Project Details
Projekt
Turbulent Behaviour of Compressible Flows
Applicant
Professor Dr. Emil Wiedemann
Subject Area
Mathematics
Term
from 2017 to 2020
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 355710483
Fluid mechanics has been a field of interest for mathematicians, physicists, and engineers alike throughout centuries. Theoretical as well as applied scientists deem the field important but also challenging. The difficulties are mostly due to effects of turbulence: Turbulent flows behave in a highly irregular way and cannot be reliably computed from the initial conditions. They exhibit microstructures with eddies appearing on smaller and smaller length scales, until kinetic energy is eventually dissipated on the smallest scale.In mathematical terms, the flow of liquids and gases is described via nonlinear partial differential equations like the Euler or Navier-Stokes equations and numerous related systems. Our lack of understanding of the phenomenon of turbulence is reflected in the many difficult open problems with regard to these equations. In particular, there is still no satisfactory existence and uniqueness theory for, say, the Euler equations.In recent years there has been significant progress in constructing solutions to the incompressible Euler equations, which model incompressible fluids like, approximately, water, and which display "turbulent" behaviour. The problem of finding turbulent solutions of compressible fluids (gases like, for instance, air), however, has so far not been solved to a similar extent. The proposed project will therefore extend the method of "convex integration", which has led to ground-breaking insights in the incompressible context, to the compressible Euler equations and related models. In view of the qualitatively different behaviour of compressible and incompressible flows, and of significant technical differences between the two systems, this is a challenging but rewarding goal. I expect this to have important applications to the existence of weak solutions, and to yield insights into the relationship between different notions of solution.In a second part of the project, it will be investigated under what circumstances turbulent behaviour like non-uniqueness and energy dissipation does not occur. Such questions are intimately connected to the regularity of the solutions. Two central principles will be studied and generalised in this context: weak-strong uniqueness and energy conservation at sufficient regularity.The requested project will contribute to our understanding, which so far is still very unsatisfactory, of turbulent phenomena in compressible flows, relying on the refinement, generalisation, and development of sophisticated mathematical techniques.
DFG Programme
Research Grants
International Connection
Czech Republic, Poland
