The Universal Depletion Functional in Dissipative Systems

This paper introduces and formalizes the Depletion Functional $\mathcal{D}$, a novel dimensionless geometric invariant that quantifies the critical balance between nonlinear flux and linear dissipation in a general class of quadratic dissipative systems, including the three-dimensional Navier-Stokes equations (NSE), Magnetohydrodynamics (MHD), and the Porous Media Equation (PME). The functional is defined in an abstract functional setting as the ratio of dual norms: $$\mathcal{D}(u)=\frac{||B(u,u)||_{V^{\prime}}}{||Lu||_{V^{\prime}}}$$ where $B(u,u)$ represents the nonlinear (inertial) term and $Lu$ is the dissipation operator. We rigorously establish that the condition $\mathbf{\mathcal{D} < 1}$ provides a universal criterion for global regularity, effectively unifying and generalizing classical conditional regularity results such as the Serrin, Beale-Kato-Majda, and Caffarelli-Kohn-Nirenberg criteria. Analytically, a uniform bound $\mathcal{D} \le c_0 < 1$ guarantees that dissipation universally controls the nonlinear energy cascade, preventing the formation of singularities. A key contribution is the demonstration of the multifaceted nature of $\mathcal{D}$: Geometric Structure: $\mathcal{D}$ is shown to be the tangent of the angle between the nonlinear flux vector and the dissipative direction within the Riemannian geometry induced by the operator $L$. The critical threshold $\mathcal{D}=1$ thus defines the boundary of the dissipative cone. Thermodynamic Principle: The regularity condition $\mathcal{D} < 1$ is proven to be mathematically equivalent to positive entropy production ($\sigma > 0$) for the system. This establishes a Thermodynamic Regularity Principle, which fundamentally connects the global well-posedness of fluid dynamics equations to the inviolability of the Second Law of Thermodynamics. This universal framework is applied to derive new structural regularity theorems (Type II regularity) for large-amplitude, highly aligned flows and is supported by a comprehensive numerical validation strategy, suggesting a profound link between turbulence dynamics and fundamental principles of geometry and physics.