On the Continuity of the Solution Map of the Euler-Poincare Equations in Besov Spaces

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-Poincare'equations is nowhere uniformly continuous in B-p,r(s)(R-d) with s > max{1+ d/2, 3/2 } and (p, r) ? (1, 8) x [1, 8). 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-Poincare' equations is non-uniformly continuous on a bounded subset of B-p,r(s)(R-d) near the origin.