The project’s goal is to explore and develop mathematical and numerical methods for analyzing stochastic transport equations. These equations are essential for understanding how random fluctuations (noise) affect dynamic systems such as turbulence, water waves, geophysical flows, and compressible fluid motion. Transport equations are a type of partial differential equations used to model the movement and dispersion of quantities like mass, energy, or chemical substances in a medium. By including stochastic elements in the models, we can take into account random influences that often arise in natural and industrial processes. The project focuses on gradient noise, where velocity fields are directly affected by noise. This allows for energy-conserving solutions, which is crucial in fluid mechanics. We use Stratonovich- and Marcus-type fluctuations to model such noise, in contrast to conventional Itô noise that continuously injects energy. So far, the project has achieved several important results: the well-posedness of a two-component Novikov system describing nonlinear wave propagation, new results on a basic hyperbolic-elliptic black-oil model, and the well-posedness of hyperbolic PDEs with discontinuous flux. These contributions provide new theoretical insight and a stronger basis for numerical simulations of complex systems.
Our project is at the cutting edge of numerical analysis of SPDEs affected by transport noise. Such equations have emerged as a focal point in mathematical research, due to their critical applications in areas like fluid dynamics, turbulence modeling, and geophysical flows. This heightened interest is driven by the need for deeper understanding and precise analysis of these complex phenomena, where SPDEs offer vital insights and predictive capabilities. What sets our research apart is its emphasis on transport-dominated systems driven by gradient noise operators. Prior numerical analysis studies have explored SPDEs with stochastic forcing, but the examination of gradient noise is still in its infancy. The complexity of gradient noise, involving anisotropic (integro-) differential operators due to stochastic calculus, presents unique challenges. There is a pressing need to develop methods that preserve the hyperbolic structure of these equations. Our project aims to pioneer this field by devising theoretically sound numerical algorithms and creating novel frameworks for stability and convergence. Our work considers supercritical transport equations, nonlinear wave equations, and the compressible Navier-Stokes equations. Research on stochastic transport equations is inspired by the famous Kraichnan turbulence model, which posits advection through a Gaussian velocity field. Yet, there is a growing consensus that noise in turbulent velocities might be better represented by a Lévy process with jumps. As the second main advancement of our project, we are breaking new ground by pioneering the analysis of SPDEs affected by Marcus-type gradient noise. This area is critically important yet hardly explored, with vast potential for groundbreaking discoveries. Marcus noise is distinct for adhering to the Newton-Leibniz chain rule, which is important in fluid dynamics, unlike the Stratonovich interpretation which fails to do so in the presence of jumps.
