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On the stability of the spherically symmetric solution to an inflow problem for an isentropic model of compressible viscous fluid

Yucong Huang, Itsuko Hashimoto, Shinya Nishibata

TL;DR

The paper studies the stability of the spherically symmetric stationary inflow solution for the multi-dimensional isentropic compressible Navier–Stokes equations in an exterior domain, with boundary inflow and a far-field nonvacuum state. It proves existence, uniqueness, and detailed decay properties of the stationary solution, including a density-extremum classification that depends on the boundary data, and establishes time-asymptotic stability under small perturbations by reformulating the problem in Lagrangian coordinates to handle boundary terms. The analysis employs a relative-energy framework and comprehensive a priori estimates in the Lagrangian setting, yielding explicit decay rates and closed energy bounds that guarantee convergence to the stationary state for small far-field density $\rho_+$ and small boundary flux. This work extends the understanding of inflow boundary problems for compressible viscous fluids, providing precise conditions under which rarefied near-vacuum states remain stable and offering a rigorous methodological blueprint for similar exterior-domain problems.

Abstract

We investigate an inflow problem for the multi-dimensional isentropic compressible Navier-Stokes equations. The fluid under consideration occupies the exterior domain of unit ball, $Ω=\{x\in\mathbb{R}^n\,\vert\, |x|\ge 1\}$, and a constant stream of mass is flowing into the domain from the boundary $\partialΩ=\{|x|=1\}$. It is shown in Hashimoto-Matsumura(2021) that if the fluid velocity at the far-field is assumed to be zero, then there exists a unique spherically symmetric stationary solution, denoted as $(\tildeρ,\tilde{u})(r)$ with $r\equiv |x|$. In this paper, we show that either $\tildeρ$ is monotone increasing or $\tildeρ$ attains a unique global minimum, and this is classified by the boundary condition of density. In addition, we also derive a set of spatial decay rates for $(\tildeρ,\tilde{u})$ which allows us to prove the time-asymptotic stability of $(\tildeρ,\tilde{u})$ using the energy method. More specifically, we prove this under small initial perturbation on $(\tildeρ,\tilde{u})$, provided that the density at the far-field is supposed to be strictly positive but suitably small, in other words, the far-field state of the fluid is not vacuum but suitably rarefied. The main difficulty for the proof is the boundary terms that appears in the a-priori estimates. We resolve this issue by reformulating the problem in Lagrangian coordinate system.

On the stability of the spherically symmetric solution to an inflow problem for an isentropic model of compressible viscous fluid

TL;DR

The paper studies the stability of the spherically symmetric stationary inflow solution for the multi-dimensional isentropic compressible Navier–Stokes equations in an exterior domain, with boundary inflow and a far-field nonvacuum state. It proves existence, uniqueness, and detailed decay properties of the stationary solution, including a density-extremum classification that depends on the boundary data, and establishes time-asymptotic stability under small perturbations by reformulating the problem in Lagrangian coordinates to handle boundary terms. The analysis employs a relative-energy framework and comprehensive a priori estimates in the Lagrangian setting, yielding explicit decay rates and closed energy bounds that guarantee convergence to the stationary state for small far-field density and small boundary flux. This work extends the understanding of inflow boundary problems for compressible viscous fluids, providing precise conditions under which rarefied near-vacuum states remain stable and offering a rigorous methodological blueprint for similar exterior-domain problems.

Abstract

We investigate an inflow problem for the multi-dimensional isentropic compressible Navier-Stokes equations. The fluid under consideration occupies the exterior domain of unit ball, , and a constant stream of mass is flowing into the domain from the boundary . It is shown in Hashimoto-Matsumura(2021) that if the fluid velocity at the far-field is assumed to be zero, then there exists a unique spherically symmetric stationary solution, denoted as with . In this paper, we show that either is monotone increasing or attains a unique global minimum, and this is classified by the boundary condition of density. In addition, we also derive a set of spatial decay rates for which allows us to prove the time-asymptotic stability of using the energy method. More specifically, we prove this under small initial perturbation on , provided that the density at the far-field is supposed to be strictly positive but suitably small, in other words, the far-field state of the fluid is not vacuum but suitably rarefied. The main difficulty for the proof is the boundary terms that appears in the a-priori estimates. We resolve this issue by reformulating the problem in Lagrangian coordinate system.
Paper Structure (18 sections, 19 theorems, 165 equations)

This paper contains 18 sections, 19 theorems, 165 equations.

Table of Contents

  1. Keywords.
  2. AMS subject classifications.
  3. Introduction
  4. Main results
  5. Related results.
  6. Outline of the paper.
  7. Notations.
  8. Stationary solution
  9. Well-posedness of the stationary solution.
  10. Decay rate for the stationary solution
  11. Classification of extremum points in density
  12. Case 1: $\eta_b \le 0$.
  13. Case 2: $\eta_b> 0$.
  14. A-priori estimates in Lagrangian coordinate
  15. Reformulation in Lagrangian Coordinate
  16. ...and 3 more sections

Key Result

Lemma 2.1

There exists $\varepsilon=\varepsilon(\rho_+,\mu,\gamma,K,n)>0$ and $C=C(\mu,\gamma,K,n)>0$ such that if $|\rho_b-\rho_+|+ u_b\le \varepsilon$, then a unique solution $(\tilde{\rho}, \tilde{u})$ to the problem st--u+ exists in a certain neighborhood of $(\rho_+, 0)$. Moreover, for $r\ge 1$ Furthermore, the sign of $\rho_b-\rho_+$ determines the behaviour of density $\tilde{\rho}(r)$ as follows

Theorems & Definitions (31)

  • Lemma 2.1
  • Theorem 2.2
  • Remark 2.3
  • Lemma 3.1: I. Hashimoto and A. Matsumura, h-m21
  • Proposition 3.2: I. Hashimoto, A. Matsumura h-m21
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • proof
  • Lemma 3.5
  • ...and 21 more