  Although the compressible fluid limit of the Boltzmann equation with cutoff
has been well investigated in [6] and [13], it still remains largely open to
obtain analogous results in case of the angular non-cutoff or even in the
grazing limit which gives the Landau equation, essentially due to the velocity
diffusion effect of collision operator such that $L^\infty$ estimates are hard
to obtain without using Sobolev embeddings. In the paper, we are concerned with
the compressible Euler and acoustic limits of the Landau equation for Coulomb
potentials in the whole space. Specifically, over any finite time interval
where the full compressible Euler system admits a smooth solution around
constant states, we construct a unique solution in a high-order weighted
Sobolev space for the Landau equation with suitable initial data and also show
the uniform estimates independent of the small Knudsen number $\varepsilon>0$,
yielding the $O(\varepsilon)$ convergence of the Landau solution to the local
Maxwellian whose fluid quantities are the given Euler solution. Moreover, the
acoustic limit for smooth solutions to the Landau equation in an optimal
scaling is also established. For the proof, by using the macro-micro
decomposition around local Maxwellians together with techniques for viscous
compressible fluid and properties of Burnett functions, we design an
$\varepsilon$-dependent energy functional to capture the dissipation in the
compressible fluid limit with feature that only the highest order derivatives
are most singular.
