# A note on the Lagrangian flow associated to a partially regular vector field

## TL;DR

This paper provides quantitative estimates for the Lagrangian flow of a partially regular vector field with mixed Sobolev regularity, ensuring well-posedness, stability, and compactness of the flow.

## Contribution

It introduces new estimates for flows driven by vector fields with mixed Sobolev and fractional Sobolev regularity, extending prior results to more complex regularity structures.

## Key findings

- Establishes well-posedness of the Lagrangian flow.
- Provides quantitative stability estimates.
- Demonstrates compactness of the flow.

## Abstract

In this paper we derive quantitative estimates for the Lagrangian flow associated to a partially regular vector field of the form $$ b(t,x_1,x_2) = (b_1(t,x_1),b_2(t,x_1,x_2)) \in {\mathbb R}^{n_1}\times{\mathbb R}^{n_2} \,, \qquad (x_1,x_2)\in{\mathbb R}^{n_1}\times{\mathbb R}^{n_2}\,. $$ We assume that the first component $b_1$ does not depend on the second variable $x_2$, and has Sobolev $W^{1,p}$ regularity in the variable $x_1$, for some $p>1$. On the other hand, the second component $b_2$ has Sobolev $W^{1,p}$ regularity in the variable $x_2$, but only fractional Sobolev $W^{\alpha,1}$ regularity in the variable $x_1$, for some $\alpha>1/2$. These estimates imply well-posedness, compactness, and quantitative stability for the Lagrangian flow associated to such a vector field.