Global $L^{\infty}$ solutions of the compressible Euler equations with spherical symmetry

Abstract

We study the compressible Euler equations with spherical symmetry surrounding a solid ball. For the spherically symmetric flow, the global existence of L ∞ entropy weak solutions has not yet obtained except a special case. In this paper, we prove the existence of global solutions in the more general case. We construct approximate solutions by using a modified Godunov scheme. The main point is to obtain an L ∞ bound for the approximate solutions.

Global existence of solutions to resonant system of isentropic gas dynamics

“…In this paper, we extend the results in [10] to the Cauchy problems (1.1)-(1.2) by using the vanishing viscosity method for general pressure function P(ρ).…”

“…γ +1 , C > 0 was obtained in [10] for the initial-boundary value problems (1.1)-(1.3) for a polytropic gas with 1 < γ < 5 3 by using a modified Godunov scheme. In this paper, we extend the results in [10] to the Cauchy problems (1.1)-(1.2) by using the vanishing viscosity method for general pressure function P(ρ).…”

Existence of Global Solutions for Unsteady Isentropic Gas Flow in a Laval Nozzle

“…The other inequality can be proved similarly. This lemma is the generalization of [21,Appendix C]. In fact, the lemma is the case…”

“…Finally, Tsuge [21] proved global existence of solutions for the spherically symmetric case (that is, A(x) = x 2 in (1.1)).…”

Resonance for the isothermal system of isentropic gas dynamics

“…Without the condition z 0 (x) ≤ 0 or w 0 (x) ≤ 0, many authors tried to prove the same uniform estimate for the spherically, symmetric solutions in the region of x > 1, where a(x) = x 2 . However, as pointed out in the paper [Ts1], there are some defects in the proofs in these papers. Instead of the uniform estimate, a reasonable estimate depending on the variable x : z(x, t) ≤ Cx…”