Global wellposedness and limit behavior for the generalized finite-depth-fluid equation with small data in critical Besov spaces \(\dot B^s_{2,1}\)
In this very interesting and technical paper, the well known equation FDF (finite-depth-fluid) introduced by \textit{R. I. Joseph} [J. Physics A 10, L225--L227 (1977; Zbl 0369.76018)] is studied. For a suitable choice of some parameters and for small enough initial data, the globally wellpossedness is proved. Two convergence phenomena are also proved: for very large depth, the solution of FDF is converging to the solution of the generalized Korteweg-de Vries euqation and for small depth, the FDF solution is near the solution of Benjamin-Ono equation. The Kato smoothing effect for a dispersive semi-groups is used to obtain some estimates of the solution, useful for the convergence procedure. The dual estimate method and the oscillatory integral techniques are also used.
scientific article; zbMATH DE number 5354958