| import OAI.Analysis.AncientFlat.Evolution.HomogeneousUniqueness | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter | |
| open scoped ContDiff ZeroAtInfty BoundedContinuousFunction Laplacian | |
| namespace ParametrixEvolution | |
| variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] | |
| [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] | |
| variable {α β : ℝ} | |
| def Evolves (A : C(ℝ,V →ᵇ Icc α β)) (b : C(ℝ,V →ᵇ ℂ)) | |
| (s T : ℝ) (d : C₀(V,ℂ)) (u : ℝ → C₀(V,ℂ)) : Prop := | |
| u s=d ∧ ContinuousOn u (Icc s T) ∧ | |
| (∀ t ∈ Ioo s T, ContDiff ℝ 4 (fun x => u t x)) ∧ | |
| (∀ t ∈ Ioo s T, ∀ n : ℕ, n≤4 → ∃ M : ℝ, ∀ x, | |
| ‖iteratedFDeriv ℝ n (fun y => u t y) x‖≤M) ∧ | |
| (∀ t ∈ Ioo s T, ∃ v : V →ᵇ ℂ, | |
| HasDerivAt (fun r => (u r).toBCF) v t ∧ ∀ x, | |
| v x=-((A t x).val : ℂ)*Δ (Δ (fun y => u t y)) x-b t x*u t x) | |
| end ParametrixEvolution | |
| end | |
| end OAI | |