File size: 3,197 Bytes
d5b562f | 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 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 | import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingBlowupObservation
import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingFilteredLimit
namespace OAI
/-! # Joint continuity of the actual physical ball observation -/
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
local notation "Radius" => {L : ℝ // 1 ≤ L}
noncomputable def expandingTorusRealization (a k L : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) :
FourierL2 →L[ℂ] C(SchrodingerTorus, ℂ) :=
LinearMap.mkContinuous
{ toFun := expandingTorusFunction a k L
map_add' := by
intro f g
ext x
simp only [expandingTorusFunction_eq_evaluation a k L ha ha1 hk hL, map_add,
ContinuousMap.add_apply]
map_smul' := by
intro c f
ext x
simp only [expandingTorusFunction_eq_evaluation a k L ha ha1 hk hL, map_smul,
ContinuousMap.smul_apply, RingHom.id_apply] }
(expandingEmbeddingBound a k) (expandingTorusFunction_norm_le a k L ha ha1 hk hL)
theorem continuous_expandingPhysical_observation (a k : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) :
Continuous (fun p : (Radius × FourierL2) × E =>
expandingTorusFunction a k p.1.1.1 p.1.2
(euclideanToTorus (p.1.1.1⁻¹ • p.2))) := by
have hi : Continuous (fun p : (Radius × FourierL2) × E =>
expandingInverseTransfer a k p.1.1.1 ha hk p.1.1.2 p.1.2) :=
((continuous_expandingInverseTransfer a k ha hk).comp
(continuous_fst.comp continuous_fst)).clm_apply (continuous_snd.comp continuous_fst)
have hf : Continuous (fun p : (Radius × FourierL2) × E =>
expandingTorusFunction a k 1
(expandingInverseTransfer a k p.1.1.1 ha hk p.1.1.2 p.1.2)) :=
(expandingTorusRealization a k 1 ha ha1 hk le_rfl).continuous.comp hi
have hy : Continuous (fun p : (Radius × FourierL2) × E =>
euclideanToTorus (p.1.1.1⁻¹ • p.2)) := by
unfold euclideanToTorus
apply continuous_pi
intro j
have hR : Continuous (fun p : (Radius × FourierL2) × E => p.1.1.1⁻¹) :=
(continuous_subtype_val.comp (continuous_fst.comp continuous_fst)).inv₀
(fun p => by change p.1.1.1 ≠ 0; linarith [p.1.1.2])
have hY : Continuous (fun p : (Radius × FourierL2) × E => p.2 j) := by fun_prop
exact (AddCircle.continuous_mk' (2 * Real.pi)).comp (hR.mul hY)
have he := continuous_eval.comp (hf.prodMk hy)
simpa only [Function.comp_def, expandingInverseTransfer_function a k _ ha ha1 hk] using he
theorem continuous_expandingPhysicalBall (a k R : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) :
Continuous (fun p : Radius × FourierL2 =>
expandingPhysicalBall a k p.1.1 R ha ha1 hk p.1.2 p.2) := by
apply ContinuousMap.continuous_of_continuous_uncurry _
have hm : Continuous (fun p : (Radius × FourierL2) × Metric.closedBall (0 : E) R =>
(p.1, (p.2 : E))) :=
continuous_fst.prodMk (continuous_subtype_val.comp continuous_snd)
have h := (continuous_expandingPhysical_observation a k ha ha1 hk).comp hm
simpa only [Function.uncurry_def, Function.comp_def, expandingPhysicalBall_apply] using h
end DefocusingNLS
end OAI
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