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Harnack inequality for parabolic equations in double-divergence form with singular lower order coefficients

Istvan Gyöngy, Seick Kim

TL;DR

The paper proves a parabolic Harnack inequality for nonnegative solutions to parabolic equations in double divergence form with singular lower-order terms. It develops a two-pronged regularization strategy: a Krylov-type c-regularization and a Zvonkin-type drift smoothing, complemented by an absorption step that embeds the lower-order terms into an augmented principal operator. This yields a robust Harnack estimate under Dini mean oscillation of the principal coefficients and Morrey-type control of the lower-order terms, improving prior results by requiring only $c \\in L_p$ with $p > d/2$ and accommodating double-divergence structure in the parabolic setting. The work thus advances the mathematical theory of parabolic density equations, including Fokker-Planck-Kolmogorov models, by providing structural conditions under which Harnack inequalities hold and hence contributing to the understanding of regularity and qualitative behavior of densities in these models.

Abstract

This paper investigates the Harnack inequality for nonnegative solutions to second-order parabolic equations in double divergence form. We impose conditions where the principal coefficients satisfy the Dini mean oscillation condition in $x$, while the drift and zeroth-order coefficients belong to specific Morrey classes. Our analysis contributes to advancing the theoretical foundations of parabolic equations in double divergence form, including Fokker-Planck-Kolmogorov equations for probability densities.

Harnack inequality for parabolic equations in double-divergence form with singular lower order coefficients

TL;DR

The paper proves a parabolic Harnack inequality for nonnegative solutions to parabolic equations in double divergence form with singular lower-order terms. It develops a two-pronged regularization strategy: a Krylov-type c-regularization and a Zvonkin-type drift smoothing, complemented by an absorption step that embeds the lower-order terms into an augmented principal operator. This yields a robust Harnack estimate under Dini mean oscillation of the principal coefficients and Morrey-type control of the lower-order terms, improving prior results by requiring only with and accommodating double-divergence structure in the parabolic setting. The work thus advances the mathematical theory of parabolic density equations, including Fokker-Planck-Kolmogorov models, by providing structural conditions under which Harnack inequalities hold and hence contributing to the understanding of regularity and qualitative behavior of densities in these models.

Abstract

This paper investigates the Harnack inequality for nonnegative solutions to second-order parabolic equations in double divergence form. We impose conditions where the principal coefficients satisfy the Dini mean oscillation condition in , while the drift and zeroth-order coefficients belong to specific Morrey classes. Our analysis contributes to advancing the theoretical foundations of parabolic equations in double divergence form, including Fokker-Planck-Kolmogorov equations for probability densities.
Paper Structure (8 sections, 10 theorems, 146 equations)

This paper contains 8 sections, 10 theorems, 146 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Proof of the main theorem
  4. Regularization of singular zeroth order coefficient
  5. Regularization of singular drift term
  6. Absorbing lower order coefficients into the principal coefficients
  7. Proof of Harnack inequality
  8. Harnack inequality for elliptic equations in double divergence form

Key Result

Theorem 1.1

Assume Conditions cond1 and cond2 hold. Let $R>0$ be a fixed number, $0<r<R/4$, and $(t_0, x_0) \in \mathbb{R}^{d+1}$. Denote $C_{r}=(t_0-r^2, t_0] \times B_{r}(x_0)$. Suppose $u \in L_1(C_{4r})$ is a nonnegative solution of Then, we have where $N$ is a constant depending only on $d$, $\delta$, $\omega_{\mathbf A}^{\mathsf x}$, $p$, $q$, $\beta$, $\mathfrak{b}$, $\mathfrak{c}$, and $R$.

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 2.4
  • Definition 2.5
  • Lemma 2.7
  • proof
  • Lemma 2.13
  • proof
  • Remark 2.19
  • Theorem 2.20
  • proof
  • ...and 15 more