{
  "abstract": "In this paper, we develop a theory of evolution dynamics in a Banach bilateral $\\mathbb{H}$-algebra and establish a well-posedness theory for the Cauchy problem of evolution equations with quaternionic variables. First, we study the geometric properties of slice structures of Banach bilateral $\\mathbb{H}$-modules. By employing different slice structures, we construct some noncommutative Banach bilateral $\\mathbb{H}$-subalgebras which are compatible with the slice regularity of the functions involved. Moreover, through the quaternionic sliced geometric path integral over Banach bilateral $\\mathbb{A}$-module, an integral representation of the solution to the homogeneous Cauchy problem with quaternionic variables is obtained in a quaternionic $σ$-ball, and for the first time, a local existence and uniqueness theorem is established for $σ$-analytic solutions. Then the local $σ$-analytic solution is extended to the entire quaternionic spherical sectorial region, thereby proving a global existence and uniqueness theorem for the Cauchy problem. Based on it, the notion of an evolution system in a quaternionic $\\mathbb{H}$-algebra is introduced and its core properties are established. In addition, the global existence and uniqueness theorems are obtained under hyperbolic-like and parabolic-like conditions. Finally, the structure of Banach bilateral $\\mathbb{H}$-modules and $\\mathbb{H}$-algebras is lifted to Banach bilateral $\\mathbb{A}$-modules and $\\mathbb{A}$-algebras, and a well-posedness theory for evolution equations in this setting is developed. This reveals, from a new perspective, the universality of well-posedness for the Cauchy problem under the generic noncommutative $\\mathbb{A}$-algebraic structures.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06721",
  "author_total": 3,
  "authors": [
    {
      "name": "Chao Wang"
    },
    {
      "name": "Guangzhou Qin"
    },
    {
      "name": "Jibin Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/chxd8g494rsxc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06721",
  "fulltext_url": null,
  "id": "chxd8g494rsxc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06721",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.CV"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.06721",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/chxd8g494rsxc.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.06721",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06721"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Evolution Dynamics and Cauchy Problems with Quaternionic Variables and Lifting to Banach Bilateral $\\mathbb{A}$-Algebras",
  "title_text": "Evolution Dynamics and Cauchy Problems with Quaternionic Variables and Lifting to Banach Bilateral 𝔸-Algebras",
  "type": "preprint",
  "updated_at": "2026-11-08T21:20:29Z",
  "url": "https://research.draconitic.com/paper/chxd8g494rsxc",
  "venue": "arXiv",
  "versions": []
}