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

Abstract

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}^s_{q,r} = B^{s+1}_{q,r}( \mathbb{R}^N_{+})\times B^s_{q,r}( \mathbb{R}^N_{+})^N$ as an underlying space, where $1 < q < \infty$, $1\leq r < \infty$, and $-1+1/q < s < 1/q$. We prove the generation of a continuous analytic semigroup $\{T(t)\}_{t\geq 0}$ on $\mathcal{H}^s_{q,r}$, and show that its generator admits maximal $L_1$ regularity. Our approach is to prove the existence of the resolvent in $\mathcal{H}^s_{q,1}$ and some new estimates for the resolvent by using $B^{s+1}_{q,1}( \mathbb{R}^N_{+}) \times B^{s\pm\sigma}_{q,1}( \mathbb{R}^N_{+})$ norms for some small $\sigma > 0$ satisfying the condition $-1+1/q < s-\sigma < s < s+\sigma < 1/q$.