TY  - JOUR
ID  - ncmm-2008-016
T1  - On scalar hyperbolic laws with discontinuous flux
A1  - Bulíček, Miroslav
A1  - Gwiazda, Piotr
A1  - Málek, Josef
A1  - Świerczewska-Gwiazda, Agnieszka
JA  - Mathematical Models and Methods in Applied Sciences
Y1  - 2011
VL  - 21
IS  - 1
SP  - 89
EP  - 113
N1  - NCMM Preprint no. 2008-016
UR  - http://www.karlin.mff.cuni.cz/ncmm/preprints/08323115158preprint.pdf
M2  - doi: 10.1142/S021820251100499X
KW  - discontinuous flux
KW  - Entropy solution
KW  - Hyperbolic scalar conservation laws
N2  - We study the Cauchy problem for scalar hyperbolic conservation laws with a flux that can have jump discontinuities. We introduce new concepts of entropy weak and measure-valued solution that are consistent with the standard ones if the flux is continuous. Having various definitions of solutions to the problem, we then answer the question what kind of properties the flux should possess in order to establish the existence and/or uniqueness of solution of a particular type. In any space dimension we establish the existence of measure-valued entropy solution for a flux having countable jump discontinuities. Under the additional assumption on the Hölder continuity of the flux at zero, we prove the uniqueness of entropy measure-valued solution, and as a consequence, we establish the existence and uniqueness of weak entropy solution. If we restrict ourselves to one spatial dimension, we prove the existence of weak solution to the problem where the flux has merely monotone jumps; in such a setting we do not require any continuity of the flux at zero.
ER  -