Scalar conservation laws with rough (stochastic) fluxes
By Pierre-Louis Lions, Benoit Perthame, Panagiotis E. Souganidis. Stochastic Partial Differential Equations Analysis and Computations, 2013
82 citations recorded by the source corpus.
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 L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1$$\end{document}-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.
Subfields: Applied Mathematics, Computational Mechanics, Finance
View Paper: https://www.semanticscholar.org/paper/c8192e5ffeb7457caa62211751ec44d4f9d146e4
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In Applied Mathematics
- Martina Hofmanov'a
- Benjamin Gess
- Panagiotis E. Souganidis
- Massimiliano Gubinelli
- Samy Tindel
- A. Deya
- Julien Vovelle
- Arnaud Debussche
In Computational Mechanics
- Benjamin Gess
- Panagiotis E. Souganidis
- Martina Hofmanov'a
- Arnaud Debussche
- Massimiliano Gubinelli
- Samy Tindel
- Julien Vovelle
- Franco Flandoli
In Finance
- Martina Hofmanov'a
- Benjamin Gess
- Panagiotis E. Souganidis
- Samy Tindel
- A. Deya
- Massimiliano Gubinelli
- Julien Vovelle
- Arnaud Debussche
Citations in the corpus
- A priori estimates for rough PDEs with application to rough conservation laws (2016, methodology): “First, Lions, Perthame and Souganidis (see [48, 50]) developed a pathwise approach for du+ div(A(x, u)) ◦ dW = 0 where W is a continuous real-valued signal and ◦ stands for the Stratonovich product in the Brownian case, then Friz and Gess (see [23]) studied du+ div f(t, x, u)dt = F (t, x, u)dt+ Λk(x”
- Degenerate parabolic stochastic partial differential equations: Quasilinear case (2013, background): “…for our model problem (1.1), we observe that already in the case of deterministic hyperbolic conservation law it is possible to find simple examples supporting the two following claims (see, e.g., [23]): (i) classical C1 solutions do not exist, (ii) weak (distributional) solutions lack uniqueness.”
- Long‐Time Behavior, Invariant Measures, and Regularizing Effects for Stochastic Scalar Conservation Laws (2014, methodology): “In contrast, [40] and its subsequent extensions by Lions, Perthame and Souganidis [42] and the authors [30] consider SSCL like du+ N ∑”