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

Abstract

We show ODE-closedness for a large class of Besov spaces B n , α , p ( R d , R d ) \mathcal {B}^{n,\alpha ,p}(\mathbb {R}^d,\mathbb {R}^d) , where n ≥ 1 , α ∈ ( 0 , 1 ] , p ∈ [ 1 , ∞ ] n \ge 1,~\alpha \in (0,1],~p \in [1,\infty ] . ODE-closedness means that pointwise time-dependent B n , α , p \mathcal {B}^{n,\alpha ,p} -vector fields u u have unique flows Φ u ∈ Id + B n , α , p ( R d , R d ) \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 , B n , α , p ( R d , R d ) ) L^1(I, \mathcal {B}^{n,\alpha ,p}(\mathbb {R}^d,\mathbb {R}^d)) . In addition, for n ≥ 2 n \ge 2 and α > β \alpha > \beta , we show continuity of the flow mapping L 1 ( I , B n , β , p ( R d , R d ) ) → C ( I , B n , α , p ( R d , R d ) ) , u → Φ u − Id 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 γ \gamma -Hölder continuity for any γ > β − α \gamma > \beta - \alpha .