On the existence of global strong solutions to 1D bilayer shallow water model

Brahima, ROAMBA; Julien, ZONGO; Mohamed Bassirou, BAMOGO; Yacouba, ZONGO; Jean De Dieu, ZABSONRE

doi:10.21608/ejmaa.2023.193654.1006

On the existence of global strong solutions to 1D bilayer shallow water model

Document Type : Regular research papers

Authors

¹ Institut Universitaire de Technologie, Université Nazi Boni, Bobo-Dioulasso, Burkina Faso.

² UFR/SEA, Université Nazi Boni, Bobo-Dioulasso, Burhina Faso.

³ UFR/SEA, Université Nazi Boni, Bobo-Dioulasso, Burkina Faso.

⁴ EPO, Université Ouaga 2S, Ouagadougou, Burkina Faso

⁵ UFR/SEA, Université Nazi boni, Bobo-Dioulasso, Burkina Faso

10.21608/ejmaa.2023.193654.1006

Abstract

Our study focuses on 1D viscous bilayers shallow water model. The model considered is represented by two superposed immiscible fluids with different physical properties. Each layer is governed by the shallow water equations in one dimension. A regularized model of the considered model has been the subject of some recent studies. Endeed, to do this, the authors considered an approximate model ( by introducing a parameter epsilon ) for which they prove the existence of global strong solutions. But note that when repsilon tends towards 0, they obtain the existence of strong solution of the stationary model. Our contribution is to extend the results of the work carried out in [Nonlinear Analysis, vol (14)2, 1216-1124, (2013)] by proving the existence of global strong solutions of the considered model. The key point of this proof is based on an estimate of a new entropy called '' mathematical BD entropy" which gives additional regularities. Also, we follow the approach used ba the autheurs in [Math Nachr. 291 (14-15), 2183-2203, (2018)] to be able the limit water heigh.

Keywords