Searcharxiv⌕ Search

arXiv subjects

Zoran Grujic

Publications and source records attributed to Zoran Grujic.

14 recordsLinked to original sources
arXiv2026

Logarithmic Depletion of Vortex Stretching and Singularity Evasion in the 3D Navier-Stokes Equations

We present a geometric-analytic mechanism for the suppression of finite-time singularities in the 3D incompressible (unforced) Navier-Stokes equations for critical point singularities exhibiting $L^{3/2, \infty}$ spatial concentration of vorticity. We demonstrate that if the vorticity direction resides locally in a logarithmically weighted space of bounded mean oscillations, $\mathrm{bmo}_{1/|\log r|}$ -- a space failing the Dini condition and thus permitting wild oscillatory defects -- the non-linear vortex stretching is fundamentally depleted. By isolating a unidirectional geometric cancellation, we recast the stretching eigenvalue as a singular integral commutator. Utilizing a localized Coifman-Rochberg-Weiss estimate coupled with dyadic BMO tail bounds, we prove the stretching potential vanishes as a logarithmic envelope on shrinking super-level sets. This depletion forces the vorticity magnitude into a sub-critical Lorentz-Zygmund space via interpolated De Giorgi energy method. The logarithmic gain is subsequently transferred to the velocity field, forcing the geometric scale of local 1D sparseness below the uniform radius of spatial analyticity, ultimately averting the finite-time blow-up via the harmonic measure maximum principle.

math.AP↗
arXiv2026

On Decay of the Local Mean Oscillations of the Vorticity Direction in Critical Navier-Stokes Flows

We isolate and analyze the geometric PDE governing the evolution of the vorticity direction in the 3D incompressible (unforced) Navier-Stokes equations (NSE), restricted to the case of a critical spatial point singularity where the vorticity magnitude concentrates as $O(|x|^{-2})$, inhabiting the critical Lorentz space $L^{3/2, \infty}$. The PDE consists of the Harmonic Map Heat Flow (HMHF) into the sphere supplemented with the fluid transport, cross-diffusion and tangential strain. The question is whether the NSE mechanics can propagate logarithmic decay of the local mean oscillations of the direction -- the condition $\xiVec \in \bmo_{1/|\log r|}$ which (in this setting) was shown in the companion paper to prevent finite time blow-up. The key observations are that the $O(|x|^{-2})$ concentration, factored out of the viscous cross-diffusion, generates an outward radial drift $4ν\, x/|x|^2$ at the core and that the HMHF nonlinearity is harmless for the quantity $\frac12|\xiVec - e|^2$, which is a subsolution on any hemisphere. This yields a transfer theorem: the $\bmo_{1/|\log r|}$ regularity of the direction at the core, uniformly up to the singular time, is controlled by a logarithmic modulus in time of the direction at the inner scale, together with the physical strain. Crucially, the strain enters the direction equation only through its tangential component $P_{\xiVec^\perp} S\xiVec$, which vanishes precisely when the direction is an eigenvector of the strain tensor. Since this includes the eigenvector carrying the maximal stretching, the result is in contrast to the classical geometric regularity criteria which are built on depleting the full vortex-stretching term and thus confined to configurations of weak stretching.

math.AP↗
arXiv2026

On taming Moffatt-Kimura vortices of doom in the viscous case

In this note we propose a two-layer viscous mechanism for preventing finite time singularity formation in the Moffatt-Kimura model of two counter-rotating vortex rings colliding at a nontrivial angle. In the first layer the scenario is recast within the framework of the study of turbulent dissipation based on a suitably defined `scale of sparseness' of the regions of intense fluid activity. Here it is found that the problem is (at worst) critical, i.e., the upper bound on the scale of sparseness of the vorticity super-level sets is comparable to the lower bound on the radius of spatial analyticity. In the second layer, an additional more subtle mechanism is identified, potentially capable of driving the scale of sparseness into the dissipation range and preventing the formation of a singularity. The mechanism originates in certain analytic cancellation properties of the vortex-stretching term in the sense of compensated compactness in Hardy spaces which then convert information on local mean oscillations of the vorticity direction (boundedness in certain log-composite weighted local bmo spaces) into log-composite faster decay of the vorticity super-level sets.

math.AP↗
arXiv2025

Time-Global Regularity of the Navier-Stokes System with Hyper-Dissipation--Turbulent Scenario

The question of whether the hyper-dissipative (HD) Napier-Stokes (NS) system can exhibit spontaneous formation of singularities in the super-critical regime--the hyper-diffusion being generated by a fractional power of the Laplacian, say $β$, confined to interval $\bigl(1, \frac{5}{4}\bigr)$--has been a major open problem in the mathematical fluid dynamics since the foundational work of J.L. Lions in 1960s. In this work, an evidence of criticality of the Laplacian is presented, more precisely, a class of plausible blow-up scenarios is ruled out as soon as $β$ is greater than one. While the framework is based on the scale of sparseness of the super-level sets of the positive and negative parts of the components of the higher-order derivatives of the velocity recently introduced by the authors, a major novelty in the current work is classification of the HD flows near a potential spatiotemporal singularity in two main categories, homogeneous (the case consistent with a near-steady behavior) and non-homogenous (the case consistent with the formation and decay of turbulence). The main theorem states that in the non-homogeneous case any $β$ greater than one prevents a singularity. In order to illustrate the impact of this result in a methodology-free setting, a two-parameter family of dynamically rescaled blow-up profiles is considered, and it is shown that as soon as $β$ is greater than one, a new region in the parameter space is ruled out. More importantly, the region is a neighborhood (in the parameter space) of the self-similar profile, i.e., the approximately self-similar blow-up, a prime suspect in possible singularity formation, is ruled out for all HD NS models.

