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Well-posedness and sharp interface limit of a non-isothermal Navier--Stokes/Allen--Cahn model

Helmut Abels, Alice Marveggio, Andrea Poiatti

TL;DR

This work develops a thermodynamically consistent non-isothermal phase-field model for two viscous incompressible fluids of equal density, coupling a Navier–Stokes system with a convective Allen–Cahn equation and a temperature equation reformulated via total energy and entropy inequalities. The authors establish local well-posedness of strong solutions with maximal regularity and a contraction-mapping argument, and prove global existence of entropic weak solutions through a novel (k,δ,m)-approximation scheme, obtaining uniform energy/entropy control and strict positivity of temperature. They then rigorously pass to the sharp-interface limit ε→0, deriving BV solutions to non-isothermal Navier–Stokes/mean curvature flow and proving a distributional motion law that couples curvature, velocity, and temperature through a thermodynamically consistent framework. Overall, the paper bridges diffuse-interface phase-field dynamics with BV sharp-interface limits under thermodynamic constraints, providing a rigorous pathway from PDE-based modeling to geometric evolution with entropy production. The results contribute a robust mathematical foundation for non-isothermal multiphase flows, including energy-conserving limits, positivity of temperature, and a rigorous description of interface motion in terms of BV solutions.

Abstract

We propose a thermodynamically consistent phase-field model for the flow of a mixture of two different viscous incompressible fluids of equal density in a bounded domain. We prove the well-posedness of local-in-time strong solutions by means of maximal regularity and contraction mapping arguments. We introduce a suitable entropic weak formulation of the problem, replacing the heat equation by the total energy inequality and an entropy production inequality, and we rigorously prove global-in-time existence of such weak solutions, developing a novel approximation scheme. We also show that an entropic weak solution to this non-isothermal phase-field model converges to a distributional (or $BV$) solution to a non-isothermal Navier--Stokes/mean curvature flow, under an energy convergence assumption.

Well-posedness and sharp interface limit of a non-isothermal Navier--Stokes/Allen--Cahn model

TL;DR

This work develops a thermodynamically consistent non-isothermal phase-field model for two viscous incompressible fluids of equal density, coupling a Navier–Stokes system with a convective Allen–Cahn equation and a temperature equation reformulated via total energy and entropy inequalities. The authors establish local well-posedness of strong solutions with maximal regularity and a contraction-mapping argument, and prove global existence of entropic weak solutions through a novel (k,δ,m)-approximation scheme, obtaining uniform energy/entropy control and strict positivity of temperature. They then rigorously pass to the sharp-interface limit ε→0, deriving BV solutions to non-isothermal Navier–Stokes/mean curvature flow and proving a distributional motion law that couples curvature, velocity, and temperature through a thermodynamically consistent framework. Overall, the paper bridges diffuse-interface phase-field dynamics with BV sharp-interface limits under thermodynamic constraints, providing a rigorous pathway from PDE-based modeling to geometric evolution with entropy production. The results contribute a robust mathematical foundation for non-isothermal multiphase flows, including energy-conserving limits, positivity of temperature, and a rigorous description of interface motion in terms of BV solutions.

Abstract

We propose a thermodynamically consistent phase-field model for the flow of a mixture of two different viscous incompressible fluids of equal density in a bounded domain. We prove the well-posedness of local-in-time strong solutions by means of maximal regularity and contraction mapping arguments. We introduce a suitable entropic weak formulation of the problem, replacing the heat equation by the total energy inequality and an entropy production inequality, and we rigorously prove global-in-time existence of such weak solutions, developing a novel approximation scheme. We also show that an entropic weak solution to this non-isothermal phase-field model converges to a distributional (or ) solution to a non-isothermal Navier--Stokes/mean curvature flow, under an energy convergence assumption.

Paper Structure

This paper contains 61 sections, 11 theorems, 333 equations.

Table of Contents

  1. Introduction
  2. Outline.
  3. Derivation of the phase-field model
  4. Notation and functional setting
  5. Main results
  6. Assumptions on the coefficients
  7. Well-posedness of local strong solutions
  8. Existence of global-in-time weak solutions
  9. $BV$ solutions for non-isothermal two-phase Navier-Stokes/mean curvature flow
  10. Local-in-time existence of strong solutions
  11. Proof of Theorem \ref{['thm:strong']}
  12. Intermediate results and proofs
  13. Proof of Lemma \ref{['prop1']}
  14. Proof of Lemma \ref{['prop2']}
  15. Global-in-time existence of weak solutions
  16. ...and 46 more sections

Key Result

Theorem 4.1

Fix $\varepsilon>0$. Let the initial data be such that $\left\Vert\varphi_{\varepsilon,0}\right\Vert _{L^\infty(\Omega)}\leq 1$ and $\inf_{x\in \Omega}\theta_0\geq c>0$. Assume $\alpha\in (0,1]$ and $\beta\geq1$. Under the assumptions stated in Section subsec:hp, there exists ${T}_\varepsilon>0$, possibly depending on $\varepsilon$, such that non-isothe with

Theorems & Definitions (18)

  • Theorem 4.1: Existence and uniqueness of a local-in-time strong solution
  • Remark 4.2
  • Definition 4.3: Weak solution to the non-isothermal Navier-Stokes/Allen-Cahn system
  • Remark 4.4
  • Theorem 4.5: Existence of global-in-time weak solutions
  • Remark 4.6
  • Corollary 4.7
  • Remark 4.8
  • Lemma 4.9: Uniform strict separation of $\theta_\varepsilon$ from zero
  • Definition 4.10: $BV$ solutions of non-isothermal two-phase Navier-Stokes/mean curvature flow
  • ...and 8 more