Math Problem Solver

Equation:

∂u/∂t + (u ⋅ ∇)u = -∇p + νΔu + f, ∇ ⋅ u = 0

Initial conditions:

u(x, 0) = u₀(x)

Boundary conditions:

u(x,t) → 0 as |x| → ∞

Under these initial and boundary conditions, are there always global, smooth and non-explosive solutions for u(x,t) and p(x,t)? Prove it.