2010/01/11 by Philippe Guyenne, David Lannes, Jean‐Claude Saut +1
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Computational Fluid Dynamics and Aerodynamics #Navier-Stokes equation solutions
paper · doi:10.1088/0951-7715/23/2/003
openalex publication_date 2010/01/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
We consider in this paper the 'shallow-water/shallow-water' asymptotic model obtained in Choi and Camassa (1999 J. Fluid Mech. 396 1–36), Craig et al (2005 Commun. Pure. Appl. Math. 58 1587–641) (one-dimensional interface) and Bona et al (2008 J. Math. Pures Appl. 89 538–66) (two-dimensional interface) from the two-layer system with rigid lid, for the description of large amplitude internal waves at the interface of two layers of immiscible fluids of different densities. For one-dimensional interfaces, this system is of hyperbolic type and its local well-posedness does not raise serious difficulties, although other issues (blow-up, loss of hyperbolicity, etc) turn out to be delicate. For two-dimensional interfaces, the system is nonlocal. Nevertheless, we prove that it conserves some properties of 'hyperbolic type' and show that the associated Cauchy problem is locally well posed in suitable Sobolev classes provided some natural restrictions are imposed on the data. These results are illustrated by numerical simulations with emphasis on the formation of shock waves.