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Global well-posedness of 3D two-fluid type model with vacuum: smallness on scaling invariant quantity

Huanyao Wen, Chanxin Xie

TL;DR

This work addresses the global well-posedness of a 3D viscous compressible two-fluid model with vacuum, where the pressure depends on two densities and the system is strongly coupled. The authors introduce a scaling-invariant combination of initial data and derive uniform a priori estimates, including a bound on $ th{P}^2$ and $ th{P}^3$, via the effective viscous flux and Zlotnik-type bounds for the densities. Under a smallness condition on this invariant quantity, they prove the existence and uniqueness of global strong solutions, without density domination assumptions. The results extend the understanding of multi-fluid models in vacuum regimes and provide a robust framework for handling pressure-velocity-density interactions in 3D.

Abstract

This paper focuses on Cauchy problem for the three-dimensional two-fluid type model, in which the presence of vacuum is permitted. Under some assumptions that the initial data satisfy appropriate regularity conditions and a compatibility constraint, and that the newly introduced scaling-invariant initial quantities $\bar P^{\frac{ 3}γ} \left(\|\sqrt{ρ_0}u_0\|_{L^2}^2+\|P_0\|_{L^1}\right) \left(\|\nabla u_0\|_{L^2}^2+\|P_0\|_{L^2}^2\right)$ and $\bar P^{\frac{6}γ+1} \left(\|\sqrt{ρ_0}u_0\|_{L^2}^2+\|P_0\|_{L^1}\right)^3 \left(\|\nabla u_0\|_{L^2}^2+\|P_0\|_{L^2}^2\right)$ are sufficiently small, the global well-posedness of strong solutions to the two-fluid type model is derived.

Global well-posedness of 3D two-fluid type model with vacuum: smallness on scaling invariant quantity

TL;DR

This work addresses the global well-posedness of a 3D viscous compressible two-fluid model with vacuum, where the pressure depends on two densities and the system is strongly coupled. The authors introduce a scaling-invariant combination of initial data and derive uniform a priori estimates, including a bound on and , via the effective viscous flux and Zlotnik-type bounds for the densities. Under a smallness condition on this invariant quantity, they prove the existence and uniqueness of global strong solutions, without density domination assumptions. The results extend the understanding of multi-fluid models in vacuum regimes and provide a robust framework for handling pressure-velocity-density interactions in 3D.

Abstract

This paper focuses on Cauchy problem for the three-dimensional two-fluid type model, in which the presence of vacuum is permitted. Under some assumptions that the initial data satisfy appropriate regularity conditions and a compatibility constraint, and that the newly introduced scaling-invariant initial quantities and are sufficiently small, the global well-posedness of strong solutions to the two-fluid type model is derived.
Paper Structure (6 sections, 13 theorems, 114 equations)

This paper contains 6 sections, 13 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Main result
  4. Proof of Theorem \ref{['theorem 1.2']}
  5. Global-in-time a priori estimates
  6. Proof of Theorem \ref{['theorem 1.2']}

Key Result

Theorem 1.2

Let the initial data $(\rho_0, n_0, u_0)$ satisfy for some positive constants $\bar{\rho} , \,\bar{n}$, and some $q\in(3,6]$, and that the initial total energy is finite, namely, and the following compatibility condition holds for some $g\in L^2(\mathbb{R}^3)$. Then there exists a positive constant $\varepsilon_0,$ depending only on $\mu,\lambda,\gamma$ and $\Gamma$ such that the initial-value

Theorems & Definitions (15)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Lemma 2.1: Local existence
  • Lemma 2.2: GN
  • Lemma 2.3: Z2000
  • Proposition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • ...and 5 more