On the Continuity of the Solution Map of the Euler–Poincaré Equations in Besov Spaces

Abstract

By constructing a series of perturbation functions through localization in the Fourier domain and using a symmetric form of the system, we show that the data-to-solution map for the Euler–Poincaré equations is nowhere uniformly continuous in $$B^s_{p,r}(\mathbb {R}^d)$$ with $$s>\max \{1+\frac{d}{2},\frac{3}{2}\}$$ and $$(p,r)\in (1,\infty )\times [1,\infty )$$ . This improves our previous result (Li et al. in Nonlinear Anal RWA 63:103420, 2022) which shows the data-to-solution map for the Euler–Poincaré equations is non-uniformly continuous on a bounded subset of $$B^s_{p,r}(\mathbb {R}^d)$$ near the origin.