On the analytic regularity of weak solutions of analytic systems of conservation laws with analytic data

1987 ◽  
pp. 154-168
Author(s):  
Paul Godin
Keyword(s):  
Conservation Laws ◽  
Weak Solutions ◽  
Regularity Of Weak Solutions ◽  
Systems Of Conservation Laws ◽  
Analytic Regularity

scholarly journals Non-convex entropies for conservation laws with involutions

2013 ◽  
Vol 371 (2005) ◽  
pp. 20120344 ◽  
Author(s):  
Constantine M. Dafermos
Keyword(s):  
Cauchy Problem ◽  
Conservation Laws ◽  
State Space ◽  
Weak Solutions ◽  
Hyperbolic Systems ◽  
Entropy Inequality ◽  
Classical Solutions ◽  
The Cauchy Problem ◽  
Systems Of Conservation Laws ◽  
Well Posed

The paper discusses systems of conservation laws endowed with involutions and contingent entropies. Under the assumption that the contingent entropy function is convex merely in the direction of a cone in state space, associated with the involution, it is shown that the Cauchy problem is locally well posed in the class of classical solutions, and that classical solutions are unique and stable even within the broader class of weak solutions that satisfy an entropy inequality. This is on a par with the classical theory of solutions to hyperbolic systems of conservation laws endowed with a convex entropy. The equations of elastodynamics provide the prototypical example for the above setting.

ADMISSIBILITY OF SELF-SIMILAR WEAK SOLUTIONS OF SYSTEMS OF CONSERVATION LAWS IN TWO SPACE VARIABLES AND TIME

2009 ◽  
Vol 06 (03) ◽  
pp. 433-481 ◽  
Author(s):  
MICHAEL SEVER
Keyword(s):  
Conservation Laws ◽  
Weak Solutions ◽  
Physical Principle ◽  
Incident Shock ◽  
Wave System ◽  
Local Entropy ◽  
Systems Of Conservation Laws ◽  
Two Space Dimensions ◽  
Self Similar ◽  
Modest Reduction

By means of an example, we postulate that the familiar local entropy conditions on discontinuities are far from sufficient to distinguish admissible weak solutions of systems of conservation laws in two space dimensions and time. We consider the familiar problem of the reflection of an incident shock by a wedge for the "nonlinear wave system", finding a plethora of self-similar weak solutions (or possibly approximate weak solutions) satisfying the usual entropy condition for this system. The multiplicity of such solutions arises from unneeded freedom in the algorithm for constructing solutions, and is directly related to a modest reduction in the assumed regularity of the solution in comparison to that assumed in previous work on this problem. We conclude that if the limit of vanishing viscosity indeed exists for such problems, an additional physical principle is needed to characterize admissible weak solutions.

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