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Space-time variational saddle point formulations of Stokes and Navier-Stokes equations. (English) Zbl 1295.35354

The authors investigate simultaneously space-time variational saddle point formulations of the incompressible Stokes and Navier-Stokes equations involving both the velocities and the pressure. They prove that the Stokes operator defined by this variational formulation is boundedly invertible beetween a suitable Hilbert space and the dual of another Hilbert space, thus obtaining well-posedness for the Stokes equations with slip and no-slip boundary conditions. Since the Navier-Stokes operator is Lipschitz continuous under appropriate conditions, the authors are able to use a fixed point argument to prove, for small data, existence for the space-time variational formulation of the Navier-Stokes equations.

MSC:

35Q30 Navier-Stokes equations
76M30 Variational methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
76D07 Stokes and related (Oseen, etc.) flows
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