  We use kinetic-theory methods to analyze Landau Fermi-liquid theory, and in
particular to investigate the number and nature of soft modes in Fermi liquids,
both in the hydrodynamic and the collisionless regimes. In the hydrodynamic
regime we show that Fermi-liquid theory is consistent with Navier-Stokes
hydrodynamics at all temperatures. The soft modes are the ones familiar from
classical hydrodynamics that are controlled by the five conservation laws;
namely, two first-sound modes, two shear diffusion modes, and one heat
diffusion mode. These modes have a particle-like spectrum and are soft, or
scale invariant, at all temperatures. In the collisionless regime we show that
the entire single-particle distribution function is soft with a continuous part
of the spectrum. This continuous soft mode, which is well known but often not
emphasized, has important physical consequences, e.g., for certain quantum
phase transitions. In addition, there are the well known soft zero-sound
excitations that describe angular fluctuations of the Fermi surface; their
spectra are particle-like. They are unrelated to conservation laws, acquire a
mass at any nonzero temperature, and their number depends on the strength of
the quasiparticle interaction. We also discuss the fates of these two families
of soft modes as the temperature changes. With increasing temperature the size
of the collisionless regime shrinks, the damping of the modes grows, and
eventually all of the collisionless modes become overdamped. In their stead the
five hydrodynamic modes appear in the hydrodynamic regime at asymptotically low
frequencies. The two families of soft modes are unrelated and have very
different physical origins.In charged Fermi liquids the first-sound modes in
the hydrodynamic regime and the l=0 zero-sound modes in the collisionless
regime get replaced by plasmons, all other modes remain soft.
