向量值Morrey空间的紧嵌入

中文摘要 AI 辅助

我们发展了具有时间Morrey控制的Banach值演化类的紧性和紧性缺陷结果。对于 \\[ \begin{aligned} \mathbb W_M^{p,\lambda}(0,T;E_0,E_1) =\{u\in\mathcal M^{p,\lambda}(0,T;E_0):\\; & u'\in\mathcal M^{p,\lambda}(0,T;E_1)\},\\\\[-1mm] &0<\lambda<1. \end{aligned} \\] 精确迹指数 \\[ \theta=1-\frac{1-\lambda}{p} \\] 同时控制连续性和紧性。若 $E_0\hookrightarrow\\!\hookrightarrow E\hookrightarrow E_1$，则有界集在\\(\emph{相同}\\)的Morrey空间 $\mathcal M^{p,\lambda}(0,T;E)$ 中是紧的，而不仅仅在 $L^p(0,T;E)$ 中。相反，在 $C([0,T];E)$ 中的紧性成立当且仅当精确迹空间 $(E_1,E_0)_{\theta,\infty}$ 紧嵌入到 $E$ 中。我们还获得了紧的低阶Hölder嵌入和一个尖锐的Hilbert三元组阈值。在无界域上，我们证明了全局Morrey紧性的一个紧性准则。在Hilbert值情形下，我们通过直接的时间平均论证建立了模空间平移的共紧性和一个平移剖面分解，其余项在每个严格次临界时间Morrey-Sobolev目标空间中消失；一个一致的小Morrey条件消除了时间指数中的损失。在双临界端点，热和全空间Stokes动力学迫使平衡抛物型剖面出现，且余项在 $\mathcal M_t^{2,\lambda}L_x^{2^*}$ 中消失。最后，在三维Navier-Stokes方程中，我们证明了经典非线性剖面分解具有尺度敏感的Morrey细化：对于每个 $2\le p<4$，余项在 $\mathcal M_t^{p,1-p/4}L_x^6$ 中很小，且正交剖面在相应的Morrey强迫空间中的相互作用消失。

英文摘要

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.

发表机构

• Cellular Products Research and Development
• Antalya Bilim University（安塔利亚 bilim 大学）

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