Existence of Global Entropy Solution for Eulerian Droplet Models and Two-phase Flow Model with Non-constant Air Velocity

Abstract

This article addresses the question concerning the existence of global entropy solution for generalized Eulerian droplet models with air velocity depending on both space and time variables. When \(f(u)=u,\) \(\kappa (t)=const.\) and \(u_a(x,t)=const.\) in (1.1), the study of the Riemann problem has been carried out by Keita and Bourgault (J Math Anal Appl 472(1):1001–1027, 2019) and Zhang et al. (Appl Anal 102(2):576–589, 2023). We show the global existence of the entropy solution to (1.1) for any strictly increasing function \(f(\cdot )\) and \(u_a(x,t)\) depending only on time with mild regularity assumptions on the initial data via shadow wave tracking approach. This represents a significant improvement over the findings of Yang (J Differ Equ 159(2):447–484, 1999). Next, by using the generalized variational principle, we prove the existence of an explicit entropy solution to (1.1) with \(f(u)=u,\) for all time \(t>0\) and initial mass \(v_0>0,\) where \(u_a(x,t)\) depends on both space and time variables, and also has an algebraic decay in the time variable. This improves the results of many authors such as Ha et al. (J Differ Equ 257(5):1333–1371, 2014), Cheng and Yang (Appl Math Lett 135(6):8, 2023) and Ding and Wang (Quart Appl Math 62(3):509–528, 2004) in various ways. Furthermore, by employing the shadow wave tracking procedure, we discuss the existence of global entropy solution to the generalized two-phase flow model with time-dependent air velocity that extends the recent results of Shen and Sun (J Differ Equ 314:1–55, 2022).