Large Time Behavior of the Strong Solution to the Two‐Fluid Incompressible Navier–Stokes–Fourier Equations

This paper develops a comprehensive framework for establishing sharp decay rates of strong solutions to the two‐fluid incompressible Navier–Stokes–Fourier system. Based on the contraction mapping argument, we prove the existence of local solutions for general initial data in and the existence of global solutions for small initial data, with the two proofs being independent of each other; the global solution is not obtained as an extension of the local one. Furthermore, we derive optimal algebraic decay rates for the solution and its first‐order spatial derivatives in spaces . We also verify faster decay rates in the intersection space . These results provide a complete characterization of the large time asymptotics for strong solutions to this coupled system, thereby extending Kato's classical theory to multi‐fluid flows.