<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>
We consider the initial-boundary value problem of compressible Navier–Stokes–Vlasov equations under a local alignment regime in a one-dimensional bounded domain. Based on the relative entropy method and compactness argument, we prove that a weak solution of the initial-boundary value problem converges to a strong solution of the limiting two-phase fluid system. This work extends in some sense the previous work of Choi and Jung [Math. Models Methods Appl. Sci. 31(11), 2213–2295 (2021)], which considered the diffusive term ∂ ξξ f ɛ in the kinetic equation. Note that the diffusion term was not considered in this paper.
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