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This paper is dedicated to the study of a one-dimensional congestion model,consisting of two different phases. In the congested phase, the pressure isfree and the dynamics is incompressible, whereas in the non-congested phase,the fluid obeys a pressureless compressible dynamics. We investigate the Cauchyproblem for initial data which are small perturbations in the non-congestedzone of travelling wave profiles. We prove two different results. First, weshow that for arbitrarily large perturbations, the Cauchy problem is locallywell-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) isthe non-congested zone. The set {x = x(t)} is a free surface, whose evolutionis coupled with the one of the solution. Second, we prove that if the initialperturbation is sufficiently small, then the solution is global. This stabilityresult relies on coercivity properties of the linearized operator around atravelling wave, and on the introduction of a new unknown which satisfiesbetter estimates than the original one. In this case, we also prove thattravelling waves are asymptotically stable.

Local and global well-posedness of one-dimensional free-congested equations