Maximal turbulence as a selection criterion for measure-valued solutions
Christian Klingenberg, Simon Markfelder, Emil Wiedemann
TL;DR
The paper tackles the non-uniqueness of measure-valued solutions in mathematical fluid dynamics by introducing a maximal-turbulence selection principle that maximizes the Jensen defect (variance) of the energy across generalized Young measures. It develops a general variational framework around the concave, upper semicontinuous functional $\mathcal{V}_f$, proves the existence of maximizers on suitable sets $M$, and shows that the mean value is uniquely determined when $f$ is strictly convex. The framework is then applied to the Euler equations, proving the existence of maximal admissible measure-valued solutions for both incompressible and isentropic compressible cases and establishing that different maximizers yield the same mean velocity (and energy), thereby providing a physically meaningful selection principle. Overall, the work offers a rigorous, general method to select physically relevant, highly turbulent measure-valued solutions and demonstrates its applicability to key fluid models.
Abstract
The quest for a good solution concept for the partial differential equations (PDEs) arising in mathematical fluid dynamics is an outstanding open problem. An important notion of solutions are the measure-valued solutions. It is well known that for many PDEs there exists a multitude of measure-valued solutions even if admissibility criteria like an energy inequality are imposed. Hence in recent years, people have tried to select the relevant solutions among all admissible measure-valued solutions or at least to rule out some solutions which are not relevant. In this paper another such criterion is studied. In particular, we aim to select generalized Young measures which are ``maximally turbulent''. To this end, we look for maximizers of a certain functional, namely the variance, or more precisely, the Jensen defect of the energy. We prove existence of such a maximizer and we show that its mean value and total energy is uniquely determined. Our theory is carried out in a very general setting which may be applied in many situations where maximally turbulent measures shall be selected among a set of generalized Young measures. Finally, we apply this general framework to the incompressible and the isentropic compressible Euler equation. Our criterion of maximal turbulence is plausible and leads to existence and uniqueness in a certain sense (in particular, the mean value and the total energy of different maximally turbulent solutions coincide).