A Thermodynamically Consistent Master Equation Framework for the Two-Dimensional Incompressible Navier–Stokes Equations: Derivation, Convergence, and Global Regularity

We present a unified and thermodynamically consistent framework for the derivation and analysis of the two-dimensional incompressible Navier–Stokes equations based on a network-type master equation. The proposed formulation originates from a discrete, conservative interaction system endowed with a dissipative structure, and is designed to satisfy fundamental physical principles including conservation laws and, in its extended form, the second law of thermodynamics. Starting from this master equation, we construct a finite-volume discretization that preserves the antisymmetric structure of nonlinear interactions and ensures discrete energy stability. We then rigorously establish the convergence of the discrete system to a continuous limit, showing that the incompressible Navier–Stokes equations arise naturally as a singular limit of the underlying thermodynamic dynamics.Using compactness arguments of Aubin–Lions type, we prove the existence of global weak solutions in two dimensions. Furthermore, by exploiting the vorticity formulation and enstrophy estimates specific to two-dimensional flows, we demonstrate global regularity and uniqueness of solutions. These results are obtained within a single, coherent framework that connects microscopic interaction models, discrete numerical structures, and continuum fluid equations. Although the global well-posedness of the two-dimensional Navier–Stokes equations are classical, the present work provides a novel perspective by deriving these results from a physically grounded master equation, thereby offering a structurally consistent bridge between discrete thermodynamic systems and continuum fluid mechanics. This approach not only clarifies the origin of the Navier–Stokes equations but also establishes a robust foundation for future extensions to more complex systems, including compressible flows and higher-dimensional turbulence.