Strong solutions for a 1D viscous bilayer shallow water model

Abstract

In this paper, we consider a viscous bilayer shallow water model in one space dimension that represents two superposed immiscible fluids. For this model, we prove the existence of strong solutions in a periodic domain. The initial heights are required to be bounded above and below away from zero and we get the same bounds for every time. Our analysis is based on the construction of approximate systems which satisfy the BD entropy and on the method developed by A. Mellet and A. Vasseur to obtain the existence of global strong solutions for the one dimensional Navier-Stokes equations.

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

“…As a reminder, it should be noted that the authors in [1] have proven that the regularized approximate system verifies the BD entropy, which gives the lower bound for the water heights. This allows them to have the existence of global strong solutions for the approximate system by using the regularity theorem for smooth data given in [2] and manages to pass to the limit.…”

“…A regularized model of the considered model has been the subject of some recent studies. 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.…”

“…Figure 1: Notation for the bilayer model This work takes its inspiration from the work done in [1]. Note that in [1], the authors showed the existence of global strong solutions of a regularized model of the model studied in this paper by adding regularizing terms at the level of the momentums equations of each layer of the form ε β ν i ∂ x (h β i ∂ x u i )) with β belongs to 0, 1 2 and ε is a small parameter. Also the studied model is associated with the initial energies:…”

“…But we noticed that when ε tends towards 0, they obtain the existence of a strong solution of the stationary model. In this paper we study the evolutionary model of the model studied in [1] when ε tends towards 0.…”

“…Other examples include [8] where the authors have proven the existence of strong solutions for one-dimensional compressible Navier-Stokes equations under the hypothesis that the initial datum is smooth and the initial density is bounded below by a positive constant. In [1] the authors proved according to the ideas developed in [8], the existence of strong solutions of one dimensional regularized bilayer model. The existence of global strong solutions to the Cauchy problem for a shallow water system in dimension N ≥ 2 has been proven in [11].…”