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. 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.The Navier-Stokes and Euler equations are the standard models for incompressible fluid dynamics. Both are a recurrent tools in computational fluid dynamics for weather forecast, micro fluidic devices [26] or industrial processes like steel production [1]. There exists a vast literature concerning the Navier-Stokes and Euler equations. In case of the Navier-Stokes equation, we only mention here the existence proof for weak solutions in three dimension by Leray [22] and the weak-strong uniqueness result due to Serrin [25]. In the context of the Euler equations, the existence of weak solutions in any space dimension is already known (see [9]) also fulfilling the energy inequality (see [10]). This result was proven via the convex integration technique. This technique grants the existence of infinitely many and also non-physical weak solutions. Additionally, it was proven for the Navier-Stokes equations via similar techniques that there exist infinitely many weak solutions that do not fulfill the energy