A non-local theory of generalized entropy solutions of the Cauchy problem for a class of hyperbolic systems of conservation laws

1999 ◽  
Vol 63 (1) ◽  
pp. 129-179 ◽  
Author(s):  
E Yu Panov
Keyword(s):  
Cauchy Problem ◽  
Conservation Laws ◽  
Hyperbolic Systems ◽  
Entropy Solutions ◽  
Local Theory ◽  
Generalized Entropy ◽  
The Cauchy Problem ◽  
Systems Of Conservation Laws ◽  
Non Local

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.

Local exact boundary controllability of entropy solutions to linearly degenerate quasilinear hyperbolic systems of conservation laws

2018 ◽  
Vol 24 (2) ◽  
pp. 793-810
Author(s):  
Tatsien Li ◽  
Lei Yu
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Systems ◽  
Front Tracking ◽  
Entropy Solutions ◽  
Exact Boundary Controllability ◽  
Boundary Controllability ◽  
Quasilinear Hyperbolic Systems ◽  
Exact Boundary ◽  
Systems Of Conservation Laws ◽  
Linearly Degenerate

In this paper, we study the local exact boundary controllability of entropy solutions to linearly degenerate quasilinear hyperbolic systems of conservation laws with characteristics of constant multiplicity. We prove the two-sided boundary controllability, the one-sided boundary controllability and the two-sided boundary controllability with fewer controls, by applying the strategy used in [T. Li and L. Yu, J. Math. Pures et Appl. 107 (2017) 1–40; L. Yu, Chinese Ann. Math., Ser. B (To appear)]. Our constructive method is based on the well-posedness of semi-global solutions constructed by the limit of ε-approximate front tracking solutions to the mixed initial-boundary value problem with general nonlinear boundary conditions, and on some further properties of both ε-approximate front tracking solutions and limit solutions.

ON EXISTENCE AND UNIQUENESS OF ENTROPY SOLUTIONS TO THE CAUCHY PROBLEM FOR A CONSERVATION LAW WITH DISCONTINUOUS FLUX

2009 ◽  
Vol 06 (03) ◽  
pp. 525-548 ◽  
Author(s):  
E. YU. PANOV
Keyword(s):  
Cauchy Problem ◽  
Unknown Function ◽  
Conservation Law ◽  
Existence And Uniqueness ◽  
Entropy Solutions ◽  
Generalized Entropy ◽  
The Cauchy Problem ◽  
Discontinuous Flux

We study the Cauchy problem for a conservation law with space discontinuous flux of generalized Audusse–Perthame form. It is shown that, after a change of unknown function, entropy solutions in the sense of Audusse–Perthame correspond to Kruzhkov's generalized entropy solutions for the transformed equation. This observation allows to use the Kruzhkov method of doubling variable (instead of rather complicated variant of this method invented by Audusse and Perthame). Applying this method for measure-valued solutions, we establish the uniqueness and the existence of entropy solutions to the problem under consideration.

scholarly journals On the Cauchy problem for scalar conservation laws in the class of Besicovitch almost periodic functions: Global well-posedness and decay property

2016 ◽  
Vol 13 (03) ◽  
pp. 633-659 ◽  
Author(s):  
Evgeny Yu. Panov
Keyword(s):  
Cauchy Problem ◽  
Conservation Laws ◽  
Almost Periodic Functions ◽  
Entropy Solutions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Scalar Conservation Law ◽  
The Cauchy Problem ◽  
Scalar Conservation ◽  
Necessary And Sufficient

We study the Cauchy problem for a multidimensional scalar conservation law with merely continuous flux vector in the class of Besicovitch almost periodic functions. The existence and uniqueness of entropy solutions are established. We also uncover the necessary and sufficient condition for the decay of almost periodic entropy solutions as the time variable [Formula: see text]. Our results are then interpreted in the framework of conservation laws on the Bohr compact.

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