Project leader: Professor Alexander Dunlap
Project manager: David Koralov
Team members: Owen Banks, Zimo Li, Jack Qian, Yifei Wang
Stochastic PDEs describe relationships between partial derivatives of functions and random noise. In this project, we study a stochastic version of Burgers equation over a closed line segment, with constant-in-time Dirichlet boundary conditions and bulk random forcing given by a Poisson point process. This equation is a relatively simple case useful for understanding a more general family of stochastic PDEs, called the KPZ equations, which are highly relevant to theoretical physics and mathematical modeling. Using PDE theory, we can reduce the problem of solving the stochastic Burgers equation to the problem of finding the path that connects two points that minimizes a certain “action” function. The action function rewards paths for passing through Poisson forcing points and for staying on boundaries, but discourages paths from traveling through space. Our project studies this optimization-style problem. In particular, we characterize how the solution to the equation evolves over time, and prove several theorems about how it changes as the boundary parameters vary.
More precisely, we are interested in studying the global entropy solution to the inviscid Burgers equation with Poisson forcing over the space-time domain $[0,1]\times \mathbb{R}$, with boundary conditions constant over time. Through a variational formulation, constructing the entropy solution of the equation reduces to finding paths that minimize a corresponding action functional. We approach this minimization problem by studying the shocks of the entropy solution, which in our setting correspond to points at which multiple distinct minimizers meet. We characterize how these shocks form and evolve through time, and how they interact with the boundaries. Combined with a perturbation argument, this allows us to describe the solution to the minimizer problem in a clean way. We also study how the solution of the PDE changes as we change the boundary condition by defining the notion of a “global shock”: a persistent shock arising from the competition of paths exiting from the left and right boundaries. We show that as we change a boundary parameter, this global shock changes in a controlled manner. Moreover, we show that only the shocks on one side of the global shock will change, as the ones on the other side remain invariant. Finally, we investigate a scheme where we derive the solution of the problem by interpolating between the two cases in which each of the boundary parameters is respectively set to 0. Using analysis, we have made some progress in showing that the solution will look like a concatenation of the two boundary cases.