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 ${\Phi }_u\in \mathrm{Id}+{\mathcal{B}}^{n,\alpha ,p}({\mathbb{R}}^d,{\mathbb{R}}^d)$. 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 $L^1(I,{\mathcal{B}}^{n,\beta ,p}({\mathbb{R}}^d,{\mathbb{R}}^d))\to C(I,{\mathcal{B}}^{n,\alpha ,p}({\mathbb{R}}^d,{\mathbb{R}}^d)),u\mapsto {\Phi }_u-\mathrm{Id}$. We even get $\gamma$-Hölder continuity for any $\gamma <\beta -\alpha$.

arXiv comment15 pages; typos corrected, Remark 1.1 included, accepted for publication in Proc. Amer. Math. Soc