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On the $\mathcal R$-boundedness of solution operators for a compressible fluid model of Korteweg type in general domains

Sri Maryani, Miho Murata

TL;DR

The paper addresses the resolvent problem for a compressible Navier-Stokes-Korteweg system with surface tension in general domains. It develops ${\mathcal{R}}$-bounded solution operator families via a localization and parametrix framework, leveraging bent-half-space results and the Weis operator-valued Fourier multiplier theorem to obtain analytic semigroup generation and maximal $L_p$-$L_q$ regularity. This enables robust local solvability theory for the associated nonlinear problem in maximal regularity spaces and provides a pathway to optimal decay and stability analyses. The approach systematically extends known results from the whole space and half-space to uniform $C^3$ domains, including the bent boundary case, by a sequence of reductions and controlled remainder terms, with implications for diffuse-interface fluid models in complex geometries.

Abstract

In this paper, we consider a resolvent problem arising from the free boundary problem for the compressible fluid model of the Korteweg type, which is called the Navier-Stokes-Korteweg system, with surface tension in general domains. The Navier-Stokes-Korteweg system describes the liquid-vapor two-phase flow with non-zero thickness phase boundaries, which is often called the diffuse interface model. Our purpose is to show the solution operator families of the resolvent problem are $\mathcal R$-bounded, which gives us the generation of analytic semigroup and the maximal regularity in the $L_p$-in-time and $L_q$-in-space setting by applying the Weis operator valued Fourier multiplier theorem.

On the $\mathcal R$-boundedness of solution operators for a compressible fluid model of Korteweg type in general domains

TL;DR

The paper addresses the resolvent problem for a compressible Navier-Stokes-Korteweg system with surface tension in general domains. It develops -bounded solution operator families via a localization and parametrix framework, leveraging bent-half-space results and the Weis operator-valued Fourier multiplier theorem to obtain analytic semigroup generation and maximal - regularity. This enables robust local solvability theory for the associated nonlinear problem in maximal regularity spaces and provides a pathway to optimal decay and stability analyses. The approach systematically extends known results from the whole space and half-space to uniform domains, including the bent boundary case, by a sequence of reductions and controlled remainder terms, with implications for diffuse-interface fluid models in complex geometries.

Abstract

In this paper, we consider a resolvent problem arising from the free boundary problem for the compressible fluid model of the Korteweg type, which is called the Navier-Stokes-Korteweg system, with surface tension in general domains. The Navier-Stokes-Korteweg system describes the liquid-vapor two-phase flow with non-zero thickness phase boundaries, which is often called the diffuse interface model. Our purpose is to show the solution operator families of the resolvent problem are -bounded, which gives us the generation of analytic semigroup and the maximal regularity in the -in-time and -in-space setting by applying the Weis operator valued Fourier multiplier theorem.

Paper Structure

This paper contains 17 sections, 18 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Model
  3. Notation and Assumptions
  4. Main theorem
  5. Problems in the whole-space and the half-space
  6. Problem in the bent half-space
  7. Reduction to a half-space problem
  8. The ${\mathcal{R}}$-boundedness for the solution operator families for \ref{['r3']}
  9. Proof of Lemma \ref{['lem:Rbound V']}
  10. Proof of Theorem \ref{['thm:main bent']}
  11. Problem in the general domain
  12. Preliminaries
  13. Localization
  14. Construction of a parametrix
  15. Proof of Theorem \ref{['thm:main general']}
  16. ...and 2 more sections

Key Result

Theorem 2.3

Let $1<q<\infty$, and let $\Omega$ be a uniform $C^3$-domain. Assume that $\gamma_j$$(j =1, 2, 3, 4)$ satisfies Assumption assumption gamma. Then there exists a constant $\lambda_0 \ge 1$ such that the following assertions hold true: $\thetag1$ For any $\lambda \in {\mathbb C}_{+, \lambda_0}$ there such that for any ${\bold F}=(d, {\bold f}, {\bold g}, k, \zeta) \in X_q(\Omega)$, are unique solu

Theorems & Definitions (27)

  • Definition 1.1
  • Remark 1.3
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Proposition 3.1
  • ...and 17 more