Maximal L₁-regularity for the compressible Navier-Stokes equations

In this paper, we consider the compressible Stokes equations with non-slip boundary conditions in the half space RN₊, which is a linearized system of the Navier-Stokes equations describing compressible viscous fluid motion. We prove the generation of a continous analytic semigroup associated with this compressible Stokes system with non-slip boundary conditions in the half space RN₊ and its L₁ in time maximal regularity. We choose the Besov space Hˢq,r = Bˢ⁺¹q,r(RN₊)× Bˢq,r(RN₊)N as an underlying space for the fluid domain, where 1 < q < ∞, 1≤ r < ∞, and $-1+1/q < s < 1/q$. Then, we prove the generation of a continuous analytic semigroup (t)≥ 0 on Hˢq,r. Moreover, (t)≥ 0 admits the L₁ in time maximal regularity on Hˢq,1(RN₊). Namely, we have $$∫_0^∞ e-γ t \|T(t)(ρ_0, u_0)\|_{Bˢ⁺¹q,1(R^N₊)× Bˢ⁺²q,1(R^N₊)}\,d t ≤ C\|(ρ_0, u_0)\|_{H^sq,1(R^N₊)}$$ for any $(ρ_0, u_0) ∈ H^sq,1(R^N₊)$. Our approach is to prove the existence of the resolvent in $H^sq,1$ and some new estimates for the resolvent by using $Bˢ⁺¹q,1(R^N₊) × Bs±σq,1(R^N₊)$ norms for some small $σ > 0$ satisfying the condition $-1+1/q < s-σ < s < s+σ < 1/q$.