Higher Regularity and Exponential Energy Decay in the Adaptive Smagorinsky Model for Turbulent Flows

This work presents a rigorous mathematical analysis of the adaptive Smagorinsky model for incompressible turbulent flows, focusing on two principal theoretical contributions. First, under the assumptions that the initial velocity belongs to the natural energy space and the external forcing is square-integrable in time, we prove that weak solutions exhibit enhanced spatial regularity. Specifically, solutions possess bounded first-order time derivatives and second-order spatial derivatives, a result stemming from the monotonicity and coercivity of the nonlinear, spatially adaptive dissipation term. This analysis accommodates discontinuous and gradient-dependent coefficients, as well as complex geometries and non-homogeneous boundary conditions, extending classical well-posedness theory. Second, in the absence of external forcing, we demonstrate that the kinetic energy of any initial perturbation decays exponentially over time. The decay rate depends explicitly on the minimal effective dissipation in the domain, incorporating both molecular viscosity and the adaptive Smagorinsky contribution. Collectively, these findings advance the theoretical understanding of the model by providing precise bounds on both solution smoothness and long-time stabilization. Beyond existence, uniqueness, and standard dissipation estimates, our results establish a rigorous foundation for stability analysis, perturbation decay, and extensions to dynamic and multiscale subgrid models. The adaptive mechanism preserves near-wall flow structures while ensuring strong damping where needed, offering both mathematical robustness and practical relevance for large eddy simulations. This work thus bridges analytical rigor with physical modeling, providing a comprehensive characterization of the adaptive Smagorinsky system and a framework for further theoretical and numerical investigations in turbulent flow modeling.