On time-dependent Besov vector fields and the regularity of their flows

TL;DR

This paper proves that a broad class of time-dependent Besov vector fields have unique flows within the same Besov space, and establishes continuity properties of the flow mapping with respect to the Besov regularity parameters.

Contribution

It demonstrates ODE-closedness for Besov spaces and continuity of the flow map, extending known results to a wider class of vector fields and regularity settings.

Findings

Unique flows exist within the Besov space for the considered vector fields.

Flow mapping is continuous between Besov spaces with different regularities.

Hölder continuity of the flow map is established for certain regularity differences.

Abstract

We show ODE-closedness for a large class of Besov spaces $\mathcal{B}^{n,\alpha,p}(\mathbb{R}^d,\mathbb{R}^d)$, where $n \geq 1,~\alpha \in (0,1],~p \in [1,\infty]$. ODE-closedness means that pointwise time-dependent $\mathcal{B}^{n,\alpha,p}$-vector fields $u$ have unique flows $\Phi_u \in \operatorname{Id} + \mathcal{B}^{n,\alpha,p}(\mathbb{R}^d,\mathbb{R}^d)$. The class of vector fields under consideration contains as a special case the class of Bochner integrable vector fields $L^1(I, \mathcal{B}^{n,\alpha,p}(\mathbb{R}^d,\mathbb{R}^d))$. In addition, for $n \geq 2$ and $\alpha < \beta$, we show continuity of the flow mapping $L^1(I,\mathcal{B}^{n,\beta,p}(\mathbb{R}^d,\mathbb{R}^d)) \rightarrow C(I,\mathcal{B}^{n,\alpha,p}(\mathbb{R}^d,\mathbb{R}^d)), ~ u \mapsto \Phi_u-\operatorname{Id}$. We even get $\gamma$-H\"older continuity for any $\gamma < \beta - \alpha$.