Uniqueness Aspects of Semi-Discretized Navier-Stokes Equations

We consider the Navier-Stokes equations on a time interval 0, T) in the weak formulation given by Here we assume ν > 0, G ⊂ ℝ³ open, bounded with C^2,1 boundary, f ∈ L² (0, T, L²), and u₀ ∈ V, where V is the closure of C^∞_0,σ in (H¹₀, ‖∇ ·‖) and (·, ·) denotes the scalar product in L². The above equations for u ∈ L²(0, T, V ∩ H²) with ∂_tu ∈ L² (0, T, L² ) can be semi-discretized with respect to time in different ways. The existence of solutions to the resulting schemes can be shown with the Galerkin method provided the viscous term is not treated fully explicitly. In the case of linear schemes, uniqueness can easily be proven with the Lax-Milgram lemma. However, schemes that are implicit in both factors of the convective term seem to cause difficulties with respect to uniqueness, even for arbitrary small step sizes. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)