Global vanishing capillarity limit of the compressible Navier-Stokes-Korteweg system

This work is dedicated to the vanishing capillarity limit of strong solutions to the compressible Navier-Stokes-Korteweg system in $ L^p $ Besov spaces. We constructed the uniform global-in-time solutions with respect to the capillarity parameter $ \kappa $ by employing the viscous effective flux. The strong solution satisfies the asymptotic formula $ (\varrho^\kappa, u^\kappa) = (\varrho^0, u^0)+\mathcal{O}(\sqrt{\kappa}) $ in the critical regularity setting for any positive times, where $ (\varrho^0, u^0) $ is the global solution to the compressible Navier-Stokes system. This is the first effort to show the global vanishing capillarity limit at the convergence rate of order $ \sqrt{\kappa} $.