Scalar conservation laws with rough (stochastic) fluxes; the spatially dependent case
Lions, Pierre-Louis; Perthame, Benoît; Souganidis, Panagiotis E. (2014), Scalar conservation laws with rough (stochastic) fluxes; the spatially dependent case, Stochastic Partial Differential Equations: Analysis and Computations, 2, 4, p. 517-538. 10.1007/s40072-014-0038-2
TypeArticle accepté pour publication ou publié
Journal nameStochastic Partial Differential Equations: Analysis and Computations
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CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Laboratoire Jacques-Louis Lions [LJLL]
Inria de Paris
Souganidis, Panagiotis E.
Department of Mathematics [Chicago]
Abstract (EN)We continue the development of the theory of pathwise stochastic entropy solutions for scalar conservation laws in \RN with quasilinear multiplicative ''rough path'' dependence by considering inhomogeneous fluxes and a single rough path like, for example, a Brownian motion. Following our previous note where we considered spatially independent fluxes, we introduce the notion of pathwise stochastic entropy solutions and prove that it is well posed, that is we establish existence, uniqueness and continuous dependence in the form of a (pathwise) L1-contraction. 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; Stochastic differential equations; stochastic conservation laws; stochastic entropy condition; kinetic formulation; dissipative solutions; rough paths.
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Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates Lions, Pierre-Louis; Perthame, Benoît; Souganidis, Panagiotis E. (1996) Article accepté pour publication ou publié