A State-Dependent Bulk Viscosity Model for Compressible Navier–Stokes Flow: Formal Analysis and Numerical Investigation

A fundamental challenge in 3D compressible Navier-Stokes equations is the prevention of finite-time singularities and the preservation of global regularity. This paper extends previous topological mapping methods—which successfully bounded solutions in the incompressible regime—to the more volatile compressible case through the integration of a state-dependent bulk viscosity, termed the Hala-Viscosity (γ). Utilizing a Successive Controlled Collapse (SCC) framework, we demonstrate that γ acts as an intrinsic kinetic governor that responds non-linearly to the divergence of the velocity field. Mathematical formalization and Particle-in-Cell (PIC) computational validation show that this operator induces a spectral gap, suppressing high-frequency “mayhem” and forcing the system onto a global attractor with a Hausdorff dimension dH < 2. The resulting Delta-Function Convergence proves that the manifold is topologically quenched into a singular, non-degenerate terminal state, or Stop-Bit, ensuring global smoothness without the need for external heuristic stabilization.