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On the $L_1$-maximal regularity in the study of free boundary problem for the compressible fluid flows

Yuko Enomoto, Yoshihiro Shibata

TL;DR

This work proves $L_1$ in-time maximal regularity for the linearized Stokes system with free boundary conditions arising from compressible viscous flows on domains with smooth boundaries. It develops a Besov-space framework and constructs operator-valued resolvent and Laplace-inverse solution operators for the Lamé equations in ${b R}^N$, half-spaces, bent half-spaces, and general domains, with uniform bounds in a parabolic sector. The spectral analysis of the Lamé system is carried out in progressively complex geometries, culminating in a global parametrix construction and resolvent estimates that yield $L_1$ maximal regularity for the evolution problem. These results enable well-posedness analyses for nonlinear free-boundary compressible flows and provide a robust toolkit for handling Lagrangian-type transforms in the presence of free surfaces.

Abstract

In this paper, we consider the Stokes equations with non-homogeneous free boundary conditions, which is obtained by the linearization procedure of the free boundary problem of the Navier-Stokes equations describing the viscous compressible fluid flows. We prove the $L_1$ maximal regularity of solutions to this Stokes equations.

On the $L_1$-maximal regularity in the study of free boundary problem for the compressible fluid flows

TL;DR

This work proves in-time maximal regularity for the linearized Stokes system with free boundary conditions arising from compressible viscous flows on domains with smooth boundaries. It develops a Besov-space framework and constructs operator-valued resolvent and Laplace-inverse solution operators for the Lamé equations in , half-spaces, bent half-spaces, and general domains, with uniform bounds in a parabolic sector. The spectral analysis of the Lamé system is carried out in progressively complex geometries, culminating in a global parametrix construction and resolvent estimates that yield maximal regularity for the evolution problem. These results enable well-posedness analyses for nonlinear free-boundary compressible flows and provide a robust toolkit for handling Lagrangian-type transforms in the presence of free surfaces.

Abstract

In this paper, we consider the Stokes equations with non-homogeneous free boundary conditions, which is obtained by the linearization procedure of the free boundary problem of the Navier-Stokes equations describing the viscous compressible fluid flows. We prove the maximal regularity of solutions to this Stokes equations.

Paper Structure

This paper contains 19 sections, 21 theorems, 384 equations.

Table of Contents

  1. Introduction
  2. Problem and Result
  3. Short History
  4. Preparations for latter sections
  5. Symbols used throughout the paper
  6. Definition of Besov spaces and some properties
  7. The estimate for the product of 2 functions and some composite functions using Besov norms
  8. Fourier multiplier theorems in ${\Bbb R}^N$
  9. Symbol classes and estimates of the integral operators in ${\Bbb R}^N_+$
  10. Estimates of operator valued holomorphic functions with respect to Besov norms
  11. $L_1$ integrability of Laplace inverse transform.
  12. The spectral analysis of the Lamé equations
  13. Estimate of solutions in ${\Bbb R}^N$
  14. Solution formulas of Lamé equations in the half-space
  15. Estimates of solution formulas in the half-space
  16. ...and 4 more sections

Key Result

Theorem 1

Assume that the following conditions 1 or 2 holds. Then, there exists a large constant $\gamma_0 > 0$ such that for any initial data $(\rho_0, {\bold u}_0) \in {\mathcal{H}}^s_{q,1}(\Omega):=B^{s}_{q,1}(\Omega)^N\times B^{s+1}_{q,1}(\Omega)^N$, and right members $F$, ${\bold G}$, and ${\bold H}$ satisfying the conditions: for some $\gamma \geq \gamma_0$, then problem Eq:Linear admits unique solu

Theorems & Definitions (42)

  • Theorem 1
  • Lemma 2
  • Remark 3
  • proof
  • Lemma 4
  • Proposition 5
  • proof
  • Proposition 6
  • Theorem 8
  • Definition 9
  • ...and 32 more