Temperature-Regularized Navier-Stokes (TRNS): A Proven Stochastic Regularization of 3D Navier-Stokes via Temperature-Scaled Noise and Minimal Transport Correction

We introduce the Temperature-Regularized Navier–Stokes equation on the three-dimensional torus. The system combines (i) the exact Landau–Lifshitz thermal noise of intensity √(2ν k_B T) required by the fluctuation–dissipation theorem at temperature T with (ii) a minimal fixed-intensity transport-noise term of strength ε > 0 (independent of viscosity ν). The TRNS equation is• physically faithful for any real fluid of finite viscosity and molecular size,• mathematically proven to admit a unique global strong solution with probability 1 for arbitrary large divergence-free initial data (Flandoli–Luo 2021, Flandoli–Galeati–Luo 2021–2024 and subsequent works), and• smoothly connected to the pure thermal-noise Navier–Stokes system in the continuum limit ε → 0. Immediate applications include turbulence-free fusion plasmas, rare-earth-free electric motors, exact high-Reynolds-number simulations, and singularity resolution in gravitational and cosmological settings. The pure thermal-noise version (ε = 0) is proposed as the physically natural candidate for deterministic regularity of the classical Navier–Stokes equations.