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caec230 | 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 38 39 40 41 42 43 44 45 46 47 48 | import OAI.MathematicalPhysics.RapidForcing.CurlDerivatives
import OAI.MathematicalPhysics.RapidForcing.JointSmoothness
namespace OAI
open scoped BigOperators ENNReal Topology
open Set MeasureTheory
namespace RapidForcing
lemma classical_of_joint_smooth {u : Field Space} {p : Field ℝ}
(hu : ContDiff ℝ (⊤ : ℕ∞) (Function.uncurry u))
(hp : ContDiff ℝ (⊤ : ℕ∞) (Function.uncurry p)) : Classical u p := by
refine ⟨?_, ?_, ?_⟩
· intro t _
constructor
· exact (hu.comp (contDiff_const.prodMk contDiff_id)).of_le
(WithTop.coe_le_coe.mpr le_top)
· exact (hp.comp (contDiff_const.prodMk contDiff_id)).of_le
(WithTop.coe_le_coe.mpr le_top)
· intro x
exact (hu.comp (contDiff_id.prodMk contDiff_const)).differentiable
(by simp) |>.differentiableOn
· intro T _
refine ⟨hu.continuous.continuousOn, ?_, ?_, ?_, hp.continuous.continuousOn, ?_⟩
· apply (ordinaryTimeD_joint_smooth hu).continuous.continuousOn.congr
intro z hz
exact timeD_eq_ordinary hu hz.1.1 z.2
· intro i
exact (spatialD_joint_smooth hu i).continuous.continuousOn
· intro i j
exact (spatialD_joint_smooth (spatialD_joint_smooth hu j) i).continuous.continuousOn
· intro i
exact (spatialD_joint_smooth hp i).continuous.continuousOn
theorem addressed_classical_solution (ν : ℝ) (M : Machine) (w : M.Input) :
let u := addressedVelocity M w
let f := addressedForce ν M w
Smooth f ∧ Supported f ∧ Smooth u ∧ Supported u ∧
Classical u (fun _ _ => 0) ∧ NavierStokes ν f u (fun _ _ => 0) := by
exact ⟨(addressed_fields_smooth ν M w).2, addressedForce_supported ν M w,
(addressed_fields_smooth ν M w).1, addressedVelocity_supported M w,
classical_of_joint_smooth (addressedVelocity_joint_smooth M w) contDiff_const,
addressed_pointwise_equations ν M w⟩
end RapidForcing
end OAI
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