Continuity of the solution map of the Euler equations in Hölder spaces and weak norm inflation in Besov spaces

Abstract

We construct an example showing that the solution map of the Euler equations is not continuous in the Hölder space from $C^{1,\alpha }$ to $L^\infty _tC^{1,\alpha }_x$ for any $0<\alpha <1$. On the other hand we show that it is continuous when restricted to the little Hölder subspace $c^{1,\alpha }$. We apply the latter to prove an ill-posedness result for solutions of the vorticity equations in Besov spaces near the critical space $B^1_{2,1}$. As a consequence we show that a sequence of best constants of the Sobolev embedding theorem near the critical function space is not continuous.