Scalar conservation laws with stochastic forcing

By Arnaud Debussche, Julien Vovelle. 2010

168 citations recorded by the source corpus.

We show that the Cauchy Problem for a randomly forced, periodic multi-dimensional scalar first-order conservation law with additive or multiplicative noise is well-posed: it admits a unique solution, characterized by a kinetic formulation of the problem, which is the limit of the solution of the stochastic parabolic approximation.

Subfields: Applied Mathematics, Finance, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/89e57a20649aa428c7d872b93add0548e6757b9d

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In Applied Mathematics

In Finance

In Statistical and Nonlinear Physics

Citations in the corpus

Degenerate parabolic stochastic partial differential equations: Quasilinear case (2013, methodology): “In the case of hyperbolic conservation laws [7], the authors defined a notion of generalized kinetic solution and obtained a comparison result showing that any generalized kinetic solution is actually a kinetic solution.”
Long‐Time Behavior, Invariant Measures, and Regularizing Effects for Stochastic Scalar Conservation Laws (2014, background): “The higher dimensional case has first studied by Chen and Frid in [11] and was subsequently generalized by Chen and Perthame [12] and Debussche and Vovelle [18].”
The Dirichlet problem for a conservation law with a multiplicative stochastic perturbation (2014, background): “Vovelle [15] proved a result of existence and uniqueness of the entropy solution to the problem posed in a d-dimensional torus and, by the way of the theory of Young measure-valued solutions, C.”
On Nonlinear Stochastic Balance Laws (2011, background): “For more general σ = σ(x, u) depending on u and for multidimensional equations in the Lp framework, the uniqueness of strong stochastic entropy solutions was first established in Feng-Nualart [9], but the existence result was restricted to one dimension; see the recent paper Debussche-Vovelle [6] fo”
Scalar conservation laws with rough (stochastic) fluxes (2013, background): “The main results of [4, 3, 7, 8], which also rely on the kinetic theory, is a well posed theory of solutions which satisfy a contraction principle in space and probability.”
Degenerate parabolic stochastic partial differential equations (2013, methodology): “Indeed, due to the parabolic term, the approach of [9] is not applicable: the comparison principle can be proved only for kinetic solutions (not generalized ones) and therefore strong convergence of approximate solutions is needed in order to prove the existence.”
THE CAUCHY PROBLEM FOR CONSERVATION LAWS WITH A MULTIPLICATIVE STOCHASTIC PERTURBATION (2012, background): “Finally, let us mention the paper by Debussche and Vovelle [7] concerning the d-dimensional problem with multiplicative noise du+ f(u)xdt = h(u)dw”
Stochastic non-isotropic degenerate parabolic–hyperbolic equations (2016, result): “In particular, Debussche, Hofmanova and Vovelle [11] and Bauzet, Vallet andWittbold [5] studied SPDE of the type (1.10) additionally including a second order quasilinear term.”