Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Almost global existence and nonlinear asymptotic stability for bubble dynamics in inviscid compressible liquid

Liangchen Zou

TL;DR

This work rigorously analyzes the full nonlinear dynamics of a spherical bubble in a compressible, inviscid liquid by formulating a free-boundary exterior Euler problem that reduces to a quasilinear wave equation with nonlinear boundary conditions. The authors develop a generalized Keel-Smith-Sogge (KSS) type energy estimate valid in exterior domains with boundary terms, and couple it with a boundary ODE for the surface variable through a careful analysis of backward pressure waves via characteristics. A bootstrap argument combines interior energy control, boundary decay, and the backward-wave analysis to establish almost global existence for small data alongside nonlinear radiative decay, with a precise bound on the bubble radius. The results extend methodological tools for nonlinear wave equations to complex free-boundary problems and have potential applicability to other exterior-domain nonlinear wave questions with challenging boundary interactions. The work highlights a novel mechanism to regain derivatives lost to quasilinear boundary interactions by leveraging a generalized weighted spacetime framework and characteristic-based boundary decompositions.

Abstract

The present paper considers the full nonlinear dynamics of a homogeneous bubble inside an unbounded isentropic compressible inviscid liquid. This model is described by a free-boundary problem of compressible Euler equations with nonlinear boundary conditions. The liquid is governed by the compressible Euler equation, while the bubble surface is determined by the kinematic and dynamic boundary conditions on the bubble-liquid interface. This classical model is of great concern in physics due to its wide applications. We begin by proving the local existence and uniqueness using energy methods under an iteration scheme. For long-time behavior, we developed a generalized weighted space-time estimate, which extends the Keel-Smith-Sogge estimate to nonlinear wave equations regardless of the boundary conditions, at the cost of the appearance of a boundary term with only lowest-order derivatives. This term is handled by using characteristics to track the backward pressure wave. Then the almost global existence and nonlinear radiative decay are proved through a bootstrap argument, which encompasses the energy estimate, the generalized Keel-Smith-Sogge estimate, and the analysis of backward pressure waves. The analysis of the backward pressure wave by characteristics involves a loss of derivative due to the quasilinear nature of the system. This is overcome by the above generalized weighted spacetime estimate with the lowest-order boundary term, which actually provides a mechanism to gain derivatives back. The coupling of these two methods is the novelty of the present paper and can not only be used for the current question but is expected to be applied to other questions regarding nonlinear wave equations with complicated boundary conditions.

Almost global existence and nonlinear asymptotic stability for bubble dynamics in inviscid compressible liquid

TL;DR

This work rigorously analyzes the full nonlinear dynamics of a spherical bubble in a compressible, inviscid liquid by formulating a free-boundary exterior Euler problem that reduces to a quasilinear wave equation with nonlinear boundary conditions. The authors develop a generalized Keel-Smith-Sogge (KSS) type energy estimate valid in exterior domains with boundary terms, and couple it with a boundary ODE for the surface variable through a careful analysis of backward pressure waves via characteristics. A bootstrap argument combines interior energy control, boundary decay, and the backward-wave analysis to establish almost global existence for small data alongside nonlinear radiative decay, with a precise bound on the bubble radius. The results extend methodological tools for nonlinear wave equations to complex free-boundary problems and have potential applicability to other exterior-domain nonlinear wave questions with challenging boundary interactions. The work highlights a novel mechanism to regain derivatives lost to quasilinear boundary interactions by leveraging a generalized weighted spacetime framework and characteristic-based boundary decompositions.

