In this paper, we study degenerate parabolic system, which is strongly coupled. We prove general existence result, but the uniqueness remains an open question. Our proof of existence is based on a crucial entropy estimate which both control the gradient of the solution and the non-negativity of the solution. Our system are of porous medium type and our method applies to models in seawater intrusion.

In this paper, we study the standard one-dimensional (non-overdamped) Frenkel-Kontorova (FK) model describing the motion of atoms in a lattice. For this model we show that for any supersonic velocity c > 1, there exist bounded traveling waves moving with velocity c. The profile of these traveling waves is a phase transition between limit states k − in −∞ and k + in +∞. Those limit states are some integers which reflect the assumed 1-periodicity of the periodic potential inside the FK model. For every c > 1, we show that we can always find k − and k + such that k + − k − is an odd integer. Furthermore for c ≥ 25 24 , we show that we can take k + − k − = 1. These traveling waves are limits of minimizers of a certain energy functional defined on a bounded interval, when the length of the interval goes to infinity. Our method of proof uses a concentration compactness type argument which is based on a cleaning lemma for minimizers of this functional.

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