Global existence of the three‐dimensional compressible Euler equations for generalized Chaplygin gas with damping

2020 ◽  
Vol 44 (2) ◽  
pp. 1176-1184
Author(s):  
Ka Luen Cheung
Keyword(s):  
Global Existence ◽  
Euler Equations ◽  
Three Dimensional ◽  
Chaplygin Gas ◽  
Compressible Euler Equations ◽  
Generalized Chaplygin Gas ◽  
Compressible Euler

scholarly journals SHOCK FORMATION IN THE COMPRESSIBLE EULER EQUATIONS AND RELATED SYSTEMS

2013 ◽  
Vol 10 (01) ◽  
pp. 149-172 ◽  
Author(s):  
GENG CHEN ◽  
ROBIN YOUNG ◽  
QINGTIAN ZHANG
Keyword(s):  
Euler Equations ◽  
Space Dimension ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Compressible Euler Equations ◽  
Shock Formation ◽  
Compressible Euler ◽  
Systems Of Conservation Laws ◽  
The One ◽  
Special Case

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∞ bound for C1 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.

scholarly journals Formation and construction of a shock wave for 3-D compressible Euler equations with the spherical initial data

2004 ◽  
Vol 175 ◽  
pp. 125-164 ◽  
Author(s):  
Huicheng Yin
Keyword(s):  
Shock Wave ◽  
Initial Data ◽  
Euler Equations ◽  
Blow Up ◽  
Three Dimensional ◽  
Compressible Euler Equations ◽  
Lipschitz Continuous ◽  
Compressible Euler ◽  
Nondegeneracy Conditions ◽  
Weak Entropy Solution

AbstractIn this paper, the problem on formation and construction of a shock wave for three dimensional compressible Euler equations with the small perturbed spherical initial data is studied. If the given smooth initial data satisfy certain nondegeneracy conditions, then from the results in [22], we know that there exists a unique blowup point at the blowup time such that the first order derivatives of a smooth solution blow up, while the solution itself is still continuous at the blowup point. From the blowup point, we construct a weak entropy solution which is not uniformly Lipschitz continuous on two sides of a shock curve. Moreover the strength of the constructed shock is zero at the blowup point and then gradually increases. Additionally, some detailed and precise estimates on the solution are obtained in a neighbourhood of the blowup point.

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