Abstract

The present paper considers the full nonlinear dynamics of a homogeneous bubble inside an unbounded isentropic compressible inviscid liquid. This model is described by a free-boundary problem of compressible Euler equations with nonlinear boundary conditions. The liquid is governed by the compressible Euler equation, while the bubble surface is determined by the kinematic and dynamic boundary conditions on the bubble-liquid interface. This classical model is of great concern in physics due to its wide applications. We begin by proving the local existence and uniqueness using energy methods under an iteration scheme. For long-time behavior, we developed a generalized weighted space-time estimate, which extends the Keel-Smith-Sogge estimate to nonlinear wave equations regardless of the boundary conditions, at the cost of the appearance of a boundary term with only lowest-order derivatives. This term is handled by using characteristics to track the backward pressure wave. Then the almost global existence and nonlinear radiative decay are proved through a bootstrap argument, which encompasses the energy estimate, the generalized Keel-Smith-Sogge estimate, and the analysis of backward pressure waves. The analysis of the backward pressure wave by characteristics involves a loss of derivative due to the quasilinear nature of the system. This is overcome by the above generalized weighted spacetime estimate with the lowest-order boundary term, which actually provides a mechanism to gain derivatives back. The coupling of these two methods is the novelty of the present paper and can not only be used for the current question but is expected to be applied to other questions regarding nonlinear wave equations with complicated boundary conditions.
Paper Structure (23 sections, 19 theorems, 562 equations, 7 figures)

This paper contains 23 sections, 19 theorems, 562 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Equations governing the liquid-bubble system
  3. Observation from linear approximation
  4. Review on previous results regarding bubble dynamics and free-boundary Euler equations
  5. Statement of the main result
  6. Comments on the results
  7. Strategy of the proof
  8. Organization of the sections
  9. Energy estimate and Local existence
  10. Setup of Bootstrap
  11. KSS type estimate with boundary terms
  12. Derivation of precise boundary ODE
  13. The analysis of backward pressure wave
  14. Estimate of (\ref{['6.5']})
  15. Estimate of (\ref{['6.6']})
  16. ...and 8 more sections

Key Result

Theorem 1.2

Let the initial value $(u_{in},\,q_{in},\,R_{in})\in H^2\times H^2\times\mathbb{R}_+$ be such that $-1<\underline{q}\leq q_{in}\leq\overline{q}<+\infty$ for all $x>0$ and constants $\underline{q}$, $\overline{q}$, and that $0<R_{in}<\overline{R}$. Moreover, we suppose that the initial value satisfie to (cnte-initial) with initial value $(u_{in},\,q_{in},\,R_{in})$. Moreover, a lower bound of $T$ c

Figures (7)

  • Figure 1: The backward and forward characteristics within the domain $\Omega_B(t_*)$
  • Figure 2: Positions of the points $X_F(1,t;\sigma_1(\eta_1,t))$, $X_F(1,t;\sigma_1(\eta_2,t))$ and $X_B(\eta_2,0;t+\sigma_1(\eta_1,t))$
  • Figure 3: Positions of the points $X_F(\xi,0;\sigma_2(\eta_1,\xi))$, $X_F(\xi,0;\sigma_2(\eta_2,\xi))$ and $X_B(\eta_2,0;\sigma_2(\eta_1,\xi))$
  • Figure 4: Positions of the points $X_F(\eta_1,0;\sigma_3(\eta_1,t_*))$, $X_F(\eta_1,0;\sigma_3(\eta_2,t_*))$ and $X_F(\eta_2,0;\sigma_3(\eta_2,t_*))$
  • Figure 5: Positions of the points $X_F(1,t_1;t_2-t_1)$, $X_F(1,t_1;\sigma_4(t_1,t_*)-t_1)$, $X_F(1,t_2;\sigma_4(t_1,t_*)-t_2)$ and $X_F(1,t_2;\sigma_4(t_2,t_*)-t_2)$
  • ...and 2 more figures

Theorems & Definitions (30)

  • Definition 1.1
  • Theorem 1.2: Local existence and uniqueness
  • Theorem 1.3: Almost global existence
  • Corollary 1.4: Pointwise decay near the surface
  • Lemma 2.1: Weighted elliptic estimate
  • proof
  • Lemma 2.2: Model question
  • Lemma 2.3: Uniform bound for mixed derivatives
  • Proposition 2.4: Uniform bounds in $H^2$
  • Lemma 2.5: Control of parameters
  • ...and 20 more