Global Classical Solutions Near Vacuum to the Initial-Boundary Value Problem of Isentropic Supersonic Flows through Divergent Ducts

TL;DR

This paper establishes the global existence and asymptotic behavior of classical solutions near vacuum for isentropic supersonic flows through divergent ducts, using hyperbolic Euler equations with a new analytical approach.

Contribution

It provides a novel proof for the global existence of classical solutions to non-dissipative hyperbolic balance laws without small data assumptions.

Findings

Global classical solutions exist near vacuum for the problem.

Asymptotic behavior of solutions is characterized along characteristics.

Numerical simulations support the theoretical results.

Abstract

In this paper, we study the global existence and asymptotic behavior of classical solutions near vacuum for the initial-boundary value problem modeling isentropic supersonic flows through divergent ducts. The governing equations are the compressible Euler equations with a small parameter, which can be written as a hyperbolic system in terms of the Riemann invariants with a non-dissipative source. We provide a new result for the global existence of classical solutions to initial-boundary value problems of non-dissipative hyperbolic balance laws without the assumption of small data. The work is based on the local existence, the maximum principle and the uniform a priori estimates obtained by the generalized Lax transformations. The asymptotic behavior of classical solutions is also shown by studying the behavior of Riemann invariants along each characteristic curve and vertical line. The results can be applied to the spherically symmetric solutions to N-dimensional compressible Euler equations. Numerical simulations are provided to support our theoretical results.