Viscous limits for piecewise smooth solutions to systems of conservation laws

1992 ◽  
Vol 121 (3) ◽  
pp. 235-265 ◽  
Author(s):  
Jonathan Goodman ◽  
Zhouping Xin
Keyword(s):  
Conservation Laws ◽  
Piecewise Smooth ◽  
Smooth Solutions ◽  
Systems Of Conservation Laws

scholarly journals The Inviscid Limits to Piecewise Smooth Solutions for a General Parabolic System

2012 ◽  
Vol 2012 ◽  
pp. 1-30
Author(s):  
Shixiang Ma
Keyword(s):  
Conservation Laws ◽  
Parabolic System ◽  
General System ◽  
Viscosity Coefficient ◽  
Limit Problem ◽  
Piecewise Smooth ◽  
Entropy Condition ◽  
Smooth Solutions ◽  
System Of Conservation Laws ◽  
Viscous Limit

We study the viscous limit problem for a general system of conservation laws. We prove that if the solution of the underlying inviscid problem is piecewise smooth with finitely many noninteracting shocks satisfying the entropy condition, then there exist solutions to the corresponding viscous system which converge to the inviscid solutions away from shock discontinuities at a rate of ε1 as the viscosity coefficient ε vanishes.

GRADIENT DRIVEN AND SINGULAR FLUX BLOWUP OF SMOOTH SOLUTIONS TO HYPERBOLIC SYSTEMS OF CONSERVATION LAWS

2004 ◽  
Vol 01 (04) ◽  
pp. 627-641 ◽  
Author(s):  
HELGE KRISTIAN JENSSEN ◽  
ROBIN YOUNG
Keyword(s):  
Initial Data ◽  
Conservation Laws ◽  
Finite Time ◽  
Hyperbolic Systems ◽  
Shock Formation ◽  
Smooth Solutions ◽  
Flux Function ◽  
Systems Of Conservation Laws ◽  
Blowup In Finite Time

We consider two new classes of examples of sup-norm blowup in finite time for strictly hyperbolic systems of conservation laws. The explosive growth in amplitude is caused either by a gradient catastrophe or by a singularity in the flux function. The examples show that solutions of uniformly strictly hyperbolic systems can remain as smooth as the initial data until the time of blowup. Consequently, blowup in amplitude is not necessarily strictly preceded by shock formation.

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