arXiv subjects

Rômulo Damasclin Chaves dos Santos

Publications and source records attributed to Rômulo Damasclin Chaves dos Santos.

32 records · Page 2Linked to original sources

arXiv2024

Dissipation and Regularity in the Smagorinsky Model with Dynamic Boundaries

This article presents an innovative extension of the Smagorinsky model incorporating dynamic boundary conditions and advanced regularity methods. We formulate the modified Navier-Stokes equations with the Smagorinsky term to model dissipation in turbulence and prove theorems concerning the existence, uniqueness, and asymptotic behavior of solutions. The first theorem establishes the existence and uniqueness of solutions in higher Sobolev spaces, considering the effect of the nonlinear Smagorinsky term and dynamic boundary conditions. The proof employs the Galerkin method and energy estimates, culminating in the application of Grönwall's theorem. The second theorem investigates the asymptotic behavior of solutions, focusing on anomalous dissipation in high-turbulence regimes. We demonstrate that the dissipated energy does not decrease with vanishing viscosity, indicating the occurrence of anomalous dissipation. The third theorem explores advanced regularity in higher Sobolev spaces, allowing for more rigorous control of nonlinear terms and ensuring improved stability conditions. The proof utilizes the energy method combined with Sobolev estimates and Grönwall's inequality. These mathematical results are fundamental for the analysis of dissipation in turbulent flows and can inspire new approaches in numerical simulations of fluids.

math.AP↗

arXiv2024

A High-Order Analytical Extension of the Corrected Smagorinsky Model for Non-Equilibrium Turbulent Flow

This study presents an extension of the corrected Smagorinsky model, incorporating advanced techniques for error estimation and regularity analysis of far-from-equilibrium turbulent flows. A new formulation that increases the model's ability to explain complex dissipative processes in turbulence is presented, using higher-order Sobolev spaces to address incompressible and compressible Navier-Stokes equations. Specifically, a refined energy dissipation mechanism that provides a more accurate representation of turbulence is introduced, particularly in the context of multifractal flow regimes. Furthermore, we derive new theoretical results on energy regularization in multifractal turbulence, contributing to the understanding of anomalous dissipation and vortex stretching in turbulent flows. The work also explores the numerical implementation of the model in the presence of challenging boundary conditions, particularly in dynamically evolving domains, where traditional methods struggle to maintain accuracy and stability. Theoretical demonstrations and analytical results are provided to validate the proposed framework, with implications for theoretical advances and practical applications in computational fluid dynamics. This approach provides a basis for more accurate simulations of turbulence, with potential applications ranging from atmospheric modeling to industrial fluid dynamics.

physics.flu-dyn↗

arXiv2024

Adaptive Dissipation in the Smagorinsky Model for Turbulence in Boundary-Driven Flows

This paper enhances the classic Smagorinsky model by introducing an innovative, adaptive dissipation term that adjusts dynamically with distance from boundary regions. This modification addresses a known limitation of the standard model over dissipation near boundaries thereby improving accuracy in turbulent flow simulations in confined or wall-adjacent areas. We present a rigorous theoretical framework for this adaptive model, including two foundational theorems. The first theorem guarantees existence and uniqueness of solutions, ensuring that the model is mathematically well-posed within the adaptive context. The second theorem provides a precise bound on the energy dissipation rate, demonstrating that dissipation remains controlled and realistic even as boundary effects vary spatially. By allowing the dissipation coefficient to decrease near boundary layers, this approach preserves the finer turbulent structures without excessive smoothing, yielding a more physically accurate representation of the flow. Future work will focus on implementing this adaptive model in computational simulations to empirically verify the theoretical predictions and assess performance in scenarios with complex boundary geometries.

physics.flu-dyn↗

arXiv2024

Stochastic Regularity in Sobolev and Besov Spaces with Variable Noise Intensity for Turbulent Fluid Dynamics

