Maximal L 1-regularity of the Navier-Stokes equations with free boundary conditions via a generalized semigroup theory

This paper develops a new approach to show the maximal regularity theorem of the Stokes equations with free boundary conditions in the half-space Rd<INF>+</INF>, d >= 2, within the L <INF>1</INF>-in-time and Bs<INF>q,1</INF>-in-space framework with (q, s) satisfying 1 1 + 1/q 1/q, where ta sq, 1 stands for either homogeneous or inhomogeneous Besov spaces. In particular, we establish a generalized semigroup theory within an L <INF>1</INF>- in-time and Bs<INF>q,1</INF>-in-space framework, which extends a classical C <INF>0</INF>-analytic semigroup theory to the case of inhomogeneous boundary conditions. The maximal L <INF>1</INF>-regularity theorem is proved by estimating the Fourier-Laplace inverse transform of the solution to the generalized Stokes resolvent problem with in- homogeneous boundary conditions, where density and interpolation arguments are used. The maximal L<INF> 1</INF>-regularity theorem is applied to show the unique existence of a local strong solution to the Navier- Stokes equations with free boundary conditions for arbitrary initial data a in Bs<INF>q,1</INF> (R d<INF>+</INF> ) d , where q and s satisfy d - 1 < q <= d and - 1 + d/q < s < 1/q, respectively. If we assume that the initial data a are small in B center dot - 1 +d/q <INF>q, 1</INF> (Rd<INF>+</INF>)d , d - 1 < q < 2d, then the unique existence of a global strong solution to the system is proved. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.