Two-component equations modelling water waves with constant vorticity
The authors first derive a two-component system of nonlinear equations \[ m_t =\alpha u_x -au_x m -u m_x -\kappa \rho\rho_x, \quad \rho_t = -u\rho_x -(a-1)u_x\rho\text{ with }m =u-u_{xx}, \] which models two-dimensional shallow water waves with constant vorticity. Then, the authors use \textit{D. G. Ebin} and \textit{J. Marsden}'s approach [Ann. Math. (2) 92, 102--163 (1970; Zbl 0211.57401)] to prove the (local) wellposedness of this system using a geometrical framework which allows us to recast this system as a geodesic flow on an infinite-dimensional manifold. Finally, a criterion is provided which guarantees the global existence of a solution.
scientific article; zbMATH DE number 6548058
scientific article