Waves and Nonlinear Phenomena

In this project we have studied several classes of nonlinear partial differential equation, including local and nonlocal, as well as deterministic and stochastic equations. These equations model several different phenomena in nature. We have showed that mathematical models for water waves, that are naturally dominated by gravity, display symmetry, localization when they are small, but exceptional crests when they are large. This is done in a number of models, and through different methods, not the least construction of such solutions to nonlinear and nonlocal equations that are determined by exact so-called dispersions relations. Furthermore, we have introduced a novel class of models for dense traffic on a multilane highway. In addition we have studied a class of nonlinear transport-diffusion models which may have very irregular solutions. Here both stochastic and deterministic equations have been studied. We have also studied the evolution of singularities for a class of equations.

Vi har bidratt til å utdanne 2 forskere med PhD, som alle vil bidra til en kompetanseheving i det norske samfunnet. Videre har vi engasjert 4 postdoktorer som både har bidratt til en kompetanseheving i Norge og økt internasjonaliseringen. Prosjektet har muliggjort at vi har kunnet delta på konferanser og kunne følge med på hva som skjer på forskningsfronten internasjonalt, og kommunisere våre egne resultater til våre kolleger. Vitenskapelig har vi økt det globale forskersamfunnets forståelse av flere fundamentale matematiske sammenhenger. Spesielt vil vi nevne gravitasjonsdrevne vannbølger, matematiske modeller for trafikkflyt, spesielt for overganger mellom modeller som beskriver individuelle kjøretøy og kollektive modeller som beskriver tett trafikk. Videre har vi utviklet en ny modell for trafikkflyt på en motorvei med flere felt. I tillegg har studert matematiske modeller for “mean field games”. Vi har også studert oppførselen til løsningen av partielle differensialligninger nær singulariteter. For ytterligere og mer presise beskrivelser av forskningsresultatene refererer til resultatrapporten.

WaNP is a 5-year FRIPRO Toppforsk project in mathematics at the Norwegian University of Science and Technology (NTNU). Its focus is basic research in nonlinear and nonlocal partial differential equations. The applicants are H. Holden (principal investigator), M. Ehrnström, U. Skre Fjordholm, K. Grunert, E. R. Jakobsen, and P. Lindqvist. Funding is sought for 4 PhD students, 2 postdoctoral fellows, and operating expenses. Collaboration with world-leading international experts are included in the project. Partial differential equations constitute the main tool to describe the physical nature. The main goal of WaNP is to analyze the interplay of singularities and nonlocal effects in a class of carefully selected wave equations. As singulairties appear, our tools to understand their behavior break down; and as nonlocal effects come into play, the study of essentially local singularities requires a global understanding of the problem. As part of our program we shall furthermore study both deterministic equations and equations with uncertainties. The key question that we ask, is How do nonlocalities affect the appearance of singularities in wave equations?