Uniqueness of a generalized entropy solution to the Cauchy problem for a quasilinear conservation law with convex flux

2010 ◽  
Vol 169 (1) ◽  
pp. 98-112
Author(s):  
E. Yu. Panov
Keyword(s):  
Cauchy Problem ◽  
Conservation Law ◽  
Entropy Solution ◽  
Generalized Entropy ◽  
The Cauchy Problem

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 Theory of Entropy Solutions of Nonlinear Degenerate Parabolic Equations

2020 ◽  
Vol 66 (2) ◽  
pp. 292-313
Author(s):  
E. Yu. Panov
Keyword(s):  
Cauchy Problem ◽  
Entropy Solution ◽  
Quasilinear Equation ◽  
Entropy Solutions ◽  
Degenerate Parabolic Equations ◽  
The Cauchy Problem ◽  
Periodic Initial Data ◽  
Degenerate Parabolic ◽  
Diffusion Function ◽  
Nonlinear Degenerate

We consider a second-order nonlinear degenerate parabolic equation in the case when the flux vector and the nonstrictly increasing diffusion function are merely continuous. In the case of zero diffusion, this equation degenerates into a first order quasilinear equation (conservation law). It is known that in the general case under consideration an entropy solution (in the sense of Kruzhkov-Carrillo) of the Cauchy problem can be non-unique. Therefore, it is important to study special entropy solutions of the Cauchy problem and to find additional conditions on the input data of the problem that are sufficient for uniqueness. In this paper, we obtain some new results in this direction. Namely, the existence of the largest and the smallest entropy solutions of the Cauchy problem is proved. With the help of this result, the uniqueness of the entropy solution with periodic initial data is established. More generally, the comparison principle is proved for entropy suband super-solutions, in the case when at least one of the initial functions is periodic. The obtained results are generalization of the results known for conservation laws to the parabolic case.

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