Vortical Memory, Structural Damping and the Geometric Limits of Blow-up in Navier–stokes
Year: 2026
Type: preprint
Abstract: VORTICAL MEMORY, STRUCTURAL DAMPING AND THE GEOMETRIC LIMITS OF BLOW-UP IN NAVIER–STOKES A Transductive Architecture for the Control and Validation of Singularity Cláudio Vicente da Silva Independent Researcher — Londrina, Paraná, Brazil November 2026 Vortical Memory, Structural Damping and the Geometric Limits of Blow-Up in Navier–Stokes: A Transductive Architecture for the Control and Validation of Singularity presents a mathematical framework for investigating vortical amplification, structural damping, geometric invariance, and the conditions associated with singularity formation in the three-dimensional Navier–Stokes equations. The study develops a transductive architecture organized around four interconnected components: the critical structural damping parameter ((\lambda_c)), the dynamical state representation ((X(t))), the geometric invariant region ((Q9)), and vortical memory ((M_\omega)). Together, these elements provide a structured approach to examining the evolution of fluid systems through the relationships among nonlinear production, viscous dissipation, damping, and accumulated dynamical effects. The mathematical formulation introduces a damped extension of the Navier–Stokes equations through the linear term (-\lambda u), representing an additional structural dissipation mechanism. From the energy equation, the study establishes an exponential energy-decay relation. The vorticity formulation incorporates the effects of stretching, transport, viscous diffusion, and structural damping, while vortical memory is represented by a time-integrated expression that records the accumulated contribution of vortical production under exponential damping. The dynamical state is defined as a composite structure incorporating velocity, vorticity, and vortical memory: [X(t)=(u(t),\omega(t),M_\omega(t))^T.] A structural control functional combines kinetic energy and vortical intensity. Based on its evolution, the critical damping parameter (\lambda_c(R)) is defined in relation to nonlinear production within a bounded region of state space. This provides a quantitative reference for evaluating the damping intensity required to control the evolution of the system within a specified functional level. The study also introduces the geometric region (Q9_R), defined through an upper bound of the structural functional. Its positive invariance is established under the condition (\lambda>\lambda_c(R)), yielding an exponential decay estimate for the functional during the regular existence of the solution. The corresponding theorem organizes the relationship between critical damping, state-space control, and geometric invariance. Vortical memory is further incorporated into an enlarged control functional, connecting instantaneous flow configurations with the accumulated effects of previous vortical production and structural damping. This formulation extends the dynamical representation by introducing a temporal component into the geometric analysis of the system. An analytical and computational validation protocol is presented, comprising the construction of incompressible initial conditions, variation of the damping parameter, determination of the critical threshold, testing of the invariance of Q9, evaluation of vortical memory, and monitoring of kinetic energy, enstrophy, and the structural functional. The resulting architecture brings together damping, memory, geometric control, and dynamical evolution in a unified mathematical representation. It provides a framework for further analytical and computational investigations into the control of vortical amplification, the preservation of bounded dynamical regions, and the geometric limits associated with blow-up in Navier–Stokes systems. Keywords: Navier–Stokes equations; vortical memory; structural damping; transduction; blow-up; geometric invariance; dynamical stability; critical damping parameter; vorticity; enstrophy; state-space analysis; mathematical modeling.
Author: Vicente da Silva, Cláudio
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225387658