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.

NONLINEAR ASYMPTOTICS FOR HYPERBOLIC INTERNAL WAVES OF SMALL WIDTH

2006 ◽  
Vol 03 (02) ◽  
pp. 269-295 ◽  
Author(s):  
OLIVIER GUES ◽  
JEFFREY RAUCH
Keyword(s):  
Internal Waves ◽  
Transition Layer ◽  
Layer Structure ◽  
Piecewise Smooth ◽  
Transmission Problem ◽  
Internal Layer ◽  
Smooth Transition ◽  
Hyperbolic Problems ◽  
Smooth Solutions ◽  
Small Width

Semilinear hyperbolic problems with source terms piecewise smooth and discontinuous across characteristic surfaces yield similarly piecewise smooth solutions. If the discontinuous source is replaced with a smooth transition layer, the discontinuity of the solution is replaced by a smooth internal layer. In this paper we describe how the layer structure of the solution can be computed from the layer structure of the source. The key idea is to use a transmission problem strategy for the problem with the smooth internal layer. That leads to an anastz different from the obvious candidates. The obvious candidates lead to overdetermined equations for correctors. With the transmission problem strategy we compute infinitely accurate expansions.

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