

Taking the $L_p$-average of the both sides of the last inequality over $B_{R^3}$ and using the Minkowski inequality, we prove the assertion of this lemma. $\quad \square$

Lemma A.2

Let $p > 1$ be a number, $w \in A_p ({\mathbb {R}}^d)$, and $f \in L_p ({\mathbb {R}}^d, w)$. Then, there exists a number $p_0 > 1$ depending only on d, p, and $[w]_{A_p}$ such that $f \in L_{p_0, \text {loc}} ({\mathbb {R}}^d).$

Proof

By Corollary 7.2.6 of [19], there exists $q \in (1, p)$ depending only on p, d, and $[w]_{A_p ({\mathbb {R}}^d)}$ such that $w \in A_{q} ({\mathbb {R}}^d).$ Let $p_0 = p/q$. Then, by this and Hölder’s inequality for any cube C,

$$\begin{aligned} \int _C |f|^{p_0} \, \mathrm{d}x \leqq \Big (\int _C |f|^p w \, \mathrm{d}x\Big )^{1/q} \Big (\int _C w^{-1/(q-1)} \, \mathrm{d}x\Big )^{(q-1)/q} < \infty . \end{aligned}$$

The lemma is proved. $\quad \square$

For numbers $p_1, \ldots , p_d \in (1, \infty )$, by $L_{p_1, \ldots , p_d} (w_1, \ldots , w_d)$ we denote the space of measurable functions with the finite norm

$$\begin{aligned}&\Vert f\Vert _{ L_{p_1, \ldots , p_d } (w_1, \ldots , w_d) } \\&\quad = \big |\int _{{\mathbb {R}}} \big | \ldots \big |\int _{{\mathbb {R}}} \big |\int _{{\mathbb {R}}} |f|^{p_1} (x) \, w_1 (x_1) \mathrm{d}x_1\big |^{\frac{p_2}{p_1} } \ldots w_d (x_d) \mathrm{d}x_d\big |^{\frac{1}{p_d}}. \end{aligned}$$

Furthermore, by $W^2_{p_1, \ldots , p_d} (w_1, \ldots , w_d)$ we mean the Sobolev space of all functions $u \in L_{p_1, \ldots , p_d} (w_1, \ldots , w_d)$ such that $D_x u, D^2_x u \in L_{p_1, \ldots , p_d} (w_1, \ldots , w_d)$.

Lemma A.3

(Interpolation inequality) Let $p_1, \ldots , p_d \in (1, \infty )$ be arbitrary numbers and $w_i \in A_{p_i} ({\mathbb {R}}), i = 1, \ldots , d$, such that $[w_i]_{ A_{p_{i}} ({\mathbb {R}}) } \leqq K, i = 1, \ldots , d,$ for some $K \geqq 1$. Then, for any $u \in W^2_{p_1, \ldots , p_d} (w_1, \ldots , w_d)$ and $\varepsilon > 0$, we have

$$\begin{aligned} \Vert D_x u\Vert \leqq \varepsilon \Vert D^2_x u\Vert + N \varepsilon ^{-1} \Vert u\Vert , \end{aligned}$$

where $\Vert \cdot \Vert = \Vert \cdot \Vert _{ L_{p_1, \ldots , p_d } (w_1, \ldots , w_d) }$ and $N = N (d, p_1, \ldots , p_d, K).$

Proof

First, by Lemma 3.8 (iii) of [17], for any $w \in A_{p_1} ({\mathbb {R}}^d)$ and $\varepsilon >0$, one has

$$\begin{aligned} \int |D_x u|^{p_1} w (x) \, \mathrm{d}x \leqq N \int |g|^{p_1} w (x) \, \mathrm{d}x, \end{aligned}$$

where

$$\begin{aligned} g (x) = \varepsilon |D^2_x u| + \varepsilon ^{-1} |u| \end{aligned}$$

and $N = N (d, p_1, [w]_{A_{p_1} ({\mathbb {R}}^d) })$. Applying a variant of the Rubio de Francia extrapolation theorem (see, for example, Theorem 7.11 of [17] and also [29]), we prove the lemma. $\quad \square$