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

TL;DR

The paper addresses maximal $L_1$-regularity for the Navier–Stokes equations with free boundary in the half-space, introducing a generalized semigroup framework to handle inhomogeneous boundary data in $L_1$-time and Besov spaces $\mathcal{B}^s_{q,1}$. It constructs solution operators $\mathcal{S}(\lambda)$ and $\mathcal{P}(\lambda)$ for the generalized Stokes resolvent problem, and proves holomorphic dependence with uniform bounds, enabling a fixed-point approach. By combining density and interpolation, the maximal $L_1$-$\mathcal{B}^s_{q,1}$ regularity is extended to the half-space, yielding local well-posedness for arbitrary initial data in $B^s_{q,1}(\mathbb{R}^d_+)^d$ and global well-posedness for small data in $\dot B^{-1+d/q}_{q,1}(\mathbb{R}^d_+)$. The results provide a robust linear theory that supports nonlinear well-posedness and pave the way for applications to hyperbolic–parabolic systems and future work involving surface tension and broader unbounded domains.

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.

Theorems & Definitions (95)

• Definition 1.1
• Remark 1.2
• Definition 1.3
• Remark 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Corollary 1.8
• Corollary 1.9
• Corollary 1.10