In order to find the weak form of the Navier–Stokes equations, firstly, consider the momentum equation ρ ρ ∂ ∂ u ∂ ∂ t − − μ μ Δ Δ u + ρ ρ ( u ⋅ ⋅ ∇ ∇ ) u + ∇ ∇ p = f multiply it for a test function v , defined in a suitable space V , and integrate both members with respect to the domain Ω Ω : ∫ ∫ Ω Ω ρ ρ ∂ ∂ u ∂ ∂ t ⋅ ⋅ v − − ∫ ∫ Ω Ω μ μ Δ Δ u ⋅ ⋅ v + ∫ ∫ Ω Ω ρ ρ ( u ⋅ ⋅ ∇ ∇ ) u ⋅ ⋅ v + ∫ ∫ Ω Ω ∇ ∇ p ⋅ ⋅ v = ∫ ∫ Ω Ω f ⋅ ⋅ v Counter-integrating by parts the diffusive and the pressure terms and by using the Gauss' theorem: − − ∫ ∫ Ω Ω μ μ Δ Δ u ⋅ ⋅ v = ∫ ∫ Ω Ω μ μ ∇ ∇ u ⋅ ⋅ ∇ ∇ v − − ∫ ∫ ∂ ∂ Ω Ω μ μ ∂ ∂ u ∂ ∂ n ^ ^ ⋅ ⋅ v ∫ ∫ Ω Ω ∇ ∇ p ⋅ ⋅ v = − − ∫ ∫ Ω Ω p ∇ ∇ ⋅ ⋅ v + ∫ ∫ ∂ ∂ Ω Ω p v ⋅ ⋅ n ^ ^