A Relaxation Formulation Stronger than the Navier–Stokes Equations and the Existence of Global Strong Solutions for Small Initial Data

This paper studies a relaxation extension of the incompressible Navier–Stokes equations in which the viscous stress tensor is treated as an independent dynamical variable with relaxation and diffusion. The resulting system forms a thermodynamically consistent extension of the Navier–Stokes equations and possesses a triple dissipation structure consisting of viscous dissipation, stress diffusion, and stress relaxation. We first establish a basic energy inequality for the extended system, showing that the relaxation structure introduces an additional dissipation mechanism that is absent in the classical Navier–Stokes equations. Next, higher-order a priori estimates are derived in Sobolev spaces with , using commutator estimates for the nonlinear convection term. Combining these estimates with a local well-posedness result obtained via a Friedrichs approximation scheme, we prove the existence and uniqueness of global strong solutions for sufficiently small initial data. Finally, we discuss the formal relaxation limit in which the stress tensor converges to the Newtonian constitutive law, recovering the incompressible Navier–Stokes equations. The results show that the relaxation formulation provides a mathematically well-posed extension of the Navier–Stokes dynamics and offers a framework for studying the stabilizing role of stress relaxation mechanisms.