Non-convergence of the Navier-Stokes equations toward the Euler equations in the endpoint Besov spaces

In this paper, we consider the inviscid limit problem to the higher dimensional incompressible Navier-Stokes equations in the whole space. It was proved in \cite[J. Funct. Anal., 276 (2019)]{GZ} that given initial data $u_0\in B_{p,r}^s$ with $1\le r<\mathrm{\infty }$, the solution of the Navier-Stokes equations converges strongly in $B_{p,r}^s$ to the solution of the Euler equations as the viscosity parameter tends to zero. In the case when $r=\mathrm{\infty }$, we prove the failure of the $B_{p,\mathrm{\infty }}^s$-convergence of the Navier-Stokes equations toward the Euler equations in the inviscid limit.