CHESNOKOV, A.A. ...et al., 2017. Stability of shear shallow water flows with free surface. SIAM Journal on Applied Mathematics, 77(3), pp.1068-1087.
Stability of inviscid shear shallow water flows with free surface is studied in the
framework of the Benney equations. This is done by investigating the generalized hyperbolicity of the integrodifferential Benney system of equations. It is shown that all shear flows having monotonic convex velocity profiles are stable. The hydrodynamic approximations of the model corresponding to the classes of flows with piecewise linear continuous and discontinuous velocity profiles are derived and studied. It is shown that these approximations possess Hamiltonian structure and a complete system
of Riemann invariants, which are found in an explicit form. Sufficient conditions for hyperbolicity of the governing equations for such multilayer flows are formulated. The generalization of the above results to the case of stratified fluid is less obvious, however, it is established that vorticity has a stabilizing effect.
This paper was accepted for publication in the journal SIAM Journal on Applied Mathematics and the definitive published version is available at https://doi.org/10.1137/16M1098164