{
  "abstract": "We develop compactness and defect-of-compactness results for Banach-valued evolution classes with Morrey control in time. For \\[ \\begin{aligned} \\mathbb W_M^{p,λ}(0,T;E_0,E_1) =\\{u\\in\\mathcal M^{p,λ}(0,T;E_0):\\;& u'\\in\\mathcal M^{p,λ}(0,T;E_1)\\},\\\\[-1mm] &0<λ<1. \\end{aligned} \\] the exact trace exponent \\[ θ=1-\\frac{1-λ}{p} \\] governs both continuity and compactness. If $E_0\\hookrightarrow\\!\\hookrightarrow E\\hookrightarrow E_1$, bounded sets are compact in the \\emph{same} Morrey space $\\mathcal M^{p,λ}(0,T;E)$, not merely in $L^p(0,T;E)$. In contrast, compactness in $C([0,T];E)$ holds if and only if the exact trace space $(E_1,E_0)_{θ,\\infty}$ embeds compactly into $E$. We also obtain compact lower-order Hölder embeddings and a sharp Hilbert-triple threshold. On unbounded domains we prove a tightness criterion for global Morrey compactness. In the Hilbert-valued case we establish, by a direct time-averaging argument, cocompactness modulo spatial translations and a translation profile decomposition whose remainder vanishes in every strictly subcritical time-Morrey--Sobolev target; a uniform little-Morrey condition removes the loss in the time exponent. At the doubly critical endpoint, heat and whole-space Stokes dynamics force balanced parabolic profiles, and the remainder vanishes in $\\mathcal M_t^{2,λ}L_x^{2^*}$. Finally, in three-dimensional Navier--Stokes we show that the classical nonlinear profile decomposition has a scale-sensitive Morrey refinement: the remainder is small in $\\mathcal M_t^{p,1-p/4}L_x^6$ for every $2\\le p<4$, and orthogonal profiles have vanishing interaction in the corresponding Morrey forcing space.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29152",
  "author_total": 2,
  "authors": [
    {
      "name": "Rishad Shahmurov"
    },
    {
      "name": "Veli Shahmurov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/71922rt9ckh55.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29152",
  "fulltext_url": null,
  "id": "71922rt9ckh55",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29152",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7214337970",
  "pdf_url": "https://arxiv.org/pdf/2609.29152",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/71922rt9ckh55.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29152",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29152"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Compact Embeddings of Vector-Valued Morrey Spaces",
  "title_text": "Compact Embeddings of Vector-Valued Morrey Spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/71922rt9ckh55",
  "venue": "arXiv",
  "versions": []
}