Localized Equation Duhamel Finish #
Part of the Caffarelli–Kohn–Nirenberg partial regularity proof.
theorem
CKN.Core.Step3.duhamel_of_suitableWeakSolution_localized
{Ω : Set Foundation.Parabolic.Vec3}
{I : Set ℝ}
{q : ℝ}
{u : Foundation.Parabolic.ParabolicPoint → Foundation.Parabolic.Vec3}
{Du : Foundation.Parabolic.ParabolicPoint → Fin 3 → Foundation.Parabolic.Vec3}
{p : Foundation.Parabolic.ParabolicPoint → ℝ}
{f : Foundation.Parabolic.ParabolicPoint → Foundation.Parabolic.Vec3}
(hsol : IsSuitableWeakSolutionIntegrable Ω I q u Du p f)
{φ : Foundation.Parabolic.Vec3 × ℝ → ℝ}
(hφ : φ ∈ spaceTimeTestFunction Ω I)
{Ω' : Set Foundation.Parabolic.Vec3}
{J : Set ℝ}
(hbox : localBox Ω I Ω' J)
(hφbox : tsupport φ ⊆ Ω' ×ˢ J)
:
∀ᵐ (z : Foundation.Parabolic.ParabolicPoint), localizedVelocity
(have this := φ;
this)
u z = duhamelPotential (localizedDivergenceG φ u Du p f) (localizedDivergenceH φ u p) z