Existence and Uniqueness of Weak Solutions to the Fractional Navier-Stokes Equations in Turbulent Flows: A Study of Boundedness and Stability
Raghad Sahib Shamsah
Dr. Raghad Sahib Shamsah, Department of Petroleum, Faculty of Engineering, University of Kerbala, P.O.Box 1125, Freaha, Karbala, Iraq.
Manuscript received on 21 July 2025 | First Revised Manuscript received on 05 August 2025 | Second Revised Manuscript received on 18 September 2025 | Manuscript Accepted on 15 October 2025 | Manuscript published on 30 October 2025 | PP: 8-17 | Volume-5 Issue-2, October 2025 | Retrieval Number: 100.1/ijam.B121105021025 | DOI: 10.54105/ijam.B1211.05021025
Open Access | Editorial and Publishing Policies | Cite | Zenodo | OJS | Indexing and Abstracting
© The Authors. Published by Lattice Science Publication (LSP). This is an open-access article under the CC-BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)
Abstract: This study employs fractional analysis to investigate the existence and uniqueness of weak solutions to the fractional NavierStokes equations in three-dimensional turbulent flows, thereby addressing the complexity inherent in turbulent regimes. Turbulence presents significant challenges in fluid dynamics, characterised by chaotic and erratic motion that confounds conventional modelling techniques. By reformulating the NavierStokes equations using fractional derivatives, we capture non-local effects and memory phenomena, thereby enhancing the mathematical representation of fluid behaviour. First, we transform the Navier-Stokes equations to demonstrate the utility of fractional analysis. The second thing we do is show that there are weak answers in some situations. This means that we can use our models with starting data that isn’t stable. This is the third thing we do. We show that these answers are unique. This proves that our models are right. Ultimately, we demonstrate that weak solutions remain bounded over time under specific conditions regarding the initial data and external forces. When the Reynolds number approaches a critical level, we investigate its stability. This helps us understand how smooth flow can become rough flow. The findings of this research not only advance the theoretical understanding of weak solutions to the fractional Navier-Stokes equations but also have practical implications for modeling complex fluid systems. By linking fractional derivatives with turbulent flows, this work contributes to the broader field of fluid dynamics, paving the way for future investigations in applied mathematics and engineering. Ultimately, this exploration enhances our understanding of turbulence and its mathematical foundations, emphasising the importance of fractional calculus in accurately modelling fluid dynamics.
Keywords: Navier-Stokes Equations, Weak Solutions, Turbulent Flows, Existence and Uniqueness, Boundedness, Stability Analysis.
Scope of the Article: Applied Mathematics