Maximal 𝐿₁ 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 ℝ^𝑁₊. We prove the generation of a continuous analytic semigroup associated with this compressible Stokes system with non-slip boundary conditions in the half space ℝ^𝑁₊ and its 𝐿₁ in time maximal regularity. We choose the Besov space ℋ^𝑠_𝑞,𝑟 = 𝐵^𝑠+1_𝑞,𝑟 (ℝ^𝑁₊) × 𝐵^𝑠_𝑞,𝑟(ℝ^𝑁₊)^𝑁 as an underlying space, where 1 < 𝑞 < ∞, 1 ≤ 𝑟 < ∞, and −1 + 1/𝑞 < 𝑠 < 1/𝑞. We prove the generation of a continuous analytic semigroup {𝑇(𝑡)}_{𝑡 ≥ 0} on ℋ^𝑠_𝑞,𝑟, and show that its generator admits maximal 𝐿₁ regularity. Our approach is to prove the existence of the resolvent in ℋ^𝑠_𝑞,1 and some new estimates for the resolvent by using 𝐵^𝑠+1_𝑞,1(ℝ^𝑁₊) × 𝐵^{𝑠 ± 𝜎}_𝑞,1(ℝ^𝑁₊) norms for some small 𝜎 > 0 satisfying the condition −1 + 1/𝑞 < 𝑠 − 𝜎 < 𝑠 < 𝑠 + 𝜎 < 1/𝑞.