Quantum-Inspired Stochastic Modeling and Regularity Analysis in Turbulent Flows

TL;DR

This paper develops a quantum-inspired mathematical framework combining stochastic analysis, fractional operators, and anisotropic modeling to better understand turbulence, regularity, and energy dissipation in fluid flows.

Contribution

It introduces a novel quantum-inspired approach with Schr"odinger-type operators and anisotropic stochastic models, providing new regularity theorems and energy dissipation insights for turbulent flows.

Findings

New regularity theorems in Sobolev-Besov spaces

Rigorous a priori estimates for solutions

Enhanced understanding of energy dissipation mechanisms

Abstract

This paper introduces a novel mathematical framework for examining the regularity and energy dissipation properties of solutions to the stochastic Navier-Stokes equations. By integrating Sobolev-Besov hybrid spaces, fractional differential operators, and quantum-inspired modeling techniques, we provide a comprehensive analysis that captures the multiscale and chaotic dynamics inherent in turbulent flows. Central to this framework is a Schr\"odinger-type operator adapted for fluid dynamics, which encapsulates quantum-scale turbulence effects, thereby elucidating the mechanisms of energy redistribution across scales. Additionally, we develop anisotropic stochastic models with direction-dependent viscosity, characterized by a pseudo-differential operator and a covariance matrix governing directional diffusion. These models more accurately reflect real-world turbulence, where viscosity varies with flow orientation, enhancing both theoretical insights and practical simulation capabilities. Our main contributions include new regularity theorems and rigorous a priori estimates for solutions in Sobolev-Besov spaces, alongside proofs of energy dissipation properties in anisotropic contexts. These findings advance the understanding of fluid turbulence by offering a refined approach to studying scale interactions, stochastic effects, and anisotropy in turbulent flows.