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This article addresses the question concerning the existence of globalentropy solution for generalized Eulerian droplet models with air velocitydepending 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 beencarried out by Keita and Bourgault [42] & Zhang et al. [38]. We show the globalexistence 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 regularityassumptions on the initial data via shadow wave tracking approach. Thisrepresents a significant improvement over the findings of Yang [26]. Next, byusing the generalized variational principle, we prove the existence of anexplicit entropy solution to (1.1) with $f(u)=u,$ for all time $t>0$ andinitial mass $v_0>0,$ where $u_a(x,t)$ depends on both space and timevariables, and also has an algebraic decay in the time variable. This improvesthe results of many authors such as Ha et al. [40], Cheng and Yang [27] & Dingand Wang [50] in various ways. Furthermore, by employing the shadow wavetracking procedure, we discuss the existence of global entropy solution to thegeneralized two-phase flow model with time-dependent air velocity that extendsthe recent results of Shen and Sun [9].

Existence of global entropy solution for Eulerian droplet models and two-phase flow model with non-constant air velocity