arXiv subjects

Rishad Shahmurov

Publications and source records attributed to Rishad Shahmurov.

At least 19 recordsLinked to original sources

arXiv2026

Axis Regularity in the 5D Corridor

We study axisymmetric Navier--Stokes flow with swirl through the five-dimensional lift of the azimuthal-vorticity equation. We prove the weighted identities behind the corridor method: zero capacity of the axis, a sign-safe Hardy inequality, the exact circulation Caccioppoli formula, five-dimensional velocity recovery, a logarithmic source-mass estimate, and negative-norm compactness of the swirl source. These estimates yield a precise conditional reduction from axis regularity to a finite set of scale-uniform closure statements. We also prove that a natural weighted quartic inequality cannot provide that closure in the proposed corridor. The paper is a rigorous reduction theorem; it does not prove unconditional regularity for arbitrary swirl.

math.AP↗

arXiv2026

Threshold Decomposition of Vorticity Stretching and Palinstrophy

We give an exact finite-time decomposition of vortex stretching by resolving the vorticity equation according to the size of the vorticity. For almost every threshold $a>0$, the stretching occurring where $|ω|>a$ equals the change of the corresponding superlevel mass plus two nonnegative viscous terms: one measuring changes in vorticity direction and one measuring dissipation across the level surface $|ω|=a$. Integrating this identity over all thresholds recovers the classical $L^p$ vorticity balance and, for $p=2$, the usual spacetime palinstrophy. At the scale-critical exponent $p=3/2$, the dissipation can be written through the nonlinear variable $Z=|ω|^{-1/4}ω$. The threshold formulation also gives a positive measure that can be used to assign overlapping high-vorticity episodes without double counting. Under additional compactness assumptions it leads, through empirical transition couplings supported on a closed physical transition relation, to a stationary Markov renormalization law; no deterministic return map is required. These are structural results; no unconditional three-dimensional regularity theorem is claimed.

math.AP↗

arXiv2026

Boundary Hyperbolic Packets in Axisymmetric Euler Flow

We study hyperbolic side-wall packets for the axisymmetric Euler equations with swirl in a periodic cylinder. In the exact odd class, $Γ=ru^θ$ and $G=ω^θ/r$ satisfy $D_tΓ=0$ and $D_tG=r^{-4}\partial_z(Γ^2)$, and the symmetry fixes the boundary point at which the relevant swirl and vorticity gradients are measured. We derive the five-dimensional side-wall Green-kernel expansion, its leading signed hyperbolic kernel, packet--core coupling, off-diagonal shear estimates, anisotropy and source-curvature identities, and a maximal conic-score formulation. Under a common admissible packet bootstrap these estimates imply the Dini system $D^+\mathfrak M\ge cB^2$, $B'\ge c\mathfrak M B$, and hence finite-time blow-up of the comparison amplitudes. The result is a conditional amplification criterion: it does not prove that one smooth Euler trajectory preserves all packet-dominance, tail, shear, anisotropy, and validity hypotheses until the comparison blow-up time. Establishing or refuting that persistence is the remaining problem for this mechanism.

math.AP↗

arXiv2026

Source-Driven Transitions in Axisymmetric Euler Flow

We study a finite-time source-driven transition in axisymmetric Euler flow with swirl. For an explicit smooth compact datum, we prove exact source, moment-return, fixed-label transport, and response identities. For the same frozen finite reference, interval arithmetic gives continuum derivative bounds and a terminal scalar-trace enclosure. Under the remaining global comparison hypotheses, these estimates imply contraction of two fixed material labels and a finite-radius outgoing level set. We state separately the unresolved steps: global defect enclosure, sufficient next-block admission, cylindrical whole-space realization, and repeated regeneration from one datum. No whole-space Euler singularity is claimed

math.AP↗

arXiv2026

Endpoint Architectures for Navier--Stokes

We compare two endpoint analyses for three-dimensional Navier--Stokes flow: a general suitable-weak framework and the axisymmetric equations with swirl. The common backend is selection of physical space--time records, nonoverlapping allocation of positive quantities, retention of the full localized state, and passage either to a compact transition law or to an explicitly named loss of compactness. The front-end theorems are different. In the general problem, bounded-critical sequences lead to symmetry/defect alternatives, while amplitude-fast normalization leads conditionally to an ancient Euler profile only after frequency tightness and a nonvanishing witness are retained. In the axisymmetric problem, logarithmic swirl records are routed to meridional energy or scale-critical signed compression, whereas first-use provenance and recharge packing force recurrent fresh service toward boundary, age, frequency, or other recession mechanisms. We state the interfaces and the remaining endpoints without identifying their currencies or reducing the general problem to the axisymmetric class. No unconditional regularity theorem or singularity construction is claimed.

