arXiv:2611.05431v1 Announce Type: new Abstract: We study the approximation of the no-flux McKean-Vlasov equation on a bounded convex domain by a whole-space equation with quadratic confinement of strength $\lambda^{-1}$ outside the domain. For a smooth bounded interaction kernel and a common bounded initial density with square-integrable gradient, we establish an $O(\lambda^{1/4})$ approximation rate. The estimate controls the $L^2$ density error uniformly on every finite time interval and the gradient error in space-time $L^2$. Without interaction, a spectral gap independent of $\lambda$ yields the $O(\lambda^{1/4})$ rate uniformly in time for the density error and uniformly over all unit time intervals for the gradient error. With interaction, we construct a counterexample showing that even weak convergence can fail to be uniform in time.
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