This paper advances the stochastic regularity theory for the Navier-Stokes equations by introducing a variable-intensity noise model within the Sobolev and Besov spaces. Traditional models usually assume constant-intensity noise, but many real-world turbulent systems exhibit fluctuations of varying intensities, which can critically affect flow regularity and energy dynamics. This work addresses this gap by formulating a new regularity theorem that quantifies the impact of stochastic perturbations with bounded variance on the energy dissipation and smoothness properties of solutions. The author employs techniques such as the Littlewood-Paley decomposition and interpolation theory, deriving rigorous bounds, and we demonstrate how variable noise intensities influence the behavior of the solution over time. This study contributes theoretically by improving the understanding of energy dissipation in the presence of stochastic perturbations, particularly under conditions relevant to turbulent flows where randomness cannot be assumed to be uniform. The findings have practical implications for more accurate modeling and prediction of turbulent systems, allowing potential adjustments in simulation parameters to better reflect the observed physical phenomena. This refined model therefore provides a fundamental basis for future work in fluid dynamics, particularly in fields where variable stochastic factors are prevalent, including meteorology, oceanography, and engineering applications involving fluid turbulence. The present approach not only extends current theoretical frameworks but also paves the way for more sophisticated computational tools in the analysis of complex and stochastic fluid systems.

physics.flu-dyn↗

arXiv2024

Energy Dissipation and Regularity in Quaternionic Fluid Dynamics using Sobolev-Besov Spaces

This study investigates the dynamics of incompressible fluid flows through quaternionic variables integrated within Sobolev-Besov spaces. Traditional mathematical models for fluid dynamics often employ Sobolev spaces to analyze the regularity of the solution to the Navier-Stokes equations. However, with the unique ability of Besov spaces to provide localized frequency analysis and handle high-frequency behaviors, these spaces offer a refined approach to address complex fluid phenomena such as turbulence and bifurcation. Quaternionic analysis further enhances this approach by representing three-dimensional rotations directly within the mathematical framework. The author presents two new theorems to advance the study of regularity and energy dissipation in fluid systems. The first theorem demonstrates that energy dissipation in quaternionic fluid systems is dominated by the higher-frequency component in Besov spaces, with contributions decaying at a rate proportional to the frequency of the quaternionic component. The second theorem provides conditions for regularity and existence of solutions in quaternionic fluid systems with external forces. By integrating these hypercomplex structures with Sobolev-Besov spaces, our work offers a new mathematically rigorous framework capable of addressing frequency-specific dissipation patterns and rotational symmetries in turbulent flows. The findings contribute to fundamental questions in fluid dynamics, particularly by improving our understanding of high Reynolds number flows, energy cascade behaviors, and quaternionic bifurcation. This framework therefore paves the way for future research on regularity in complex fluid dynamics.

math.AP↗

arXiv2024

Advanced Theoretical Analysis of Stability and Convergence in Computational Fluid Dynamics for Computer Graphics

Mathematical modeling of fluid dynamics for computer graphics requires high levels of theoretical rigor to ensure visually plausible and computationally efficient simulations. This paper presents an in-depth theoretical framework analyzing the mathematical properties, such as stability, convergence, and error bounds, of numerical schemes used in fluid simulation. Conditions for stability in semi-Lagrangian and particle-based methods were derived, demonstrating that these methods remain stable under certain conditions. Furthermore, convergence rates for Navier-Stokes discretizations were obtained, showing that numerical solutions converge to analytical solutions as spatial resolution and time step decrease. Furthermore, new theoretical results were introduced on the maintenance of incompressibility and conservation of vorticity, which are crucial for the physical accuracy of simulations. The findings serve as a mathematical foundation for future research in adaptive fluid simulation, guiding the development of robust simulation techniques for real-time graphics applications.

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arXiv2024

Enhanced Diffusion in Anisotropic and Non-uniform Geometries

Enhanced diffusion, which describes the accelerated spread of passive scalars due to the interaction between advection and molecular diffusion, has been extensively studied in simplified geometries, such as uniform shear and radial flows. However, many real-world applications occur in complex, anisotropic domains where standard assumptions do not hold. This paper extends the theory of enhanced diffusion to anisotropic and non-uniform geometries, where scaling in the \(x\)- and \(y\)-directions follows distinct, power-law relationships, and the velocity field exhibits spatially varying regularity. We define a generalized framework for anisotropic diffusion enhancement and rigorously derive new scaling laws for diffusion rates in these settings. Specifically, we show that in a domain with anisotropic scaling functions \( f(x) \sim |x|^p \) and \( g(y) \sim |y|^q \), the enhanced diffusion rate \( r(κ) \) for a passive scalar satisfies \( r(κ) = C κ^{\frac{pq}{p+q+2}} \), where \( C \) depends on the regularity and scaling properties of the domain but remains independent of the diffusivity \( κ\). This theoretical result is validated through detailed proofs leveraging stochastic process techniques and variance estimation. Our findings offer a new perspective on the role of domain geometry and anisotropy in diffusion processes, with potential applications in environmental science, engineering, and beyond. This framework lays the groundwork for future studies in both theoretical and applied settings, enabling a deeper understanding of transport phenomena in complex domains.