math.AP↗

arXiv2026

Critical Structure of Axisymmetric Navier--Stokes with Swirl

We study the axisymmetric Navier--Stokes equations with swirl in the variables $F=u^θ/r$, $G=ω^θ/r$, and $Γ=ru^θ$. Their distinct scaling degrees lead to several critical transition surfaces rather than a single scalar exponent. We prove mesoscopic and temporal critical-face identities, source/contact trace estimates, compact finite-depth preparation obstructions, an affine six-power flattening barrier, a fixed-old-current compression ceiling, and compactness for rescaled strain generators with bounded action and bounded variation. A sharp radial estimate gives exponential control of the time occupied by large swirl-energy records; on parabolic time intervals it yields logarithmic routing from total kinetic records to the meridional component, and an independent cone argument routes non-meridional Type-II records to a scale-critical signed compression action. The source-free circulation also satisfies an exact quadratic dissipation identity, while scalar-weight calculations show that no radial power simultaneously yields a compression-free continuity law and a scale-zero current action. The results exclude several compact and passively prepared recurrence mechanisms under explicit hypotheses; meridional Type-II records, signed critical compression, noncompact high-frequency behavior, and boundary-entry alternatives remain.

math.AP↗

arXiv2026

Endpoint Structure in Three-Dimensional Navier--Stokes

We study possible singular endpoints of the three-dimensional incompressible Navier--Stokes equations by parabolic rescaling. Weak limits can lose information through pressure, nonlinear products, concentration, spatial tails, or symmetry, so we keep these quantities as part of one augmented endpoint state and use a fixed whole-space pressure representative throughout the shrinking chain. We prove compact endpoint realization, nonoverlapping assignment of positive defects, two-generation inheritance, rotational alternatives, and a high-critical kinetic normalization centered at the original singular point. We show that positive rotational distance alone does not coerce a productive rotational Reynolds stress, and we give an explicit conditional compactness theorem under which frequency-tight amplitude-fast records pass strongly to a nonzero ancient Euler profile. We then study a possible Euler limit that is invisible under positive heat evolution. Exact heat-orbit nullity forces cancellation separately on each interaction-energy shell and isotropic covariance on every Fourier sphere; independently, any nonzero $L^2$ field normalized along its long heat orbit loses all mass from every fixed spatial ball. Frequency anti-concentration, the accumulating heat-chain passage, and retention of a singular witness remain open; no unconditional regularity theorem is claimed.

math.AP↗

arXiv2026

Critical Compression in Axisymmetric Swirl

We study two scale-critical inward-compression mechanisms in smooth axisymmetric Navier--Stokes flow with swirl. For the weighted swirl family Y_d=r^d u^theta, -1<=d<1, an exact L^p balance yields a signed radial-compression action. If p>=3/(1-d), finite L^p mass and a bounded signed action imply continuation. On the critical diagonal Y_q=r^(1-3/q)u^theta, q>=3/2, the integrating factor is universal, exp(3 int Ubar_q); H^1 data automatically supply the critical mass for 2<=q<=3, while bounded circulation extends this to every finite q>3. Independently, a sliding Petrovskii displacement criterion reaches the endpoint coefficient 2 and admits a quantified iterated-log correction, while a pulse train shows terminal-only control is insufficient. For profiles with the corresponding weighted moments, the instantaneous Hodge response to partial_z(F^2) is non-amplifying for power-weighted compressive F^2-work throughout 1<=alpha<=5. Thus a finite-time singularity must escape a continuum of critical signed-compression detectors and recurrently cross the Petrovskii wall.

math.AP↗

arXiv2026

Vorticity-Response Blowup in Axisymmetric Models

We study how the orientation and elliptic depth of swirl-generated vorticity response affect singularity formation in axisymmetric model equations. For a sign-reversed Hou--Li model on the whole line, the dynamics factorize into two real Riccati--diffusion channels. For every viscosity $ν>0$, every nontrivial smooth datum in an infinite-dimensional cone of even, nonnegative, monotone channel data develops finite-time amplitude blowup; in the inviscid case, characteristics give the exact blowup time. For the same smooth swirling inviscid datum, the standard response sign is global whereas the reversed sign blows up. We then introduce a three-parameter 5D response family separating transport geometry, response orientation, and elliptic depth. Exact divergence, energy, and scaling identities show that physical three-dimensional incompressibility selects one geometry, weighted 5D incompressibility selects another, and Navier--Stokes scaling makes two elliptic inversions critical. The frozen physical response is critical and dissipative, while a natural one-inversion 5D-solenoidal model is supercritical and energy-pumping. These results identify response orientation and causal depth as load-bearing structure in axisymmetric model dynamics.

