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Global solutions to the compressible Euler equations with geometrical structure. (English) Zbl 0857.76073
Summary: We prove the existence of global solutions to the Euler equations of compressible isentropic gas dynamics with geometrical structure, including transonic nozzle flow and spherically symmetric flow. Due to the presence of the geometrical source terms, the existence results themselves are new, especially as they pertain to radial flow in an unbounded region, \(|\vec x|\geq 1\), and to transonic nozzle flow. Arbitrary data with \(L^\infty\) bounds are allowed in these results. A shock capturing numerical scheme is introduced to compute such flows and to construct approximate solutions. The convergence and consistency of the approximate solutions generated from this scheme to the global solutions are proved with the aid of a compensated compactness framework.

76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35Q35 PDEs in connection with fluid mechanics
76M20 Finite difference methods applied to problems in fluid mechanics
76H05 Transonic flows
Full Text: DOI
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