math.AP↗
arXiv2025

On persistence of spatial analyticity in the hyper-dissipative Navier-Stokes models

The goal of this note is to demonstrate that as soon as the hyper-diffusion exponent is greater than one, a class of finite time blow-up scenarios consistent with the analytic structure of the flow (prior to the possible blow-up time) can be ruled out. The argument is self-contained, in spirit of the regularity theory of the hyper-dissipative Navier-Stokes system in turbulent regime developed by Grujic and Xu.

math.AP↗
arXiv2025

Asymptotic Criticality of the Navier-Stokes Regularity Problem

The problem of global-in-time regularity for the 3D Navier-Stokes equations, i.e., the question of whether a smooth flow can exhibit spontaneous formation of singularities, is a fundamental open problem in mathematical physics. Due to the super-criticality of the equations, the problem has been super-critical in the sense that there has been a scaling gap between any regularity criterion and the corresponding \emph{a priori} bound (regardless of the functional setup utilized). The purpose of this work is to present a mathematical framework--based on a suitably defined scale of sparseness of the super-level sets of the positive and negative parts of the components of the higher-order spatial derivatives of the velocity field--in which the scaling gap between the regularity class and the corresponding \emph{a priori} bound vanishes as the order of the derivative goes to infinity.

math.AP↗
arXiv2019

Analysis method for detecting topological defect dark matter with a global magnetometer network

The Global Network of Optical Magnetometers for Exotic physics searches (GNOME) is a network of time-synchronized, geographically separated, optically pumped atomic magnetometers that is being used to search for correlated transient signals heralding exotic physics. GNOME is sensitive to exotic couplings of atomic spins to certain classes of dark matter candidates, such as axions. This work presents a data analysis procedure to search for axion dark matter in the form of topological defects: specifically, walls separating domains of discrete degenerate vacua in the axion field. An axion domain wall crossing the Earth creates a distinctive signal pattern in the network that can be distinguished from random noise. The reliability of the analysis procedure and the sensitivity of the GNOME to domain-wall crossings is studied using simulated data.

astro-ph.IM↗
arXiv2019

A Regularity Criterion for Solutions to the 3D NSE in `Dynamically Restricted' Local Morrey Spaces

It is shown that a local-in-time strong solution $u$ to the 3D Navier-Stokes equations remains regular on an interval $(0,T)$ provided a smallness $ε_0$-condition on $u$ in a lower time-restricted local Morrey space is stipulated; more precisely, $$\sup_{t\in(0,T)} \ \sup_{x \in \mathbb{R}^3, \ η(t) \le r \le 1} \ \frac{1}{r^α} \int_{B_r(x)} |u(y,t)|^p dy \le ε_0$$ where $η$ is a dynamic dissipation scale consistent with the turbulence phenomenology and $α$ and $p$ are suitable parameters. Such regularity criterion guarantees the volumetric sparseness of local spatial structure of intense vorticity components, preventing the formation of the finite-time blow up at $T$ under the framework of $Z_α$-sparseness classes introduced in [Bradshaw, Farhat and Grujic, ARMA, 2018].

math.AP↗
arXiv2018

Local near-Beltrami structure and depletion of the nonlinearity in the 3D Navier-Stokes flows

Computational simulations of turbulent flows indicate that the regions of low dissipation/enstrophy production feature high degree of local alignment between the velocity and the vorticty, i.e., the flow is locally near-Beltrami. Hence one could envision a geometric scenario in which the persistence of the local near-Beltramy property might be consistent with a (possible) finite-time singularity formation. The goal of this note is to show that this scenario is in fact prohibited if the sine of the angle between the velocity and the vorticty is small enough with respect to the local enstrophy.

math.AP↗
arXiv2016

Frequency localized regularity criteria for the 3D Navier-Stokes equations

Two regularity criteria are established to highlight which Littlewood-Paley frequencies play an essential role in possible singularity formation in a Leray-Hopf weak solution to the Navier-Stokes equations in three spatial dimensions. One of these is a frequency localized refinement of known Ladyzhenskaya-Prodi-Serrin-type regularity criteria restricted to a finite window of frequencies the lower bound of which diverges to $+\infty$ as $t$ approaches an initial singular time.

math.AP↗
arXiv2013

A note on the surface quasi-geostrophic temperature variance cascade

In this note we examine the dynamical role played by inertial forces on the surface temperature (or buoyancy) variance in strongly rotating, stratified flows with uniform potential vorticity fields. In particular, using a dynamic, multi-scale averaging process, we identify a sufficient condition for the existence of a direct temperature variance cascade across an inertial range. While the result is consistent with the physical and numerical theories of SQG turbulence, the condition which triggers the cascade is more exotic, a fact reflecting the non-locality introduced by fractional dissipation.

math-ph↗
arXiv2013

Blow-up scenarios for 3D NSE exhibiting sub-criticality with respect to the scaling of one-dimensional local sparseness

It is shown that, if the vorticity magnitude associated with a (presumed singular) three-dimensional incompressible Navier-Stokes flow blows-up in a manner exhibiting certain {\em time dependent local structure}, then {\em time independent} estimates on the $L^1$ norm of $|ω|\log\sqrt{1+ |ω|^2}$ follow. The implication is that the volume of the region of high vorticity decays at a rate of greater order than a rate connected to the critical scaling of one-dimensional local sparseness and, consequently, the solution becomes sub-critical.

math.AP↗