Loss of Uniform Convergence for Solutions of the Navier--Stokes Equations in the Inviscid Limit

In this paper, we consider the inviscid limit problem to the higher dimensional incompressible Navier–Stokes equations in the whole space. It is shown in [Guo, Li, Yin: J. Funct. Anal., 276 (2019)] that given initial data $u_0\in B_{p,r}^s$ and for some $T>0$, the solutions of the Navier–Stokes equations converge strongly in $L_T^{\mathrm{\infty }}{B}_{p,r}^s$ to the Euler equations as the viscosity parameter tends to zero. We furthermore prove the failure of the uniform (with respect to the initial data) $B_{p,r}^s$ convergence in the inviscid limit of a family of solutions of the Navier-Stokes equations towards a solution of the Euler equations.