  Numerous evolution equations with nonlocal convolution-type interactions have
been proposed. In some cases, a convolution was imposed as the velocity in the
advection term. Motivated by analyzing these equations, we approximate
advective nonlocal interactions as local ones, thereby converting the effect of
nonlocality. In this study, we investigate whether the solution to the nonlocal
Fokker-Planck equation can be approximated using the Keller-Segel system. By
singular limit analysis, we show that this approximation is feasible for the
Fokker-Planck equation with any potential and that the convergence rate is
specified. Moreover, we provide an explicit formula for determining the
coefficient of the Lagrange interpolation polynomial with Chebyshev nodes.
Using this formula, the Keller-Segel system parameters for the approximation
are explicitly specified by the shape of the potential in the Fokker-Planck
equation. Consequently, we demonstrate the relationship between advective
nonlocal interactions and a local dynamical system.
