Scalar conservation laws with rough (stochastic) fluxes
Souganidis, Panagiotis E.; Perthame, Benoît; Lions, Pierre-Louis (2013), Scalar conservation laws with rough (stochastic) fluxes, Stochastic Partial Differential Equations: Analysis and Computations, 1, 4, p. 664-686. http://dx.doi.org/10.1007/s40072-013-0021-3
TypeArticle accepté pour publication ou publié
External document linkhttp://hal.archives-ouvertes.fr/hal-00859393
Journal nameStochastic Partial Differential Equations: Analysis and Computations
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Abstract (EN)We develop a pathwise theory for scalar conservation laws with quasilinear multiplicative rough path dependence, a special case being stochastic conservation laws with quasilinear stochastic dependence. We introduce the notion of pathwise stochastic entropy solutions, which is closed with the local uniform limits of paths, and prove that it is well posed, i.e., we establish existence, uniqueness and continuous dependence, in the form of pathwise $L^1$-contraction, as well as some explicit estimates. Our approach is motivated by the theory of stochastic viscosity solutions, which was introduced and developed by two of the authors, to study fully nonlinear first- and second-order stochastic pde with multiplicative noise. This theory 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.
Subjects / Keywordsrough paths; dissipative solutions; kinetic formulation; stochastic entropy condition; stochastic conservation laws; Stochastic differential equations
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