Maximal $L_1$-regularity of the Navier-Stokes equations with free boundary conditions via a generalized semigroup theory

Abstract

This paper develops a new approach to show the maximal regularity theorem of the Stokes equations with free boundary conditions in the half-space ${\mathbb{R}}_{+}^d$, $d\ge 2$, within the $L_1$-in-time and ${\mathcal{B}}_{q,1}^s$-in-space framework with $(q,s)$ satisfying $1<q<\infty$ and $1+1/q<s<1/q$, where ${\mathcal{B}}_{q,1}^s$ stands for either homogeneous or inhomogeneous Besov spaces. In particular, we establish a generalized semigroup theory within an $L_1$-in-time and ${\mathcal{B}}_{q,1}^s$-in-space framework, which extends a classical $C_0$-analytic semigroup theory to the case of inhomogeneous boundary conditions. The maximal $L_1$-regularity theorem is proved by estimating the Fourier--Laplace inverse transform of the solution to the generalized Stokes resolvent problem with inhomogeneous boundary conditions, where density and interpolation arguments are used. The maximal $L_1$-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 $\mathbf{a}$ in $B_{q,1}^s({\mathbb{R}}_{+}^d{)}^d$, where $q$ and $s$ satisfy $d-1<q\le d$ and $-1+d/q<s<1/q$, respectively. If we assume that the initial data $\mathbf{a}$ are small in ${\dot{B}}_{q,1}^{1+d/q}({\mathbb{R}}_{+}^d{)}^d$, $d1<q<2d$, then the unique existence of a global strong solution to the system is proved.

Minor version. Accepted for publication in Journal of Differential Equations

References in corpus (3)

• On homogeneous Sobolev and Besov spaces on the whole and the half space
• Global wellposedness of the 3D compressible Navier-Stokes equations with free surface in the maximal regularity class
• $L_1$ approach to the compressible viscous fluid flows in the half-space