Fractional Space-Time Variational Formulations of (Navier--) Stokes Equations

Abstract

Well-posed space-time variational formulations in fractional order Bochner--Sobolev spaces are proposed for parabolic partial differential equations, and in particular for the instationary Stokes and Navier--Stokes equations on bounded Lipschitz domains. The latter formulations include the pressure variable as a primal unknown and so account for the incompressibility constraint via a Lagrange multiplier. The proposed new variational formulations can be the basis of adaptive numerical solution methods that converge with the best possible rate, which, by exploiting the tensor product structure of a Bochner space, equals the rate of best approximation for the corresponding stationary problem. Unbounded time intervals are admissible in many cases, permitting an optimal adaptive solution of long-term evolution problems.