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 \[ \begin{aligned} \mathbb W_M^{p,λ}(0,T;E_0,E_1) =\{u\in\mathcal M^{p,λ}(0,T;E_0):\;& u'\in\mathcal M^{p,λ}(0,T;E_1)\},\\[-1mm] &0<λ<1. \end{aligned} \] 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 ${\mathcal{M}}^{p,\lambda }(0,T;E)$, 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 $(E_1,E_0{)}_{\theta ,\infty }$ 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 ${\mathcal{M}}_t^{2,\lambda }{L}_x^{2^{\ast }}$. 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.