Entropy and the stability of classical solutions of hyperbolic systems of conservation laws

1996 ◽  
pp. 48-69 ◽  
Author(s):  
C. M. Dafermos
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Systems ◽  
Classical Solutions ◽  
Systems Of Conservation Laws ◽  
The Stability

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.

scholarly journals On the uniqueness of solutions to hyperbolic systems of conservation laws

2021 ◽  
Vol 291 ◽  
pp. 110-153
Author(s):  
Shyam Sundar Ghoshal ◽  
Animesh Jana ◽  
Konstantinos Koumatos
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Systems ◽  
Uniqueness Of Solutions ◽  
Systems Of Conservation Laws

High-order relaxation approaches for adjoint-based optimal control problems governed by nonlinear hyperbolic systems of conservation laws

2016 ◽  
Vol 24 (1) ◽  
Author(s):  
Elimboto M. Yohana ◽  
Mapundi K. Banda
Keyword(s):  
Optimal Control ◽  
Conservation Laws ◽  
Optimal Control Problems ◽  
Initial Conditions ◽  
Hyperbolic Systems ◽  
Control Problems ◽  
Computational Investigation ◽  
Backward Problem ◽  
Systems Of Conservation Laws ◽  
The Cost

AbstractA computational investigation of optimal control problems which are constrained by hyperbolic systems of conservation laws is presented. The general framework is to employ the adjoint-based optimization to minimize the cost functional of matching-type between the optimal and the target solution. Extension of the numerical schemes to second-order accuracy for systems for the forward and backward problem are applied. In addition a comparative study of two relaxation approaches as solvers for hyperbolic systems is undertaken. In particular optimal control of the 1-D Riemann problem of Euler equations of gas dynamics is studied. The initial values are used as control parameters. The numerical flow obtained by optimal initial conditions matches accurately with observations.

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