On the existence of weak solutions in the context of multidimensional incompressible fluid dynamics

Abstract

We define the concept of energy-variational solutions for the Navier--Stokes and Euler equations. The underlying relative energy inequality holds as an equality for classical solutions and if the additional variable vanishes, these solutions are equivalent to the weak formulation with the strong energy inequality. By introducing an additional defect variable in time, all restrictions and all concatenations of energy-variational solutions are again energy-variational solutions. Via the criterion of maximal dissipation, a unique solution is selected that is not only continuously depending on the data but also turns out to be a unique weak solution.