On the existence of energy‐variational 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 and prove their existence in any space dimension. It is shown that the concept of energy‐variational solutions enjoys several desirable properties. Energy‐variational solutions are not only known to exist and coincide with local strong solutions, but the operator, mapping the data to the set of energy‐variational solutions, is additionally continuous and all restrictions and all concatenations of energy‐variational solutions are energy‐variational solutions again. Finally, different selection criteria for these solutions are discussed.