# Stochastic conservation laws

## About the project

Random effects are of paramount importance in mathematical models, and Stochastic Partial Differential Equations (SPDEs) provide an essential tool in modeling, analysis, and prediction of numerous physical, biological, and economic systems. In some nonlinear situations the presence of noise leads to complex phenomena not seen in deterministic systems. Nonlinear SPDEs possess solutions exhibiting singularities, oscillations, and/or concentration effects, which in the real world are reflected in the appearance of shock waves, turbulence, material defects, etc. This immediately poses very fundamental questions such as what is the nature and effect of noise on singularities and can the solution be continued past singularity formation? Questions like these are intimately tied to several core issues, such as understanding what we actually mean by a solution; the development of theories about existence, uniqueness and stability of solutions; and construction of numerical algorithms. Over the last few decades these questions have been given satisfactory answers for several classes of deterministic nonlinear partial differential equations. However, the situation is dramatically different for stochastic equations. This project targets a wide range of fundamental questions concerning mathematical analysis and numerical algorithms for random/stochastic conservation laws and related nonlinear SPDEs.

An overall goal of the project is to develop concepts and techniques for the analysis of random phenomena modeled by nonlinear SPDEs with solutions possessing low regularity (shock waves). This will be carried out through a combination of mathematical and numerical analysis. It will involve several branches of mathematics, including partial differential equations; functional analysis; stochastic analysis, using a range of techniques such as Ito calculus, martingale theory, Malliavin calculus, entropy and viscosity analysis, a priori estimates, (compensated) compactness analysis, weak convergence methods, and homogenization.

Some relevant research topics include:

- Establish well-posedness theory for conservation laws and mixed hyperbolic-parabolic equations with multiplicative noise based on non-Gaussian (Levy) stochastic processes.
- Convergence and error analysis of numerical methods for stochastic conservation laws with Gaussian and non-Gaussian forcing.
- Development and analysis of numerical methods for stochastic conservation laws with rough stochastic flux in the sense of Lions-Perthame-Souganidis.
- Establish a well-posedness theory for degenerate convection-diffusion equations with random diffusion matrices.
- Convergence analysis of "multi-level Monte-Carlo"-based numerical methods for random convection-diffusion equations.
- Existence of weak solutions to systems of conservation laws with random flux and small data (via front tracking).

## Financing

Research council of Norway, Independent projects - project number 250674. Total budget approx 12,6 mill NOK.