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Professor Mária Lukácová-Medvidová, Professor for Numerical Mathematics, University of Mainz
Dissipative solutions of compressible fluid flows
In this talk I present an overview of our recent results on generalized solutions, the so-called dissipative solutions, of compressible fluid flows. I will concentrate on the inviscid flows, the Euler equations, and mention also the relevant results obtained for viscous compressible flows, governed by the compressible Navier-Stokes equations.
The existence of dissipative solutions will be shown by the convergence analysis of a suitable, invariant-domain preserving finite volume scheme. In the case that the strong solution to the above equations exists, the dissipative solutions coincide with the strong solution on its lifespan. In this case we can also apply the relative entropy to derive rigorous error estimates between numerical solutions and the exact strong solution.
If the strong solution does not exist we apply a new tool, the so-called K-convergence, and show the strong convergence of the empirical means of numerical solutions to a dissipative solution. The latter is the expected value of the dissipative measure-valued solutions and satisfies a weak formulation of the Euler equations modulo the Reynolds stress tensor. We will discuss a relation of the dissipative solutions to the Kolmogorov hypothesis and turbulence.
The present research has been partially supported by TRR 146 Multiscale simulation methods for soft matter systems, TRR 165 Waves to Weather funded by the German Science Foundation and by the Gutenberg Research College.