On the Mathematical Analysis of Generalized Quantum-Nabla Fractional Fluid Models with Dissipative Nonlinearities

Abstract

We investigate a nonlinear fluid system governed by the generalized quantum-Caputo nabla fractional operator, capturing nonlocal memory effects in velocity, shear stress, and fluidity. The system is formulated with polynomial nonlinearities and modeled over the unit disk. We establish a general existence and uniqueness theorem for mild solutions in the function spaces H¹(D)³, H²(D)³, and ℓ^∞(D)³, based on fixed-point theory and the integral representation of the fractional operators. Under mild dissipativity assumptions, we prove boundedness and asymptotic stability using generalized (q, τ )-Mittag-Leffler decay. Furthermore, we present illustrative examples for each functional space and validate the theoretical results with numerical simulations. The findings provide a rigorous and flexible framework for modeling fractional fluid dynamics with memory-driven dissipation.