{
  "id": "2408.09514",
  "version": "v1",
  "published": "2024-08-18T15:39:59.000Z",
  "updated": "2024-08-18T15:39:59.000Z",
  "title": "Regularity Propagation of Global Weak Solutions to a Navier-Stokes-Cahn-Hilliard System for Incompressible Two-phase Flows with Chemotaxis and Active Transport",
  "authors": [
    "Jingning He",
    "Hao Wu"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We analyze a diffuse interface model that describes the dynamics of incompressible viscous two-phase flows, incorporating mechanisms such as chemotaxis, active transport, and long-range interactions of Oono's type. The evolution system couples the Navier-Stokes equations for the volume-averaged fluid velocity $\\bm{v}$, a convective Cahn-Hilliard equation for the phase-field variable $\\varphi$, and an advection-diffusion equation for the density of a chemical substance $\\sigma$. For the initial-boundary value problem with a physically relevant singular potential in three dimensions, we demonstrate that every global weak solution $(\\bm{v}, \\varphi, \\sigma)$ exhibits a propagation of regularity over time. Specifically, after an arbitrary positive time, the phase-field $\\varphi$ transitions into a strong solution, whereas the chemical density $\\sigma$ only partially regularizes. Subsequently, the velocity field $\\bm{v}$ becomes regular after a sufficiently large time, followed by a further regularization of the chemical density $\\sigma$, which in turn enhances the spatial regularity of $\\varphi$. Furthermore, we show that every global weak solution stabilizes towards a single equilibrium as $t\\to +\\infty$. Our analysis uncovers the influence of chemotaxis, active transport, and long-range interactions on the propagation of regularity at different stages of time. The proof relies on several key points, including a novel regularity result for a convective Cahn-Hilliard-diffusion system with minimal regularity requirements on the velocity field $\\bm{v}$, the strict separation property of $\\varphi$ for large times, as well as two conditional uniqueness results pertaining to the full system and its subsystem for $(\\varphi, \\sigma)$ with a given velocity, respectively.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-18T15:39:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35A01",
      "35A02",
      "35K35",
      "35Q92",
      "76D05"
    ],
    "keywords": [
      "incompressible two-phase flows",
      "active transport",
      "regularity propagation",
      "navier-stokes-cahn-hilliard system",
      "chemotaxis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}