Project Details
Projekt
Computational Convex Integration
Applicants
Dr. Peter Korn; Professor Dr. László Székelyhidi
Subject Area
Mathematics
Term
since 2023
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 500072446
Convex integration (CI) was developed in the past decade as a constructive technique for the incompressible Euler equations in connection with Onsager’s conjecture. Because of the high flexibility of the method, it has been extended to prove non-uniqueness statements for weak solutions of several other inviscid (inhomogeneous/compressible Euler, active scalar transport, magneto-hydrodynamics, Boussinesq equations) and viscous models (Navier-Stokes, advection-diffusion equations). In this project we focus on the Euler equations, because they constitute the simplest model with for which a comprehensive theoretical background concerning CI exists. Our goal is to transform the conceptual CI algorithm into a proven computational scheme, enabling us for the first time to directly compute weak solutions of the Euler equations with small-scale structure that are compatible with Kolmogorov’s K41 theory, whose existence was conjectured by L.Onsager (1949). Such a scheme would be an important breakthrough in the science of hydrodynamic turbulence from both theoretical and computational perspectives. Specifically we shall: • extend recent work on K41 and intermittent weak solutions by designing a new construction scheme based on a space-time wavelet-type decomposition, thereby putting all available con- structions in the literature on the same footing, • develop a numerical implementation of the construction of weak solutions with the ability to go beyond temporal scales currently available in HPC turbulence simulations, • compare numerical convergence rates to available theoretical bounds, • study for the first time Lagrangian properties of K41 and intermittent weak solutions, and • explore the numerical construction of weak solutions with multifractal properties, with an eye towards bounds on structure function exponents. The basic method of constructing weak solutions by CI follows Nash's original approach for rough isometric immersions. The latter has been implemented numerically before for the purpose of visualizing the Nash embedding within the geometrical setting. The algorithm is described in the pertinent publication and the source code is available from http://hevea-project.fr. Although CI for fluid mechanics has been derived from Nash’s approach, there are a number of substantial differences, such as having multi-dimensional building blocks (as opposed to 1D spirals), having a PDE in divergence form (requiring the use of nonlocal operators), the use of super-exponential growth of frequencies, and the presence of (non)linear advective terms. Thus, a 1-1 correspondence of the existing numerical implementation is not possible, but at least it serves as a proof of concept. Moreover, our aims go beyond mere solution visualization by involving a comprehensive study of various Euler/Lagrange properties of small-scale isotropic turbulence, as as well as going beyond isotropic turbulence with an eye towards applications in Project 2 of the Research Unit.
DFG Programme
Research Units
Co-Investigator
Professor Dr.-Ing. Rupert Klein