math.AP↗

arXiv2026

Hardy Criticality in Axisymmetric Swirl

For smooth axisymmetric Navier--Stokes flow with swirl, set $F=u^θ/r$, $Γ=ru^θ$, $G=ω^θ/r$, and $U=u^r/r$. Weighted swirl energies are classical; we determine the sharp two-parameter coercivity spectrum of the exact power-weight family $\int |F|^p r^α\,dr\,dz$, $p>1$, $α>1$. The full diffusion form has a positive gradient gap for $α<2p+1$, loses every uniform gap at $α=2p+1$, and is indefinite above it. Exactly at the same threshold the radial-strain term cancels, and the identity becomes the classical $L^p$ circulation energy for $Γ$. Thus stretching cancellation and Hardy criticality coincide within the power family. We also characterize $r^3drdz$ by conservative $F$-transport, self-adjoint five-dimensional diffusion, and a flat $G$-source pairing, and obtain the optimal strain form-bound threshold within the Hardy-coercive powers.

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↗

arXiv2020

The regularity properties and blow-up for convolution wave equations and applications

In this paper, the Cauchy problem for linear and nonlinear convolution wave equations are studied.The equation involves convolution terms with a general kernel functions whose Fourier transform are operator functions defined in a Banach space E together with some growth conditions. Here, assuming enough smoothness on the initial data and the operator functions, the local, global existence, uniqueness and regularity properties of solutions are established in terms of fractional powers of given sectorial operator functon. Furthermore, conditions for finite time blow-up are provided. By choosing the space E and the operators, the regularity properties the wide class of nonlocal wave equations in the field of physics are obtained.

math.AP↗

arXiv2019

Existence local and global solution of multipoint Cauchy problem for nonlocal nonlinear equations

In this paper, the multipoint Cauchy problem for nonlocal nonlinear wave type equatıons are studied.The equation involves a convolution integral operator with a general kernel function whose Fourier transform is nonnegative. We establish local and global existence and uniqueness of solutions assuming enough smoothness on the initial data together with some growth conditions on the nonlinear term

math.AP↗

arXiv2019

The local and global dynamics of a general cancer tumor growth model with multiphase structure

We present a phase-space analysis of a mathematical model of tumor growth with an immune responses. We consider mathematical analysis of the model equations with multipoint initial condition regarding to dissipativity, boundedness of solutions, invariance of non-negativity, local and global stability and the basins of attractions. We derive some features of behavior of one of three-dimensional tumor growth models with dynamics described in terms of densities of three cells populations: tumor cells, healthy host cells and effector immune cells. We found sufficient conditions, under which trajectories from the positive domain of feasible multipoint initial conditions tend to one of equilibrium points. Here, cases of the small tumor mass equilibrium points-the healthy equilibrium point, the "death" equilibrium point have been examined. Biological implications of our results are discussed.

math.DS↗

arXiv2019

The integral Cauchy problem for generalized Boussinesq equations with general leading parts

In this paper, the integral initial value problems for Boussinesq type equations are studied. The equation include the general differential operators. The existence, uniqueness and regularity properties of solution of these problems are obtained. By choosing differential operators including in the equation, the regularity properties of the Cauchy problem for different type of Boussinesg equations are studied.

math.AP↗

arXiv2017

Operator-valued multipliers in vector-valued weighted Besov spaces and applications

The operator-valued multiplier theorems in weighted abstract Besov spaces are studied. These results permit us to show embedding theorems in weighted Besov-Lions type spaces. The most regular class of interpolation space is found such that the mixed differential operator is bounded from Besov-Lions space to this and Ehrling-Nirenberg-Gagliardo type sharp estimates are established. By using these results the separability properties of degenerate differential operators are studied. Especially, we prove that the associated differential operators are positive and also are generators of analytic semigroups. Moreover, regularity properties for abstract elliptic equation, Cauchy problem for degenerate abstract parabolic equation and the infinite systems of degenerate parabolic equations are studied.

math.AP↗

arXiv2009

On Integral Operators with Operator Valued Kernels

Here Lq-Lp boundedness of integral operator with operator-valued kernels is studied and the main result is applied to convolution operators. Using these results Besov space regularity for Fourier multiplier operator is established.

math.FA↗

arXiv2009

Lp regularity for convolution operator equations in Banach spaces

Here we utilize operator--valued Lq-Lp Fourier multiplier theorems to establish lower bound estimates for large class of elliptic integro-differential equations in Rd. Moreover, we investigate separability properties of parabolic convolution operator equations that arise in heat conduction problems in materials with fading memory. Finally, we give some remarks on optimal regularity of elliptic differential equations and Cauchy problem for parabolic equations.

math.AP↗