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.