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- ItemOn the existence of weak solutions in the context of multidimensional incompressible fluid dynamics(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Lasarzik, RobertWe define the concept of energy-variational solutions for the Navier--Stokes and Euler equations. This concept is shown to be equivalent to weak solutions with energy conservation. Via a standard Galerkin discretization, we prove the existence of energy-variational solutions and thus weak solutions in any space dimension for the Navier--Stokes equations. In the limit of vanishing viscosity the same assertions are deduced for the incompressible Euler system. Via the selection criterion of maximal dissipation we deduce well-posedness for these equations.
- ItemMaximal dissipative solutions for incompressible fluid dynamics(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Lasarzik, RobertWe introduce the new concept of maximal dissipative solutions for the Navier--Stokes and Euler equations and show that these solutions exist and the solution set is closed and convex. The concept of maximal dissipative solutions coincides with the concept of weak solutions as long as the weak solutions inherits enough regularity to be unique. A maximal dissipative solution is defined as the minimizer of a convex functional and we argue that this definition bears several advantages.