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12dee50 | 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 | 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
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