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Quantitative estimates for flows of regular Lagrangian flows for Hörmander singular kernels and LD vector fields

Henrique Borrin

TL;DR

The paper addresses the well-posedness of regular Lagrangian flows for non-smooth vector fields whose derivatives have a Hörmander-convolution structure with summable data, under a controlled growth condition. The main contribution is a fundamental estimate that quantifies flow stability under $L^1$ perturbations of the vector field, achieved by a refined grand maximal operator framework and a Calderón-Zygmund decomposition tailored to LD_loc vector fields. This extends the Bouchut-Crippa approach to a broader class of singular kernels and provides a mean-value inequality for vector fields with symmetric derivatives, enabling renormalized flow analysis and the associated transport and continuity equations. The results significantly broaden the class of vector fields for which existence, uniqueness, and stability of regular Lagrangian flows can be established, with potential impact on PDEs involving Hörmander-type singular integrals.

Abstract

In this paper, we obtain quantitative estimates of regular Lagrangian flows associated to vector fields whose derivative can be written as convolution of a fundamental singular kernel satisfying the ``Hörmander'' condition convoluted with summable function in spacetime.

Quantitative estimates for flows of regular Lagrangian flows for Hörmander singular kernels and LD vector fields

TL;DR

The paper addresses the well-posedness of regular Lagrangian flows for non-smooth vector fields whose derivatives have a Hörmander-convolution structure with summable data, under a controlled growth condition. The main contribution is a fundamental estimate that quantifies flow stability under perturbations of the vector field, achieved by a refined grand maximal operator framework and a Calderón-Zygmund decomposition tailored to LD_loc vector fields. This extends the Bouchut-Crippa approach to a broader class of singular kernels and provides a mean-value inequality for vector fields with symmetric derivatives, enabling renormalized flow analysis and the associated transport and continuity equations. The results significantly broaden the class of vector fields for which existence, uniqueness, and stability of regular Lagrangian flows can be established, with potential impact on PDEs involving Hörmander-type singular integrals.

Abstract

In this paper, we obtain quantitative estimates of regular Lagrangian flows associated to vector fields whose derivative can be written as convolution of a fundamental singular kernel satisfying the ``Hörmander'' condition convoluted with summable function in spacetime.
Paper Structure (4 sections, 7 theorems, 99 equations)

This paper contains 4 sections, 7 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and proof of
  3. Grand maximal operator and Hörmander condition
  4. Mean value inequality with symmetric derivative

Key Result

Theorem 1.1

Let $\boldsymbol{b},\,\bar{\boldsymbol{b}}\in L_{\operatorname{loc}}^1((0,T)\times\mathbb{R}^d))$ be vector fields satisfying growth. Moreover, assume that $\boldsymbol{b}$ with derivatives as in derivativeconvolution, with singular kernels $\Gamma_{ijk,R}$ as in singularkernel and summable function

Theorems & Definitions (13)

  • Definition 1.1: Fundamental singular kernel
  • Definition 1.2: Regular Lagrangian flow
  • Theorem 1.1: Fundamental estimate
  • Proposition 1.1
  • Definition 2.1
  • Lemma 2.1
  • Lemma 2.2
  • proof : Proof of \ref{['fundamental']}.
  • Theorem 2.1: Calderón-Zygmund decomposition
  • Proposition 3.1: Weak $L^1$ estimate
  • ...and 3 more