Communications in Mathematical Sciences

Volume 13 (2015)

Number 6

Scalar conservation laws with multiple rough fluxes

Pages: 1569 – 1597



Benjamin Gess (Department of Mathematics, University of Chicago, Illinois, U.S.A.)

Panagiotis E. Souganidis (Department of Mathematics, University of Chicago, Illinois, U.S.A.)


We study pathwise entropy solutions for scalar conservation laws with inhomogeneous fluxes and quasilinear multiplicative rough path dependence. This extends the previous work of Lions, Perthame, and Souganidis who considered spatially independent and inhomogeneous fluxes with multiple paths and a single driving singular path respectively. The approach is motivated by the theory of stochastic viscosity solutions which relies on special test functions constructed by inverting locally the flow of the stochastic characteristics. For conservation laws, this is best implemented at the level of the kinetic formulation which we follow here.


stochastic scalar conservation laws, rough paths, random dynamical systems, kinetic solutions

2010 Mathematics Subject Classification

35L65, 35R60, 60H15

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Published 13 May 2015