Balance laws with space-dependent nonlocalities: modeling, simulation and uncertainty quantification (NonLoc)

In recent years, conservation laws with nonlocal flux have gained growing attention for a wide field of applications appearing in fluid mechanics problems such as gas dynamics, granular flow, sedimentation, aggregation phenomena, crowd motions, and traffic flows.

In general, space-dependent nonlocal balance laws are intended to cover macroscopically the interplay of nonlocal interactions occurring at the microscopic level and are typically characterized by nonlinear flux functions and source terms depending on space-integrals of the unknows.

The analysis and numerical approximations of nonlocal equations present several challenging open problems, ranging from well-posedness results to the design of suitable high-resolution numerical schemes and uncertainty quantification. Based on the encouraging results obtained so far, we aim at further advance the analytical and numerical study of these nonlocal equations, while also carefully investigating the model hierarchies between nonlocal and local hyperbolic balance laws.

Except for well-posedness results, the analysis and numerical approximation of nonlocal balance laws remain under-explored at present, in particular in multi-dimensions and for systems. Aiming at filling the gaps between the theory and the applications, this project addresses in particular the formal derivation of nonlocal equations, concepts to show existence and uniqueness of solutions, transitions from nonlocal to local hyperbolic balance laws, the development of efficient, high-order finite volume numerical schemes and the investigation of stochastic influences in the initial data or flux function.

Summarizing, the main impact of the project is to push forward major mathematical advances in the theory and numerical approximation of highly non-standard problems to put the basis for innovative tools to handle modern applications in fluid mechanics.