Compact Embeddings of Vector-Valued Morrey Spaces
Abstract
We develop compactness and defect-of-compactness results for Banach-valued evolution classes with Morrey control in time. For 𝕎Mp,λ(0,T;E0,E1) ={u∈ℳp,λ(0,T;E0): & u'∈ℳp,λ(0,T;E1)}, [−1mm] &0<λ<1. the exact trace exponent θ=1−(1−λ)/p governs both continuity and compactness. If E0↪ ↪ E↪ E1, bounded sets are compact in the \emph{same} Morrey space ℳp,λ(0,T;E), not merely in Lp(0,T;E). In contrast, compactness in $C([0,T];E)$ holds if and only if the exact trace space (E1,E0)θ,∞ 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 ℳt2,λLx2*. 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 ℳtp,1−p/4Lx6 for every 2≤ p<4, and orthogonal profiles have vanishing interaction in the corresponding Morrey forcing space.
Related
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- Exact Traces for Vector-Valued Morrey Spaces
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- Saturating spaces for the squared-gradient nonlinearity on compact Riemannian manifolds via Nash embeddings
BibTeX
@misc{shahmurov2026compact,
title = {{Compact Embeddings of Vector-Valued Morrey Spaces}},
author = {Rishad Shahmurov and Veli Shahmurov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29152},
eprint = {2609.29152},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29152},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Compact Embeddings of Vector-Valued Morrey Spaces
AU - Rishad Shahmurov
AU - Veli Shahmurov
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29152
UR - https://arxiv.org/abs/2609.29152
AB - We develop compactness and defect-of-compactness results for Banach-valued evolution classes with Morrey control in time. For 𝕎_M^p,λ(0,T;E_0,E_1) ={u∈ℳ^p,λ(0,T;E_0): & u'∈ℳ^p,λ(0,T;E_1)}, [−1mm] &0<λ<1. the exact trace exponent θ=1−(1−λ)/p governs both continuity and compactness. If E_0↪ ↪ E↪ E_1, bounded sets are compact in the \emph{same} Morrey space 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,∞ 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 ℳ_t^2,λL_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 ℳ_t^p,1−p/4L_x^6 for every 2≤ p<4, and orthogonal profiles have vanishing interaction in the corresponding Morrey forcing space.
ER -