Turbulent Behaviour of Compressible Flows
Final Report Abstract
Turbulent flows appear in a wide range of applications, such as oceanography, weather prediction, astrophysics, medicine, or aerospace engineering. Despite its obvious relevance, turbulence is still not very well understood on the foundational level. This is true even more for compressible flows, for which fundamental mathematical questions remain open. It is widely believed that the behaviour of turbulent flows should eventually be traced back to the fundamental laws of classical mechanics, such as Newton’s laws. In a continuum setting, this leads to nonlinear systems of partial differential equations such as the Euler and Navier-Stokes equations, both of which are available in incompressible and compressible versions. In this project, we focused on the analysis of the isentropic compressible Euler equations, whose mathematical theory is still open to a large extent. Basic open questions about this system include the following: • Does there exist a unique solution, in a sense to be specified, for any reasonable initial data for all times? • Do solutions conserve or dissipate energy, and are there sufficient criteria for energy conservation? • How can the genuinely compressible phenomenon of vacuum formation be handled mathematically? This project has yielded substantial progress in some aspects of these questions. It consisted of three parts: Energy conservation with possible vacuum; weak semicontinuity for convexly constrained variational problems related to the Euler equations and other physical models; and the construction of solutions via the method of convex integration. The following results have emerged from the project: • Weak solutions conserve energy under certain regularity assumptions, even when vacuum occurs; this is the first result on energy conservation with vacuum under realistic conditions. • Functionals whose arguments take values in a convex set are weakly lower semicontinuous if and only if they satisfy a natural quasiconvexity condition; the question was motivated from the case of divergence-free positive tensors, of which the Euler equations are an instance. • The best possible convex integration scheme for the compressible Euler equations is essentially the known one, which relies heavily on the incompressible framework; this was surprising, and maybe a bit disappointing, but helped clarify substantially the potential role of convex integration in compressible fluid dynamics. A lot remains to be done, of course, but we have made several contributions to clarify important analytical aspects of compressible flow. The results and methods developed in this project have already given rise to further investigations, e.g., on weak-strong uniqueness for Euler with vacuum, or on the relationship between weak and measure-valued solutions.
Publications
- Lower semi-sontinuity for A -quasiconvex functionals under convex restrictions, 2021
W. D. Skipper, E. Wiedemann
(See online at https://doi.org/10.48550/arXiv.1909.11543) - Energy conservation for the compressible Euler and Navier-Stokes equations with vacuum. Anal. PDE 13(3):789–811, 2020
Akramov, T. Debiec, J. W. D. Skipper, E. Wiedemann
(See online at https://doi.org/10.2140/apde.2020.13.789)
