Local well-posedness of the incompressible Euler equations in [formula omitted] and the inviscid limit of the Navier–Stokes equations

Abstract

We prove the inviscid limit of the incompressible Navier–Stokes equations in the same topology of Besov spaces as the initial data. The proof is based on proving the continuous dependence of the Navier–Stokes equations uniformly with respect to the viscosity. To show the latter, we rely on some Bona–Smith type argument in the Lp setting. Our obtained result implies a new result that the Cauchy problem of the Euler equations is locally well-posed in the borderline Besov space B∞,11(Rd), d≥2, in the sense of Hadmard, which is an open problem left in recent works by Bourgain and Li in [3,4] and by Misiołek and Yoneda in [12–14].