GLOBAL EXISTENCE OF SHOCK FRONT SOLUTIONS TO THE AXIALLY SYMMETRIC PISTON PROBLEM FOR COMPRESSIBLE FLUIDS

2004 ◽  
Vol 01 (01) ◽  
pp. 51-84 ◽  
Author(s):  
SHUXING CHEN ◽  
ZEJUN WANG ◽  
YONGQIAN ZHANG
Keyword(s):  
Global Existence ◽  
Shock Front ◽  
Compressible Fluids ◽  
Axially Symmetric ◽  
Piston Problem ◽  
Front Solution ◽  
Glimm Scheme ◽  
Front Solutions ◽  
Shock Front Solution

In this paper, we study the axially symmetric piston problem for compressible fluids when the velocity of the piston is a perturbation of a constant. Under the assumptions that both the velocity of the piston and the density of the gas outside the piston are small, we prove the global existence of a shock front solution by using a modified Glimm scheme.

Asymptotic decay to relaxation shock fronts in two dimensions

2001 ◽  
Vol 131 (6) ◽  
pp. 1385-1410 ◽  
Author(s):  
Hailiang Liu
Keyword(s):  
Shock Front ◽  
Initial Perturbation ◽  
Two Dimensions ◽  
Decay Rates ◽  
Front Solution ◽  
Asymptotic Decay ◽  
Planar Shock ◽  
Spatial Dimensions ◽  
Shock Front Solution ◽  
Shock Fronts

We prove nonlinear stability of planar shock fronts for certain relaxation systems in two spatial dimensions. If the subcharacteristic condition is assumed and the initial perturbation is sufficiently small and the mass carried by the perturbations is not necessarily finite, then the solution converges to a shifted planar shock front solution as time t ↑ ∞. The asymptotic phase shift of shock fronts is, in general, non-zero and governed by a similarity solution to the heat equation. The asymptotic decay rate to the shock front is proved to be t−1/4 in L∞(R2) without imposing extra decay rates in space for the initial perturbations. The proofs are based on an elementary weighted energy analysis to the error equation.

scholarly journals A remark on the Glimm scheme for inhomogeneous hyperbolic systems of balance laws

2015 ◽  
Vol 12 (04) ◽  
pp. 787-797 ◽  
Author(s):  
Cleopatra Christoforou
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Weak Solutions ◽  
State Vector ◽  
Equilibrium Solution ◽  
Hyperbolic Systems ◽  
Balance Laws ◽  
Glimm Scheme ◽  
Systems Of Balance Laws ◽  
The Cauchy Problem

General hyperbolic systems of balance laws with inhomogeneous flux and source are studied. Global existence of entropy weak solutions to the Cauchy problem is established for small BV data under appropriate assumptions on the decay of the flux and the source with respect to space and time. There is neither a hypothesis about equilibrium solution nor about the dependence of the source on the state vector as previous results have assumed.

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