Free boundary problems of the incompressible Navier–Stokes equations with non-flat initial surface in the critical Besov space

Global well-posedness of the Navier–Stokes equations with a free boundary condition is considered in the scaling critical homogeneous Besov spaces \({\dot{B}}_{p,1}^{-1+n/p}({\mathbb {R}}^n_+)\) ${\dot{B}}_{p,1}^{-1+n/p}({\mathbb{R}}_{+}^n)$ with \(n-1< p< 2n-1\) $n-1<p<2n-1$ . To show the global well-posedness, we establish end-point maximal \(L^1\) $L^1$ -regularity for the initial-boundary value problem of the Stokes equations. Such an estimate is obtained via related estimate for the initial-boundary value problem of the heat equation with the inhomogeneous Neumann data as well as the pressure estimate in the critical Besov space framework. The proof heavily depends on the explicit expression of the fundamental integral kernel of the Lagrange transformed linearized Stokes equations and the almost orthogonal estimates with the space-time Littlewood–Paley dyadic decompositions. Our result here improves the initial space and boundary state than previous results by Danchin–Hieber–Mucha–Tolksdorf (Free boundary problems via Da Prato–Grisvard theory. arXiv:2011.07918v2) and ourselves (Ogawa and Shimizu in J Evol Equ 22(30):67, 2022; Ogawa and Shimizu in J Math Soc Jpn. arXiv:2211.06952v3).