{
  "id": "2406.19805",
  "version": "v1",
  "published": "2024-06-28T10:30:20.000Z",
  "updated": "2024-06-28T10:30:20.000Z",
  "title": "The $L^p$-$L^q$ maximal regularity for the Beris-Edward model in the half-space",
  "authors": [
    "Daniele Barbera",
    "Miho Murata"
  ],
  "comment": "56 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we consider the model describing viscous incompressible liquid crystal flows, called the Beris-Edwards model, in the half-space.This model is a coupled system by the Navier-Stokes equations with the evolution equation of the director fields $Q$. The purpose of this paper is to prove that the linearized problem has a unique solution satisfying the maximal $L^p$ -$L^q$ regularity estimates, which is essential for the study of quasi-linear parabolic or parabolic-hyperbolic equations. Our method relies on the $\\mathcal R$-boundedness of the solution operator families to the resolvent problem in order to apply operator-valued Fourier multiplier theorems. Consequently, we also have the local well-posedness for the Beris-Edwards model with small initial data.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-28T10:30:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "76A15",
      "35Q30",
      "35Q35"
    ],
    "keywords": [
      "maximal regularity",
      "beris-edward model",
      "half-space",
      "apply operator-valued fourier multiplier theorems",
      "small initial data"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 56,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}