Scalar conservation laws with rough (stochastic) fluxes

Pierre Louis Lions (1, 2) , Benoît Perthame (3, 4) , Panagiotis E. Souganidis (5)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 Collège de France - Chaire Équations aux dérivées partielles et applications
3 LJLL - Laboratoire Jacques-Louis Lions
4 BANG - Nonlinear Analysis for Biology and Geophysical flows
5 Department of Mathematics [Chicago]

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.

Mots clés: Stochastic differential equations; stochastic conservation laws; stochastic entropy condition; kinetic formulation; dissipative solutions; rough paths

Domaines: Equations aux dérivées partielles [math.AP]; Probabilités [math.PR]

Pierre Louis Lions, Benoît Perthame, Panagiotis E. Souganidis. Scalar conservation laws with rough (stochastic) fluxes. Stochastics and Partial Differential Equations: Analysis and Computations, 2013, 1 (4), pp.664-686. ⟨10.1007/s40072-013-0021-3⟩. ⟨hal-00859393⟩