math.AP↗

arXiv2024

Stochastic Trajectories and Spectral Boundary Conditions for Enhanced Diffusion in Immersed Boundary Problems

This work presents a comprehensive framework for enhanced diffusion modeling in fluid-structure interactions by combining the Immersed Boundary Method (IBM) with stochastic trajectories and high-order spectral boundary conditions. Using semi-Lagrangian schemes, this approach captures complex diffusion dynamics at moving interfaces, integrating probabilistic methods that reflect multi-scale fluctuations. In addition to a rigorous mathematical foundation that includes stability proofs, this model exhibits reduced numerical diffusion errors and improved stability in long-term simulations. Comparative studies highlight its effectiveness in multi-scale scenarios that require precision in interface dynamics. Focusing on various shear and circular flows, including those with Hölder and Lipschitz regularities and critical points, we establish sharp bounds on effective diffusion rates using specific initial data examples. This dual exploration in enhanced diffusion highlights how flow regularity and critical points influence dissipation. These findings advance both the theoretical understanding and practical applications of enhanced diffusion in fluid dynamics, offering new insights into diffusion rate optimization through interface dynamics and flow structure regularities. Future research can further refine the IBM framework by exploring alternative probabilistic methods to improve interface accuracy, opening up the potential for improved modeling in applications that require precise control over mixing rates and dissipation processes.

math.AP↗

arXiv2024

Hybrid Adaptive Dual Reciprocity Method for Efficient Solution of Large-Scale Non-Linear Boundary Conditions

This article proposes a hybrid adaptive numerical method based on the Dual Reciprocity Method (DRM) to solve problems with non-linear boundary conditions and large-scale problems, named Hybrid Adaptive Dual Reciprocity Method (H-DRM). The method uses a combination of DRM to handle non-homogeneous terms, iterative techniques to deal with non-linear boundary conditions, and an adaptive multiscale approach for large-scale problems. Additionally, the H-DRM incorporates local finite elements in critical regions of the domain. This method aims to improve computational efficiency and accuracy for problems involving complex geometry and non-linearities at the boundary, offering a robust solution for physical and engineering problems. Demonstrations and computational results are presented, validating the effectiveness of the method compared to other known methods through an iterative process of 7 million iterations.

math.NA↗

arXiv2024

Vortex Formation and Dissipation in Chaotic Flows: A Hypercomplex Approach

This article presents a comprehensive analysis of the formation and dissipation of vortices within chaotic fluid flows, leveraging the framework of Sobolev and Besov spaces on Riemannian manifolds. Building upon the Navier-Stokes equations, we introduce a hypercomplex bifurcation approach to characterize the regularity and critical thresholds at which vortices emerge and dissipate in chaotic settings. We explore the role of differential geometry and bifurcation theory in vortex dynamics, providing a rigorous mathematical foundation for understanding these phenomena. Our approach addresses spectral decomposition, asymptotic stability, and dissipation thresholds, offering critical insights into the mechanisms of vortex formation and dissipation. Additionally, we introduce two new theorems that further elucidate the role of geometric stability and bifurcations in vortex dynamics. The first theorem demonstrates the geometric stability of vortices on manifolds with positive Ricci curvature, while the second theorem analyzes the bifurcation points that lead to the formation or dissipation of vortex structures. This research contributes to the existing literature by providing a more complete mathematical picture of the underlying mechanisms of vortex dynamics in turbulent fluid flows.

physics.flu-dyn↗

arXiv2024

High-Order Dynamic Integration Method (HODIM) for Modeling Turbulent Fluid Dynamics

