Derivation and well-posedness of Boussinesq/Boussinesq systems for internal waves

Cung The Anh¹

¹ Department of Mathematics Hanoi University of Education 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam

Annales Polonici Mathematici, Tome 96 (2009) no. 2, pp. 127-161

Résumé

We consider the propagation of internal waves at the interface between two layers of immiscrible fluids of different densities, under the rigid lid assumption, with the presence of surface tension and with uneven bottoms. We are interested in the case where the flow has a Boussinesq structure in both the upper and lower fluid domains. Following the global strategy introduced recently by Bona, Lannes and Saut [J. Math. Pures Appl. 89 (2008)], we derive an asymptotic model in this regime, namely the Boussinesq/Boussinesq systems. Then using a contraction-mapping argument and energy methods, we prove that the derived systems which are linearly well-posed are in fact locally nonlinearly well-posed in suitable Sobolev classes. We recover and extend some known results on asymptotic models and well-posedness, for both surface waves and internal waves.

Détail

DOI : 10.4064/ap96-2-3

Keywords: consider propagation internal waves interface between layers immiscrible fluids different densities under rigid lid assumption presence surface tension uneven bottoms interested where flow has boussinesq structure upper lower fluid domains following global strategy introduced recently bona lannes saut math pures appl derive asymptotic model regime namely boussinesq boussinesq systems using contraction mapping argument energy methods prove derived systems which linearly well posed locally nonlinearly well posed suitable sobolev classes recover extend known results asymptotic models well posedness surface waves internal waves