Shock formation in the compressible Euler equations and related systems. (English) Zbl 1452.35137
Summary: We prove shock formation results for the compressible Euler equations and related systems of conservation laws in one space dimension, or three dimensions with spherical symmetry. We establish an \(L^{\infty}\) bound for \(C^{1}\) solutions of the one-dimensional (1D) Euler equations, and use this to improve recent shock formation results of the authors. We prove analogous shock formation results for 1D magnetohydrodynamics (MHD) with orthogonal magnetic field, and for compressible flow in a variable area duct, which has as a special case spherically symmetric three-dimensional (3D) flow on the exterior of a ball.
MSC:
| 35Q31 | Euler equations |
| 35L65 | Hyperbolic conservation laws |
| 35L67 | Shocks and singularities for hyperbolic equations |
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
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