Ill-posedness for the Euler-Poincaré equations in Besov spaces

We prove that the initial value problem for the Euler-Poincar & eacute; equations is not locally well-posed for initial data in the Besov space B-p,infinity(sigma) whenever sigma > 2 + max{1 + d/p, 3/2}. By presenting a new construction of initial data u(0), we prove the corresponding solution of Euler-Poincar & eacute; equations starting from u(0) is discontinuous at t = 0 in the norm of B-p,infinity(sigma), which implies the ill-posedness. Since this problem is locally well-posed in the Besov space B-p,r(sigma) for r < infinity and the same sigma, our result suggests that well-posedness does not hold at the endpoint r = infinity. (c) 2023 Elsevier Ltd. All rights reserved.