Hydrodynamic limit of the Boltzmann equation

In a scaling ${\partial }_{t}f+v\cdot {\nabla }_{x}f={\epsilon }^{-1}Q(f,f)$ with small Knudsen number, the leading collision balance formally forces a local Maxwellian. The conserved moments then give the Euler equations with a monatomic ideal-gas closure. The next nonequilibrium correction produces viscous and heat-conduction terms. These substitutions are formal: rigorous limits need uniform estimates, initial and boundary conditions, and sufficient convergence.