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Well-posedness for a diffuse interface model of non-Newtonian two-phase flows

Fang Li, Duan Xingyu, Guo Zhenhua

TL;DR

This work analyzes a diffuse-interface NSCH model for two-phase, incompressible non-Newtonian flows with density-dependent viscosity and a Landau-type free energy in 3D. It develops a robust semi-Galerkin approximation and leverages monotonicity to prove global existence of weak solutions even when the initial density may vanish, and to obtain local-in-time strong solutions on the periodic domain for a power-law index range ${5/2}<p<3$. The approach combines energy estimates, compactness via Simon’s lemma, and careful treatment of nonlinear diffusion and coupling terms to pass to the limit in the nonlinearities. These results provide a rigorous foundation for the mathematical well-posedness of non-Newtonian diffuse-interface fluid models and offer insights into the dynamics of density-contrast two-phase flows with phase-field coupling.

Abstract

The evolution of two partially miscible, nonhomogeneous, incompressible viscous fluids of non-Newtonian type, can be governed by the Navier-Stokes-Cahn-Hilliard system. In the present work, we prove the global existence of weak solutions for the case of initial density containing zero and the concentration depending viscosity with free energy potential equal to the Landau potential in a bounded domain of three dimensions. Furthermore, we show that a strong solutions exist locally in time in the case of three dimensions periodic domain ${\mathbb T}^3.$ The proof relies on a suitable semi-Galerkin scheme and the monotonicity method.

Well-posedness for a diffuse interface model of non-Newtonian two-phase flows

TL;DR

This work analyzes a diffuse-interface NSCH model for two-phase, incompressible non-Newtonian flows with density-dependent viscosity and a Landau-type free energy in 3D. It develops a robust semi-Galerkin approximation and leverages monotonicity to prove global existence of weak solutions even when the initial density may vanish, and to obtain local-in-time strong solutions on the periodic domain for a power-law index range . The approach combines energy estimates, compactness via Simon’s lemma, and careful treatment of nonlinear diffusion and coupling terms to pass to the limit in the nonlinearities. These results provide a rigorous foundation for the mathematical well-posedness of non-Newtonian diffuse-interface fluid models and offer insights into the dynamics of density-contrast two-phase flows with phase-field coupling.

Abstract

The evolution of two partially miscible, nonhomogeneous, incompressible viscous fluids of non-Newtonian type, can be governed by the Navier-Stokes-Cahn-Hilliard system. In the present work, we prove the global existence of weak solutions for the case of initial density containing zero and the concentration depending viscosity with free energy potential equal to the Landau potential in a bounded domain of three dimensions. Furthermore, we show that a strong solutions exist locally in time in the case of three dimensions periodic domain The proof relies on a suitable semi-Galerkin scheme and the monotonicity method.

Paper Structure

This paper contains 11 sections, 23 theorems, 273 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm2-1']}
  4. The semi-Galerkin scheme
  5. Uniform estimates
  6. Passage to the limit
  7. Proof of Theorem \ref{['thm2-2']}
  8. Approximate scheme and initial date
  9. Uniform estimate
  10. Passage to the limit
  11. Appendix

Key Result

Theorem 1.1

Let $\Omega\subset {\mathbb R}^3$ be a bounded domain with $\partial\Omega\in {\mathcal{C}}^2,$$T$ be a positive time and $\in(\frac{5}{2},+\infty).$ Suppose that the initial data $(\rho_0,{\mathbf u}_0, \phi_0,\mu_0)$ satisfies Then the problem E1-1-E1-4 admits a global weak solution $(\rho,{\mathbf u}, \phi,\mu)$ on $[0,T]$ satisfying the following regularity properties and the following equal

Theorems & Definitions (43)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 33 more