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

Jou-Chun Kuo

TL;DR

The paper establishes maximal $L_1$-regularity for the linearized compressible Navier–Stokes (Stokes) system in the half-space by constructing a continuous analytic semigroup on Besov-type spaces $\mathcal{H}^s_{q,r}(\mathbb R^N_{+})$. It develops a resolvent framework, analyzes the associated complex Lamé equations, and uses a blend of Fourier multiplier theory and interpolation to obtain sharp Besov-space estimates. The results rely on decomposing resolvent representations into half-space and boundary contributions, with careful symbol bounds for stability across sectors in the complex plane. This maximal-regularity setting provides a robust foundation for treating nonlinear problems (via Lagrangian coordinates) in unbounded domains and informs potential global well-posedness analyses for the full compressible Navier–Stokes system.

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 $B^{s+1}_{q,1}( \mathbb{R}^N_{+}) \times B^{s\pmσ}_{q,1}( \mathbb{R}^N_{+})$ norms for some small $σ> 0$ satisfying the condition $-1+1/q < s-σ< s < s+σ< 1/q$.

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

TL;DR

The paper establishes maximal -regularity for the linearized compressible Navier–Stokes (Stokes) system in the half-space by constructing a continuous analytic semigroup on Besov-type spaces . It develops a resolvent framework, analyzes the associated complex Lamé equations, and uses a blend of Fourier multiplier theory and interpolation to obtain sharp Besov-space estimates. The results rely on decomposing resolvent representations into half-space and boundary contributions, with careful symbol bounds for stability across sectors in the complex plane. This maximal-regularity setting provides a robust foundation for treating nonlinear problems (via Lagrangian coordinates) in unbounded domains and informs potential global well-posedness analyses for the full compressible Navier–Stokes system.

Abstract

In this paper, we consider the linearized compressible Navier-Stokes equations with non-slip boundary conditions in the half space . We prove the generation of a continous analytic semigroup associated with this compressible Stokes system with non-slip boundary conditions in the half space and its in time maximal regularity. We choose the Besov space as an underlying space, where , , and . We prove the generation of a continuous analytic semigroup on , and show that its generator admits maximal regularity. Our approach is to prove the existence of the resolvent in and some new estimates for the resolvent by using norms for some small satisfying the condition .
Paper Structure (17 sections, 28 theorems, 318 equations)

This paper contains 17 sections, 28 theorems, 318 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Function spaces on $\mathbb R^N$
  5. Function spaces on $\mathbb R^N_+$
  6. Class of multipliers
  7. Interpolation of small $\ell^p$ spaces of vector-valued sequences
  8. Technical tools
  9. Solution formula
  10. Whole space case
  11. Half space case
  12. Estimates of solution operators in the whole space
  13. Estimates of solution formulas of complex Lamé equations
  14. Spectral analysis based on interpolation theory
  15. A proof of Theorem \ref{['thm:6.1']}.
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $1 < q < \infty$, $-1+1/q < s < 1/q$, and $1 \leq r < \infty$. Then, the operator ${\mathcal{A}}^s_{q,r}$ generates a continuous analytic semigroup $\{T(t)\}_{t\geq0}$ on ${\mathcal{H}}^s_{q,r}(\mathbb R^N_+)$. Moreover, there exists a large $\omega_0\geq1$ such that, for any $\omega\geq\omega_0

Theorems & Definitions (45)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • Lemma 3.1
  • Lemma 3.2
  • ...and 35 more