Exponential decay for 2D reduced gravity two-and-a-half layer model with quantum potential and drag force

Abstract

<p style='text-indent:20px;'>In this paper, we study the global weak solutions to a reduced gravity two-and-a-half layer model with quantum potential and drag force in two-dimensional torus. Inspired by Bresch, Gisclon, Lacroix-Violet [Arch. Ration. Mech. Anal. (233):975-1025, 2019] and Bresch, Gisclon, Lacroix-Violet, Vasseur [J. Math. Fluid Mech., 24(11):16, 2022], we prove that the weak solutions decay exponentially in time to equilibrium state. In order to prove the decay property of weak solutions, we obtain the relative entropy inequality of weak solutions and equilibrium solutions by defining the relative entropy functional. Considering that the structure of reduced gravity two-and-a-half layer model is more complicated than the compressible Navier-Stokes equations due to the presence of cross terms <inline-formula><tex-math id="M1">\begin{document}$ h_{1}\nabla h_{2} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ h_{2}\nabla h_{1} $\end{document}</tex-math></inline-formula>, we need to estimate the cross term in relative entropy.</p>