Quantitative Closure Analysis toward Ideal Fluids

We establish the incompressible low–Mach/high–Reynolds limit for the Boltzmann equation for a broad class of initial data, without recourse to any asymptotic expansion. Exploiting the local Maxwellian manifold and the macro–micro decomposition in a new quasi-linear analysis, we derive quantitative estimates for the purely microscopic fluctuation, as well as bounds for the kinetic vorticity and the entropic fluctuation in terms of the initial data. As a consequence, in two space dimensions, the rescaled velocity and temperature converge to a global solution of the incompressible Euler equations coupled to a transported temperature, within the frameworks of DiPerna–Lions–Majda and Delort.