  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 [42] & Zhang et al. [38]. 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 [26]. 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. [40], Cheng and Yang [27] & Ding
and Wang [50] 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 [9].
