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On a thermodynamically consistent diffuse interface model for incompressible two-phase flows with unmatched densities: Energy equality and Lyapunov stability

Harald Garcke, Maoyin Lv, Hao Wu

Abstract

We consider the initial-boundary value problem of a thermodynamically consistent diffuse interface model for incompressible two-phase flows with unmatched densities in a bounded domain $Ω\subset\mathbb{R}^3$. Our first aim is to study the energy equality for global weak solutions by establishing mixed $L_t^qL_x^r$-regularity conditions on the velocity field and its gradient, under which the global weak solution conserves its energy for all times. The proof is based on the propagation of regularity for weak solutions to the convective Cahn-Hilliard equation with a physically relevant Flory-Huggins-type potential, combined with global mollification and boundary cut-off techniques. Next, we prove the existence and uniqueness of global strong solutions in the general setting with non-constant gradient energy coefficient and non-degenerate mobility, provided that the initial velocity is sufficiently small and the initial phase-field variable is a sufficiently small perturbation of a local minimizer of the free energy. This yields Lyapunov stability for each steady state consisting of a zero velocity together with a local energy minimizer. The proof relies on the energy equality for (local) strong solutions and the Łojasiewicz-Simon approach.

On a thermodynamically consistent diffuse interface model for incompressible two-phase flows with unmatched densities: Energy equality and Lyapunov stability

Abstract

We consider the initial-boundary value problem of a thermodynamically consistent diffuse interface model for incompressible two-phase flows with unmatched densities in a bounded domain . Our first aim is to study the energy equality for global weak solutions by establishing mixed -regularity conditions on the velocity field and its gradient, under which the global weak solution conserves its energy for all times. The proof is based on the propagation of regularity for weak solutions to the convective Cahn-Hilliard equation with a physically relevant Flory-Huggins-type potential, combined with global mollification and boundary cut-off techniques. Next, we prove the existence and uniqueness of global strong solutions in the general setting with non-constant gradient energy coefficient and non-degenerate mobility, provided that the initial velocity is sufficiently small and the initial phase-field variable is a sufficiently small perturbation of a local minimizer of the free energy. This yields Lyapunov stability for each steady state consisting of a zero velocity together with a local energy minimizer. The proof relies on the energy equality for (local) strong solutions and the Łojasiewicz-Simon approach.

Paper Structure

This paper contains 18 sections, 15 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. Regularity criteria for the energy equality
  3. Global strong solutions and Lyapunov stability of steady states
  4. Main Results
  5. Preliminaries
  6. Main results
  7. Regularity Criterion on Energy Equality
  8. Approximated formulas
  9. Passage to the limit as ɛ 0
  10. Passage to the limit as δ, τ 0
  11. Lyapunov Stability
  12. Bounds for initial data near (0,φ_*)
  13. Refined estimates
  14. Global strong solutions and Lyapunov stability
  15. Useful Tools
  16. ...and 3 more sections

Key Result

Proposition 2.1

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^3$. Assume that the gradient energy coefficient $a(\cdot)$ and the non-degenerate mobility $b(\cdot)$ are both positive constants (equal to one without loss of generality). For any given initial data $\mathbf{v}_0\in \mathbf{L}_\sigma^2(\Omega)$

Theorems & Definitions (20)

  • Proposition 2.1
  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 10 more