| import OAI.Analysis.AncientFlat.Evolution.EvolutionCore |
|
|
| 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] |
| variable {α β : ℝ} |
| lemma bilaplacian_add {f g : V → ℂ} (hf : ContDiff ℝ 4 f) (hg : ContDiff ℝ 4 g) : |
| Δ (Δ (fun x => f x+g x))=fun x => Δ (Δ f) x+Δ (Δ g) x := by |
| have hh : Δ (fun x => f x+g x)=fun x => Δ f x+Δ g x := |
| funext (fun x => (hf.of_le (by norm_num)).contDiffAt.laplacian_add |
| (hg.of_le (by norm_num)).contDiffAt) |
| rw [hh] |
| exact funext (fun x => (ClassicalIBP.contDiff_laplacian f 2 hf).contDiffAt.laplacian_add |
| (ClassicalIBP.contDiff_laplacian g 2 hg).contDiffAt) |
| lemma bilaplacian_smul {f : V → ℂ} (hf : ContDiff ℝ 4 f) (c : ℂ) : |
| Δ (Δ (fun x => c • f x))=fun x => c • Δ (Δ f) x := by |
| have hh : Δ (fun x => c • f x)=fun x => c • Δ f x := |
| funext (fun x => InnerProductSpace.laplacian_smul c (hf.of_le (by norm_num)).contDiffAt) |
| rw [hh] |
| exact funext (fun x => InnerProductSpace.laplacian_smul c |
| (ClassicalIBP.contDiff_laplacian f 2 hf).contDiffAt) |
| variable {A : C(ℝ,V →ᵇ Icc α β)} {b : C(ℝ,V →ᵇ ℂ)} |
| {s T : ℝ} {d e : C₀(V,ℂ)} {u v : ℝ → C₀(V,ℂ)} |
| lemma Evolves.add (hu : Evolves A b s T d u) (hv : Evolves A b s T e v) : |
| Evolves A b s T (d+e) (fun t => u t+v t) := by |
| refine ⟨by simp [hu.1,hv.1],hu.2.1.add hv.2.1,?_,?_,?_⟩ |
| · intro t ht |
| exact (hu.2.2.1 t ht).add (hv.2.2.1 t ht) |
| · intro t ht n hn |
| obtain ⟨M,hM⟩ := hu.2.2.2.1 t ht n hn |
| obtain ⟨N,hN⟩ := hv.2.2.2.1 t ht n hn |
| refine ⟨M+N,fun x => ?_⟩ |
| have hcu : ContDiff ℝ n (fun y => u t y) := (hu.2.2.1 t ht).of_le (by exact_mod_cast hn) |
| have hcv : ContDiff ℝ n (fun y => v t y) := (hv.2.2.1 t ht).of_le (by exact_mod_cast hn) |
| change ‖iteratedFDeriv ℝ n (fun y => u t y+v t y) x‖≤_ |
| rw [fun_iteratedFDeriv_add hcu hcv] |
| exact (norm_add_le _ _).trans (add_le_add (hM x) (hN x)) |
| · intro t ht |
| obtain ⟨p,hp,hpe⟩ := hu.2.2.2.2 t ht |
| obtain ⟨q,hq,hqe⟩ := hv.2.2.2.2 t ht |
| refine ⟨p+q,hp.add hq,fun x => ?_⟩ |
| change p x+q x= -((A t x).val : ℂ)*Δ (Δ (fun y => u t y+v t y)) x-b t x*(u t x+v t x) |
| rw [hpe,hqe,bilaplacian_add (hu.2.2.1 t ht) (hv.2.2.1 t ht)] |
| ring |
| lemma Evolves.smul (hu : Evolves A b s T d u) (c : ℂ) : |
| Evolves A b s T (c•d) (fun t => c•u t) := by |
| refine ⟨by simp [hu.1],hu.2.1.const_smul c,?_,?_,?_⟩ |
| · intro t ht |
| exact (hu.2.2.1 t ht).const_smul c |
| · intro t ht n hn |
| obtain ⟨M,hM⟩ := hu.2.2.2.1 t ht n hn |
| refine ⟨‖c‖*M,fun x => ?_⟩ |
| have hc : ContDiff ℝ n (fun y => u t y) := (hu.2.2.1 t ht).of_le (by exact_mod_cast hn) |
| change ‖iteratedFDeriv ℝ n (fun y => c • u t y) x‖≤_ |
| rw [iteratedFDeriv_const_smul_apply' hc.contDiffAt] |
| simpa only [Pi.smul_apply,norm_smul] using mul_le_mul_of_nonneg_left (hM x) (norm_nonneg c) |
| · intro t ht |
| obtain ⟨p,hp,hpe⟩ := hu.2.2.2.2 t ht |
| refine ⟨c•p,hp.const_smul c,fun x => ?_⟩ |
| change c • p x= -((A t x).val : ℂ)*Δ (Δ (fun y => c•u t y)) x-b t x*(c•u t x) |
| rw [hpe,bilaplacian_smul (hu.2.2.1 t ht) c] |
| simp only [smul_eq_mul] |
| ring |
| end ParametrixEvolution |
| |
| end |
| |
| end OAI |
| |
