  In the paper, we are concerned with the large time asymptotics toward the
viscous contact waves for solutions of the Landau equation with physically
realistic Coulomb interactions. Precisely, for the corresponding Cauchy problem
in the spatially one-dimensional setting, we construct the unique
global-in-time solution near a local Maxwellian whose fluid quantities are the
viscous contact waves in the sense of hydrodynamics and also prove that the
solution tends toward such local Maxwellian in large time. The result is proved
by elaborate energy estimates and seems the first one on the dynamical
stability of contact waves for the Landau equation. One key point of the proof
is to introduce a new time-velocity weight function that includes an
exponential factor of the form $\exp (q(t)\langle \xi\rangle^2)$ with $$
q(t):=q_1-q_2\int_0^tq_3(s)\,ds, $$ where $q_1$ and $q_2$ are given positive
constants and $q_3(\cdot)$ is defined by the energy dissipation rate of the
solution itself. The time derivative of such weight function is able to induce
an extra quartic dissipation term for treating the large-velocity growth in the
nonlinear estimates due to degeneration of the linearized Landau operator in
the Coulomb case. Note that in our problem the explicit time-decay of solutions
around contact waves is unavailable but no longer needed under the crucial use
of the above weight function, which is different from the situation in [11,
14].
