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

Abstract:

We show ODE-closedness for a large class of Besov spaces $\mathcal {B}^{n,\alpha ,p}(\mathbb {R}^d,\mathbb {R}^d)$, where $n \ge 1,~\alpha \in (0,1],~p \in [1,\infty ]$. 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 $n \ge 2$ and $\alpha < \beta$, we show continuity of the flow mapping $L^1(I,\mathcal {B}^{n,\beta ,p}(\mathbb {R}^d,\mathbb {R}^d)) \rightarrow C(I,\mathcal {B}^{n,\alpha ,p}(\mathbb {R}^d,\mathbb {R}^d)), ~ u \to \Phi _u-\operatorname {Id}$. We even get $\gamma$-Hölder continuity for any $\gamma < \beta - \alpha$.