arXiv subjects

Veli Shahmurov

Publications and source records attributed to Veli Shahmurov.

4 recordsLinked to original sources

arXiv2026

Exact Traces for Vector-Valued Morrey Spaces

Trace spaces determine exactly which initial values are compatible with an evolution class. We identify the exact trace generated by vector-valued Morrey control in time and show that it differs essentially from the classical $L^p$ theory. For a Banach couple $X_1\hookrightarrow X_0$, the trace of the natural Morrey evolution class is the weak real-interpolation space $(X_0,X_1)_{θ,\infty}$, where $θ=1-(1-λ)/p$. Every element of this space occurs as a trace through a bounded extension operator. The result is sharp in both interpolation parameters: in general the smoothness exponent cannot be increased and the fine index $\infty$ cannot be replaced by any finite index. Thus Morrey control changes the exact trace mechanism rather than merely strengthening an integrability estimate. At the limiting endpoint, bounded mean oscillation (BMO) control yields finite-index interpolation traces together with a logarithmic modulus of continuity. The result isolates the precise initial-data space naturally associated with local, scale-sensitive time regularity and provides a trace framework suited to evolution equations with Morrey-type maximal regularity. \keywords{Morrey spaces \and trace spaces \and real interpolation \and evolution equations \and bounded mean oscillation}

math.AP↗

arXiv2026

Compact Embeddings of Vector-Valued Morrey Spaces

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.

math.AP↗

arXiv2026

A Morrey-to-Lebesgue Equivalence for Convolution Calderón--Zygmund Operators

We prove that, for vector-valued convolution Calderón--Zygmund operators, boundedness on a single nontrivial Morrey space is equivalent to the corresponding global $L^p$ boundedness. Thus one Morrey scale already contains the full finite-$p$ boundedness information. The implication from Morrey to $L^p$ is obtained by a separated-copy amplification argument that reconstructs the global norm from a single scale-local estimate; the converse is proved in the same vector-valued framework by a local/far-field decomposition. As a consequence, boundedness of the vector-valued Hilbert transform on one nontrivial Morrey space is equivalent to the UMD property. The result shows that Morrey estimates do not bypass the classical Banach-space obstruction: they detect it exactly.

math.AP↗

arXiv2026

Nonlocal Hyperdissipative Perturbations of the Three Dimensional Navier-Stokes System

We study the three-dimensional incompressible Navier-Stokes system on $\mathbb{R}^3$ with an additional dissipative nonlocal term \[ \partial_t u + (u\cdot\nabla)u + \nabla p = νΔu + Lu, \qquad {\rm div}\, u = 0, \] where $L$ is a self-adjoint Fourier multiplier whose symbol is comparable to $-|ξ|^{2α}$ for some $α>1$. We first identify a sharp Fourier-symbol criterion distinguishing lower-order convolution perturbations from genuinely regularizing nonlocal corrections. In the resulting hyperdissipative class we prove the exact $L^2$ energy identity, global weak solvability for every $α>1$, and local strong well-posedness in $H^s(\mathbb{R}^3)$ for $s>\frac52$. We then show that the Lions exponent $α=\frac54$ remains the critical energy-growth threshold in this nonlocal setting: if $α\ge \frac54$, every $H^s$ solution is global, while for every $α>1$ one has global strong solvability for sufficiently small $H^s$ data. Finally, for the vanishing-hyperdissipation approximation of the classical three-dimensional Navier-Stokes equations, we prove a near-singular divergence principle: if the classical flow blows up at a first singular time $T_*$ in a continuation norm $X$, then the corresponding regularized family cannot remain uniformly bounded in $X$ on any interval approaching $T_*$. This identifies the precise point at which the fixed-parameter global theory degenerates in the Navier-Stokes limit.

math.AP↗