Lubrication and shallow-water systems Bernis-Friedman and BD entropies

Didier Bresch,Pascal Noble,Khawla Msheik,Xi Song,Mathieu Colin,Philippe Helluy,Nicolas Seguin,Jean-Marc Hérard

doi:10.1051/proc/202069001

Abstract

This paper concerns the results recently announced by the authors, in C.R. Acad. Sciences Maths volume 357, Issue 1, 1-6 (2019), which make the link between the BD entropy introduced by D. Bresch and B. Desjardins for the viscous shallow-water equations and the Bernis-Friedman (called BF in our paper) dissipative entropy introduced to study the lubrication equations. More precisely different dissipative BF entropies are obtained from the BD entropies playing with drag terms and capillarity formula for viscous shallow water type equations. This is the main idea in the paper which makes the link between two communities. The limit processes employ the standard compactness arguments taking care of the control in the drag terms. It allows in one dimension for instance to prove global existence of nonnegative weak solutions for lubrication equations starting from the global existence of nonnegative weak solutions for appropriate viscous shallow-water equations (for which we refer to appropriate references). It also allows to prove global existence of nonnegative weak solutions for fourth-order equation including the Derrida-Lebowitz-Speer-Spohn equation starting from compressible Navier-Stokes type equations.

Highlights

This paper concerns the results recently announced by the authors, in C.R
Order nonlinear degenerate parabolic equations with F (h) = hn and D(h) = 0 with n > 1 and suggested a new entropy inequality- referred to by BF entropy- which provides additional estimates serving for increasing the regularity of the weak solution obtained
The key tool in our method is to prove an existence result for system (1) by passing the limit of ε to zero in both the weak formulation of (3) as well as the BD-entropy

Summary

Mathematical justification

The first essential step is the a priori estimate which is maintained by the uniform bounds that both the energy and BD-entropy offer. (1) using uniform boundedness of ∂xhε in H1((0, T ) × Ω), we get the strong convergence and weak convergence of hε to h in L2((0, T ) × Ω). (5) As the for fifth and sixth terms, we use the results obtained in (21), (31), the strong convergence of hε to h in C(0, T, L2(Ω)) and L2(0, T, H1(Ω)), and the weak convergence of ∂x2hε in L2(0, T, L2(Ω)), we get t hε∂x2hε∂xφ − h∂x2h∂xφ dx dt =. As for the sixth term, we have deduced that ∂xhε converges strongly to ∂xh in L2(0, T ; L2(Ω), we get the strong and weak convergence of ( ∂xhε) to ( ∂xh) in L1((0, T ) × Ω), t (∂xhε)2 ∂xφ dx dt −→.

BF-entropy information include in the limit part of the BD-entropy

The limit of a viscous compressible system with a general drag term

Ansatz between n and m on the physical basis

Convergence of the BD-entropy

Limit problem