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 [Guo et al. J. Funct. Anal. 276:2821–2830, 2019] that given initial data \(u_0\in B^{s}_{p,r}\) with \(1\le r<\infty\), the solution of the Navier–Stokes equations converges strongly in \(B^{s}_{p,r}\) to the solution of the Euler equations as the viscosity parameter tends to zero. In the case when \(r=\infty\), we prove the failure of the \(B^{s}_{p,\infty }\)-convergence of the Navier-Stokes equations toward the Euler equations in the inviscid limit.