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37bcf52 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 | import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.InitializedFluid
namespace OAI
/-! The flat-coordinate force depends only on a germ. No smoothness away
from the point is needed for this locality statement. -/
noncomputable section
open Filter
open scoped Topology
open ShearFlows
namespace ForcedComputation
theorem force_congr_germ {U V : Velocity} {t : ℝ} {x : Space}
(he : U =ᶠ[𝓝 (t, x)] V) (ν : ℝ) : force ν U (t, x) = force ν V (t, x) := by
have ht : (fun s => U (s, x)) =ᶠ[𝓝 t] fun s => V (s, x) :=
he.comp_tendsto (continuous_id.prodMk continuous_const).continuousAt
have hx : (fun z => U (t, z)) =ᶠ[𝓝 x] fun z => V (t, z) :=
he.comp_tendsto (continuous_const.prodMk continuous_id).continuousAt
have hd (j : Fin 3) : derivative (fun z => U (t, z)) j =ᶠ[𝓝 x]
derivative (fun z => V (t, z)) j :=
(hx.fderiv (𝕜 := ℝ)).mono (fun z hz => congrArg (fun A => A (basis j)) hz)
have hl : laplacian (fun z => U (t, z)) x = laplacian (fun z => V (t, z)) x := by
apply Finset.sum_congr rfl
intro j _
exact congrArg (fun A => A (basis j)) ((hd j).fderiv_eq (𝕜 := ℝ))
change deriv (fun s => U (s, x)) t - ν • laplacian (fun z => U (t, z)) x =
deriv (fun s => V (s, x)) t - ν • laplacian (fun z => V (t, z)) x
rw [ht.deriv_eq, hl]
end ForcedComputation
end
end OAI
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