Ill-posedness and inviscid limit of basic equations of fluid dynamics in Besov spaces

In this paper, we consider the Cauchy problem to the basic equations of fluid dynamics on the torus. Firstly, we construct a new initial data and provide a simple proof on the ill-posedness of $B_{p,\mathrm{\infty }}^s$ solution of the Euler equations and the surface quasi-geostrophic equation, which covers the results obtained by Cheskidov-Shvydkoy \cite{CS} and Misiołek-Yoneda \cite{MY}. Secondly, we prove the failure of the $B_{p,\mathrm{\infty }}^s$-convergence in the inviscid limit for both the Navier-Stokes equations and the surface quasi-geostrophic equation.