The slice H² bound of a strong solution from its time derivative #
On one time slice, a solenoidal vector field f ∈ H²(ℝ³) solving the stationary
Navier–Stokes system Δf = g + (f·∇)f + ∇q with g ∈ L² and a pressure q ∈ H¹
satisfies ‖D²f‖ ≤ 2‖g‖ + C‖∇f‖³ (prop:lps-smoothing).
The proof tests the equation with Δf: the Hessian and the Laplacian have the same
L² norm, the Laplacian of a solenoidal field lies in J, so the pressure gradient
drops out, and the convection term is bounded by ‖f‖_{L⁶}‖∇f‖_{L³}, which the
Sobolev and interpolation inequalities control by ‖∇f‖^{3/2}‖D²f‖^{1/2}. Young's
inequality absorbs the Hessian factor.
A square-integrable pressure with square-integrable weak gradient is orthogonal to
the solenoidal space J: ⟨∇q, w⟩ = 0 for every w ∈ J (prop:lps-smoothing).
A square-integrable vector field whose weak divergence vanishes almost everywhere
is weakly solenoidal (prop:lps-smoothing).
The convection field (f·∇)f of an H² vector field on ℝ³ is square integrable,
and its Euclidean L² norm is at most K ‖∇f‖^{3/2} ‖D²f‖^{1/2}: the velocity is in
L⁶ by the Sobolev inequality and the gradient in L³ by interpolation between L²
and L⁶ (prop:lps-smoothing).
The slice H² bound (prop:lps-smoothing): if f ∈ H²(ℝ³) is solenoidal and
g - Δf + (f·∇)f + ∇q = 0 almost everywhere with g ∈ L² and q ∈ L² having a
square-integrable weak gradient, then ‖D²f‖ ≤ 2‖g‖ + C‖∇f‖³ for the Euclidean L²
norms, with C universal. The second-order family of f uses ordered words, first
letter applied first.