$L_1$ approach to the compressible viscous fluid flows in the half-space

The paper is devoted to the proof of the local well-posedness for the Navier–Stokes equations describing the motion of isotropic barotoropic compressible viscous fluid flow with nonslip boundary conditions, where the half-space $\mathbb{R}_+^N = \{x=(x_1, \ldots, x_N) \in \mathbb{R}^N \mid x_N>0\}$ ($N \geq 2$) is the fluid domain. The density part of the solutions belongs to
$$ W^1_1((0, T), B^s_{q,1}(\mathbb{R}_+^N)) \cap L_1((0, T), B^{s+1}_{q,1}(\mathbb{R}_+^N)) $$
and the velocity part of them belongs to
$$ W^1_1((0, T), B^{s}_{q,1}(\mathbb{R}_+^N)^N) \cap L_1((0, T), B^{s+2}_{q,1}(\mathbb{R}_+^N)), $$
where $B^\mu_{q,1}(\mathbb{R}_+^N)$ denotes the standard Besov space on $\mathbb{R}_+^N$. Namely, the equations are solved in the $L_1$ in time and $B^{s+1}_{q,1}(\mathbb{R}_+^N) \times B^s_{q,1}(\mathbb{R}_+^N)^N$ in space maximal regularity framework. The Lagrange transformation is used to eliminate the convection term $\mathbf{v}\cdot\nabla\rho$, and an analytic semigroup approach is invoked. Only the strict positivity of the initial mass density is assumed. An essential assumption is that $-1+N/q \leq s < 1/q $ and $N-1 < q < \infty$. Here, $N/q$ is the crucial order to obtain $\|\nabla \mathbf{u}\|_{L_\infty} \leq C\|\nabla\mathbf{u}\|_{B^{N/q}_{q,1}}$.