[Paper Review] Scalar conservation laws with rough (stochastic) fluxes

Pierre‐Louis Lions, Benoı̂t Perthame+1 more|arXiv (Cornell University)|Sep 8, 2013

Fluid Dynamics and Turbulent Flows16 references4 citations

TL;DR

This paper establishes a pathwise theory for scalar conservation laws with quasilinear multiplicative rough path dependence, introducing the concept of pathwise stochastic entropy solutions. It proves existence, uniqueness, and continuous dependence via pathwise $L^1$-contraction, extending stochastic viscosity solution techniques to conservation laws using kinetic formulations and flow inversion.

ABSTRACT

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.

Motivation & Objective

• To develop a robust, pathwise theory for scalar conservation laws with quasilinear multiplicative rough path dependence, particularly in the stochastic case.
• To define and rigorously establish the notion of pathwise stochastic entropy solutions, ensuring stability under local uniform convergence of paths.
• To extend the stochastic viscosity solution framework—previously used for Hamilton-Jacobi equations—to the context of conservation laws.
• To overcome the limitations of Itô calculus in expectation-based approaches by focusing on pathwise $L^1$-contraction instead of almost sure estimates.
• To provide a stable and intrinsic uniqueness framework for solutions despite the presence of shocks and discontinuities in the flux.

Proposed method

• Adopt the kinetic formulation of conservation laws to handle nonlinearity and shocks more effectively than classical weak solutions.
• Introduce pathwise stochastic entropy solutions by adapting test functions constructed via local inversion of the stochastic characteristics flow.
• Use the Doss-Sussmann-type transformation to relate the rough path SCL to a deterministic kinetic equation under a change of variables.
• Establish well-posedness through pathwise $L^1$-contraction, ensuring continuous dependence on initial data and paths.
• Leverage techniques from stochastic viscosity theory, particularly the use of special test functions derived from the flow of characteristics.
• Prove stability and intrinsic uniqueness by analyzing the kinetic measure and its behavior under perturbations of the path and initial data.

Experimental results

Research questions

• RQ1 Can a pathwise theory of entropy solutions be developed for scalar conservation laws with rough (stochastic) fluxes, ensuring stability and uniqueness?
• RQ2 How can the stochastic viscosity solution framework be adapted to the context of conservation laws, particularly through the kinetic formulation?
• RQ3 Why is the Itô calculus approach insufficient for establishing pathwise uniqueness and stability in this setting?
• RQ4 What is the role of the Doss-Sussmann transformation in relating rough path SCLs to deterministic kinetic equations?
• RQ5 Under what conditions is the shock structure preserved under the transformation from the rough path equation to the kinetic formulation?

Key findings

• The paper establishes existence and uniqueness of pathwise stochastic entropy solutions for scalar conservation laws with quasilinear multiplicative rough path dependence.
• Pathwise $L^1$-contraction is proven as the key stability property, ensuring continuous dependence on initial data and paths in the local uniform topology.
• The theory is closed under local uniform limits of paths, making it robust and suitable for approximations of Brownian motion or other rough paths.
• The approach avoids the use of Itô calculus for expectation-based estimates, which fail to provide necessary bounds for shock preservation.
• The transformation $u = ilde{ ho}(v,t)$ via the flow of characteristics does not preserve shocks in general unless the forcing is linear, invalidating a naive approach.
• Explicit estimates are derived that control the $L^1$-norm of differences between solutions, confirming the intrinsic stability of the solution concept.