On the local Smoothness of Solutions of the Navier-Stokes Equations

Original · EN

We consider the Cauchy problem for incompressible Navier-Stokes equations uₜ+u∇ₓu-Δu+∇ p=0, div u=0 in Rᵈ × R+ with initial data a∈ Lᵈ(Rᵈ), and study in some detail the smoothing effect of the equation. We prove that for T<∞ and for any positive integers n and m we have tᵐ⁺ⁿ/²DᵐₜDⁿₓ u∈ Lᵈ⁺²(Rᵈ× (0,T)), as long as the uₗᵈ⁺²ₓ,ₜ₍ᵣᵈ× ₍₀,ₜ₎₎ stays finite.