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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

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scl-xdependent_April_3.pdf (314.9Kb)
Type
Article accepté pour publication ou publié
Date
2014
Journal name
Stochastic Partial Differential Equations: Analysis and Computations
Volume
2
Number
4
Publisher
M. Dekker
Pages
517-538
Publication identifier
10.1007/s40072-014-0038-2
Metadata
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Author(s)
Lions, Pierre-Louis

CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Perthame, Benoît cc
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 / Keywords
rough paths; Stochastic differential equations; stochastic conservation laws; stochastic entropy condition; kinetic formulation; dissipative solutions; rough paths.

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