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

TL;DR

This paper introduces a new generalized semigroup approach to establish maximal $L_1$-regularity for the Stokes and Navier-Stokes equations with free boundary conditions in half-space, enabling existence results for solutions with less regular initial data.

Contribution

It develops a generalized semigroup theory within an $L_1$-in-time and Besov space framework, extending classical methods to inhomogeneous boundary conditions for the first time.

Findings

Proves maximal $L_1$-regularity for the Stokes equations with free boundary conditions.

Establishes local existence of strong solutions for Navier-Stokes with arbitrary initial data.

Shows global existence under small initial data assumptions.

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 $\mathcal B^s_{q, 1}$-in-space framework with $(q, s)$ satisfying $1 < q < \infty$ and $1 + 1 / q < s < 1 / q$, where $\mathcal B^s_{q, 1}$ stands for either homogeneous or inhomogeneous Besov spaces. In particular, we establish a generalized semigroup theory within an $L_1$-in-time and $\mathcal B^s_{q,1}$-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 $\boldsymbol a$ in $B^s_{q, 1} (\mathbb R^d_+)^d$, 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 $\boldsymbol a$ are small in $\dot B^{1 + d / q}_{q, 1} (\mathbb R^d_+)^d$, $d 1 < q < 2 d$, then the unique existence of a global strong solution to the system is proved.