WIAS Preprint No. 2834, (2021)

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


  • Lasarzik, Robert
    ORCID: 0000-0002-1677-6925

2020 Mathematics Subject Classification

  • 35D99 35Q30 35Q31 76D05


  • Existence, uniqueness, Navier--Stokes, Euler, incompressible, fluid dynamics, weak solutions, dissipative solutions




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.

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