This research explores the development and application of the High-Order Dynamic Integration Method for solving integro-differential equations, with a specific focus on turbulent fluid dynamics. Traditional numerical methods, such as the Finite Difference Method and the Finite Volume Method, have been widely employed in fluid dynamics but struggle to accurately capture the complexities of turbulence, particularly in high Reynolds number regimes. These methods often require significant computational resources and are prone to errors in nonlinear dynamic systems. The High-Order Dynamic Integration Method addresses these challenges by integrating higher-order interpolation techniques with dynamic adaptation strategies, significantly enhancing accuracy and computational efficiency. Through rigorous numerical analysis, this method demonstrates superior performance over the Finite Difference Method and the Finite Volume Method in handling the nonlinear behaviors characteristic of turbulent flows. Furthermore, the High-Order Dynamic Integration Method achieves this without a substantial increase in computational cost, making it a highly efficient tool for simulations in computational fluid dynamics. The research validates the capabilities of the High-Order Dynamic Integration Method through a series of benchmark tests and case studies. Results indicate a marked improvement in both accuracy and stability, particularly in simulations of high-Reynolds-number flows, where traditional methods often falter. This innovative approach offers a robust and efficient alternative for solving complex fluid dynamics problems, contributing to advances in the field of numerical methods and computational fluid dynamics.

physics.flu-dyn↗

arXiv2024

Mountain-Pass Solutions in 2D Turbulence with Generalized Sobolev Operators

This work extends the study of mean field equations arising in two-dimensional (2D) turbulence by introducing generalized weighted Sobolev operators. Employing variational methods, particularly the mountain pass theorem and a refined blow-up analysis, we establish the existence of nontrivial solutions under broader boundary conditions than those considered in previous studies. In contrast to the unweighted approach developed by Ricciardi (2006), the incorporation of variable weights \(ρ(x)\) enables a more accurate representation of complex geometric domains. This generalization addresses cases where the geometry of the manifold or external factors induce local fluctuations, thereby enhancing the applicability of mean field models to real-world turbulence phenomena. The weighted Sobolev framework also strengthens control over the nonlinearity in the problem, reducing the risk of blow-up and ensuring the convergence of solutions in the \(H^1(M)\) space. The analysis reveals new conditions for the existence of solutions, including situations where classical methods may fail due to a lack of compactness. The significant advancement lies in the flexibility of the model; the inclusion of weights allows for a more precise description of physical systems in varied settings, representing a key advantage over the original unweighted problem. These results contribute to a deeper understanding of the role of geometric and physical constraints in turbulent behavior and pave new pathways for future research on nonlinear partial differential equations in complex domains.

math.AP↗

arXiv2024

Quantum-Inspired Stochastic Modeling and Regularity in Turbulent Fluid Dynamics

This paper presents an innovative framework for analyzing the regularity of solutions to the stochastic Navier-Stokes equations by integrating Sobolev-Besov hybrid spaces with fractional operators and quantum-inspired dynamics. We propose new regularity theorems that address the multiscale and chaotic nature of fluid flows, offering novel insights into energy dissipation mechanisms. The introduction of a Schrödinger-like operator into the fluid dynamics model captures quantum-scale turbulence effects, enhancing our understanding of energy redistribution across different scales. The results also include the development of anisotropic stochastic models that account for direction-dependent viscosity, improving the representation of real-world turbulent flows. These advances in stochastic modeling and regularity analysis provide a comprehensive toolset for tackling complex fluid dynamics problems. The findings are applicable to fields such as engineering, quantum turbulence, and environmental sciences. Future directions include the numerical implementation of the proposed models and the use of machine learning techniques to optimize parameters for enhanced simulation accuracy.

physics.flu-dyn↗

arXiv2024

Hypercomplex Dynamics and Turbulent Flows in Sobolev and Besov Functional Spaces

This paper presents a rigorous study of advanced functional spaces, with a focus on Sobolev and Besov spaces, to investigate key aspects of fluid dynamics, including the regularity of solutions to the Navier-Stokes equations, hypercomplex bifurcations, and turbulence. We offer a comprehensive analysis of Sobolev embedding theorems in fractional spaces and apply bifurcation theory within quaternionic dynamical systems to better understand the complex behaviors in fluid systems. Additionally, the research delves into energy dissipation mechanisms in turbulent flows through the framework of Besov spaces. Key mathematical tools, such as interpolation theory, Littlewood-Paley decomposition, and energy cascade models, are integrated to develop a robust theoretical approach to these problems. By addressing challenges related to the existence and smoothness of solutions, this work contributes to the ongoing exploration of the open Navier-Stokes problem, providing new insights into the intricate relationship between fluid dynamics and functional spaces.

math.AP↗