| 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 | |