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The Diffusion of Passive Tracers in Laminar Shear Flow
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Diffuse interface model for two-phase flows on evolving surfaces with different densities: global well-posedness.

Helmut Abels1, Harald Garcke1, Andrea Poiatti2

  • 1Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany.

Calculus of Variations and Partial Differential Equations
|May 7, 2025
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Summary
This summary is machine-generated.

This study proves the global existence and uniqueness of solutions for a two-phase fluid flow model on evolving surfaces. It also confirms the separation of fluid phases over time.

Keywords:
35D3535Q3035Q3576D0576T06

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Area of Science:

  • Fluid dynamics
  • Partial differential equations
  • Geometric analysis

Background:

  • The Navier-Stokes/Cahn-Hilliard system models two-phase fluid flow with interfaces.
  • Analyzing such systems on evolving surfaces presents significant mathematical challenges.
  • Understanding fluid behavior on dynamic surfaces is crucial for various applications.

Purpose of the Study:

  • To establish global-in-time existence and uniqueness of strong solutions.
  • To analyze a diffuse interface model for two-phase flow on a 2D evolving surface.
  • To investigate the behavior of fluids with different densities and singular potentials.

Main Methods:

  • Utilizing prior results on local well-posedness.
  • Developing novel regularity results for the convective Cahn-Hilliard equation.
  • Deriving higher-order energy estimates to extend local solutions globally.

Main Results:

  • Global-in-time existence and uniqueness of strong solutions are proven.
  • The instantaneous strict separation property of pure phases is established.
  • The analysis covers systems with differing densities and logarithmic potentials.

Conclusions:

  • The mathematical framework supports the long-term behavior of complex fluid systems on dynamic surfaces.
  • The findings provide a rigorous foundation for diffuse interface models in fluid dynamics.
  • This work advances the understanding of partial differential equations on evolving manifolds.