From Network Master Equations to Navier–Stokes Dynamics: A Relaxation Framework for the Continuum Limit of Interacting Systems

The Navier–Stokes equations provide the fundamental continuum description of viscous fluid motion, yet their derivation from discrete interacting systems remains an important theoretical challenge. In this study we propose a network-based master equation framework for fluid dynamics and demonstrate how Navier–Stokes–type equations emerge from interacting systems through a relaxation mechanism. The system is formulated as a set of nodes exchanging mass, momentum, and energy along network edges. The evolution of node states is governed by a master equation that incorporates both conservative fluxes and entropy-producing dissipative interactions. Under appropriate structural assumptions, the resulting discrete dynamics preserve global conservation laws while satisfying a discrete form of the second law of thermodynamics. By analyzing the continuum limit of the network system, we show that the master equation converges to a conservation-law-type partial differential equation. A relaxation extension is then introduced to represent nonequilibrium stresses through auxiliary variables. The resulting relaxation system possesses an extended entropy structure that yields uniform a priori estimates. Using compactness arguments based on the Aubin–Lions theorem, we establish the strong convergence of velocity fields and prove that a subsequence of solutions converges to a Leray–Hopf weak solution of the incompressible Navier–Stokes equations. In particular, the forcing generated by residual stresses vanishes in the limit due to the dissipative structure of the extended system. The present framework provides a unified perspective linking discrete network dynamics, relaxation systems, and continuum fluid mechanics. It suggests a new pathway for understanding how classical hydrodynamic equations may arise from interacting systems beyond the traditional kinetic-theory setting.