Wave Phenomena and Stability - a Shocking Combination

In this project we studied mathematical equations that govern the motion of waves. The equations are of the type non-linear partial differential equations. A very fascinating phenomenon in this context, which, e.g., can be observed close to a shore, is wave breaking. As we know from our daily experience, a lot of energy concentrates in a single point for a moment, but some of this energy is going to disappear immediately afterwards and hence affects the future shape of the wave. A central question for such equations is stability: What are the future consequences for the wave profile of a small perturbation of the initial data? In this project we considered stability questions for some special equations, which model wave phenomena. We studied the evolution of singularities and constructed metrics and numerical methods, which take into account the influence of the wave phenomena on the solutions. Furthermore, the limit of particle models has been studied. When the number of particles in such models tends to infinity, the resulting equation describes, e.g., wave phenomena. In addition, we had a look at the coupling of differential models, which occur, e.g., in epidemiology or traffic modelling.

Vi har bidratt til kompetanseheving i Norge ved å utdanne 1 forsker med PhD og ved å engasjere 1 postdoktor, som i tillegg har bidratt i økt internasjonalisering. Prosjektet har muliggjort at vi har kunnet delta på konferanser og kommunisere våre egne resultater til våre kolleger. I tillegg kunne vi følge med på hva som skjer på forskningsfronten internasjonalt og intensivere eksisterende og påbegynne nye samarbeider nasjonalt og internasjonalt. Vitenskapelig har vi økt det globale forskersamfunnets forståelse av matematiske ligninger som beskriver spredning av bølger. Vi har også studert partikkel modeller og deres sammenheng med partielle differensialligninger. Videre har vi sett på koblete modeller som opptrer, f. eks., i epidemiologi eller matematiske modeller for trafikkflyt.

WaPheS is a 4-year Unge Forskertalenter 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 principal investigator is Katrin Grunert. Funding is sought for 1 PhD student, 1 postdoctoral fellow, and operating expenses. Collaborations with world-leading experts are included in the project. Partial differential equations turn up in the description of various phenomena that can be observed in everyday life. The main goal of WaPheS is to increase the understanding of shock formation and wave breaking as well as to trace their impact on stability results in the case of nonlinear and nonlocal partial differential equations. Both the formation of shocks and the breaking of waves are characterised by singularities turning up and the loss of the uniqueness of solutions. Thus classical methods to describe solutions globally break down and have to be replaced by tailor-made solution concepts. The key question that we ask is How do nonlinear terms affect global solution concepts and their stability?