  This paper is dedicated to the study of a one-dimensional congestion model,
consisting of two different phases. In the congested phase, the pressure is
free and the dynamics is incompressible, whereas in the non-congested phase,
the fluid obeys a pressureless compressible dynamics. We investigate the Cauchy
problem for initial data which are small perturbations in the non-congested
zone of travelling wave profiles. We prove two different results. First, we
show that for arbitrarily large perturbations, the Cauchy problem is locally
well-posed in weighted Sobolev spaces. The solution we obtain takes the form
(vs, us)(t, x -- x(t)), where x < x(t) is the congested zone and x > x(t) is
the non-congested zone. The set {x = x(t)} is a free surface, whose evolution
is coupled with the one of the solution. Second, we prove that if the initial
perturbation is sufficiently small, then the solution is global. This stability
result relies on coercivity properties of the linearized operator around a
travelling wave, and on the introduction of a new unknown which satisfies
better estimates than the original one. In this case, we also prove that
travelling waves are asymptotically stable.
