Analysis of Stability (2+1)-Dimensional Fuzzy Navier-Stokes Equations Based on Lipschitz Condition

Abstract

We apply a rigorous stability analysis to the two-dimensional fuzzy Navier-Stokes equations that are formulated under the α-cut representation. For this study, we used the nonlinear advection-pressure operator to test the Lipschitz condition of the fuzzy velocity field on the initial fuzzy data in order to prove its continuous dependence. The fuzzy version of the Navier-Stokes model introduces uncertainty in fundamental physical parameters including viscosity, density, and external forces. We express the system in abstract operator form, while also arriving at a differential inequality that governs the evolution of perturbation between two fuzzy solutions. By Lipschitz continuity and Grönwall’s inequality, the differential between any two fuzzy velocity solutions is shown to be bounded by an exponential function of time. Hence, it is demonstrated that at every α-level the fuzzy Navier-Stokes system is stable, such that the overall fuzzy solution will inherit the stability property. This provides a theoretical foundation for analyzing uncertain or imprecise fluid dynamics models, which can be expanded to energy stability or numerical verification in further research.