Ill-posedness for the Euler equations in Besov spaces

Abstract

In the paper, we consider the Cauchy problem to the Euler equations in Rd with d≥2. We construct an initial data u0∈Bp,∞σ showing that the corresponding solution map of the Euler equations starting from u0 is discontinuous at t=0 in the metric of Bp,∞σ, which implies the ill-posedness for this equation in Bp,∞σ. We generalize the periodic result of Cheskidov and Shvydkoy (2010).