Irregularity and Noise In Continuity Equations (INICE)
About the project
Partial differential equations (PDEs) appear in the modeling of a large number of natural phenomena, such as the flow of air in the atmosphere; water waves propagating over the oceans; electrical signals in a circuit; and even the collective behavior of large groups of animals such as birds, fish or humans. Given data about the current state of the system, the solution of the PDE informs us of the future behavior of the system. For a given PDE and some class of input data, the mathematician's task is to determine whether the solution exists, whether it is unique, what its qualitative properties are, and whether it can be computed, either exactly or approximately.
It is known that many important PDEs suffer from nonuniqueness – the PDE can predict several different outcomes for the same input data. A selection principle is then required to single out the physically correct solution from the multitude of solution. Such selection principles have been very successful for many classes of PDEs, but for others (perhaps most importantly, for the Euler equations for modeling gas flow), no known selection principle is able to systematically single out only one "correct" solution.
The goal of the project Irregularity and Noise In Continuity Equations (INICE) is to further the understanding of selection principles for PDEs, more specifically socalled continuity equations. We draw inspiration and techniques from the field of stochastic differential equations, equations which incorporate noise, and which have been much more successful in the question of uniqueness than traditional PDEs have.
Financing
Research council of Norway, Independent projects  project number 301538. Total budget approx 9.5 mill NOK.
Publications

Kirkeby, Adrian (2022). Exact and approximate solutions to the Helmholtz, Schrödinger and wave equation in R3 with radial data. Wave motion. ISSN 01652125. 108. doi: 10.1016/j.wavemoti.2021.102841.