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 $L_t^{\infty }{C}_x^{1,\alpha }$ 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 $B_{2,1}^1$. As a consequence we show that a sequence of best constants of the Sobolev embedding theorem near the critical function space is not continuous.

References in corpus (3)

• Ill-posedness results in critical spaces for some equations arising in hydrodynamics
• Ill-posedness of the incompressible Euler equations in the $C^1$ space
• Local ill-posedness of the Euler equations in $B_{\infty ,1}^1$

Cited by in corpus (1)

• Local well-posedness of the incompressible Euler equations in $B_{\infty ,1}^1$ and the inviscid limit of the Navier-Stokes equations