Semi-discretization for Stochastic Scalar Conservation Laws with Multiple Rough Fluxes

Abstract

International audienceWe develop a semi-discretization approximation for scalar conservation laws with multiple rough time dependence in inhomogeneous fluxes. The method is based on Brenier's transport-collapse algorithm and uses characteristics defined in the setting of rough paths. We prove strong L 1-convergence for inhomogeneous fluxes and provide a rate of convergence for homogeneous one's. The approximation scheme as well as the proofs are based on the recently developed theory of path-wise entropy solutions and uses the kinetic formulation which allows to define globally the (rough) characteristics

A priori estimates for rough PDEs with application to rough conservation laws

“…• Mild formulation of rough Burgers equations with spatially irregular noise have been first introduced by Hairer and Weber [41,40] and Hairer, Maas and Weber [39] leading to the groundbreaking work of Hairer on the Kardar-Parisi-Zhang equation [38]. • Solutions of conservation laws with rough fluxes have been studied via flow transformation by Friz and Gess [26] and via the transformed test function approach by Lions, Perthame and Souganidis [54,52,53], Gess and Souganidis [32,31], Gess, Souganidis and Perthame [30] and Hofmanová [44].…”

Nonlinear diffusion equations with nonlinear gradient noise

“…Only in recent years, in a series of works [LPS13, LPS14, GS17a, GS17b, GS17c, GS15] a kinetic approach to (simpler versions of) (1.1) was developed based on rough path methods, cf. also [HKRSs18,GPS15,GG18,GGLS18]. for numerical methods and regularity/qualitative properties of the solutions.…”

Non-equilibrium large deviations and parabolic-hyperbolic PDE with irregular drift

“…Stochastic PDE with conservative noise have been considered in the framework of stochastic scalar conservation laws by Lions-Perthame-Souganidis [49,50,51], Friz-Gess [23], Gess-Souganidis [29,30], later extended to parabolic-hyperbolic stochastic PDE with conservative noise by Gess-Souganidis in [31], Fehrman-Gess [20], Dareiotis-Gess [10]. Approaches to their numerical treatment have been developed by Gess-Perthame-Souganidis [28], Hoel-Karlsen-Risebro-Storrosten [36,37]. Related results on the level of stochastic Hamilton-Jacobi equations, based on entirely different methods, have been developed by Lions-Souganidis [53,54,55,56] (see Souganidis [62] for a recent account on the theory), with extensions by Friz-Gassiat-Lions-Souganidis [22] and detailed study of fine properties in [26,27,57].…”