Global existence of classical solutions for the gas flows near vacuum through ducts expanding with space and time

In this study, we examine the global existence of classical solutions for the gas flows near vacuum through ducts expanding with space and time. It is described by the initial-boundary value problem of compressible Euler equations together with a sufficiently small variable parameter, which can be viewed as a hyperbolic system of balance laws whose source is a non-dissipative term when Riemann invariants are applied. We prove a couple of global existence theorems of classical solutions under the suitable conditions of expanding ducts and the initial-boundary data whose C⁰ norms can be large. The analysis depends primarily on the local existence theorem and on uniform a priori estimates, which are obtained by giving the maximum principle and introducing new generalized Lax transformations. Furthermore, the limit behavior of expanding ducts at infinity and the long-time behavior of global classical solutions along all characteristic curves and vertical lines are also determined. Lastly, we explore the feasibility of the initial value problem for such expanding ducts.