Maximal $L_1$-regularity for the linearized compressible Navier-Stokes equations

In this paper, we consider the linearized compressible Navier-Stokes equations with non-slip boundary conditions in the half space ${\mathbb{R}}_{+}^N$. We prove the generation of a continous analytic semigroup associated with this compressible Stokes system with non-slip boundary conditions in the half space ${\mathbb{R}}_{+}^N$ and its $L_1$ in time maximal regularity. We choose the Besov space ${\mathcal{H}}_{q,r}^s=B_{q,r}^{s+1}({\mathbb{R}}_{+}^N)\times B_{q,r}^s({\mathbb{R}}_{+}^N{)}^N$ as an underlying space, where $1<q<\mathrm{\infty }$, $1\le r<\mathrm{\infty }$, and $-1+1\mathrm{/}q<s<1\mathrm{/}q$. We prove the generation of a continuous analytic semigroup $\{T(t){\}}_{t\ge 0}$ on ${\mathcal{H}}_{q,r}^s$, and show that its generator admits maximal $L_1$ regularity. Our approach is to prove the existence of the resolvent in ${\mathcal{H}}_{q,1}^s$ and some new estimates for the resolvent by using $B_{q,1}^{s+1}({\mathbb{R}}_{+}^N)\times B_{q,1}^{s\pm \sigma }({\mathbb{R}}_{+}^N)$ norms for some small $\sigma >0$ satisfying the condition $-1+1\mathrm{/}q<s-\sigma <s<s+\sigma <1\mathrm{/}q$.