Existence & Smoothness of the Navier–stokes Equation

Equation (1) is just Newton’s law f = ma for a fluid element subject to the external force f = (fi(x, t))1 i n and to the forces arising from pressure and friction. Equation (2) just says that the fluid is incompressible. For physically reasonable solutions, we want to make sure u(x, t) does now grow large as |x| → ∞. Hence, we will restrict attention to forces f and initial conditions u◦ that satisfy ∣∣∂α x u◦(x)∣∣ CαK(1 + |x|) on R, for any α and K (4) and ∣∣∂α x ∂ t f(x, t)∣∣ CαmK(1 + |x|+ t) on R × [0,∞), for any α,m,K . (5)