File size: 2,892 Bytes
37bcf52 | 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 71 72 73 74 75 76 77 78 79 80 81 82 | import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Flow.FlowPointwise
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
import Mathlib.Topology.Order.ProjIcc
namespace OAI
/-! Integration on the fixed unit interval as a bounded linear operator.
This permits elapsed time to be a parameter of the Picard equation. -/
noncomputable section
namespace ForcedComputation.Flow
open Set MeasureTheory
open scoped Topology
abbrev UnitTime := Icc (0 : ℝ) 1
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
def extendPath (u : C(UnitTime, E)) : C(ℝ, E) :=
u.comp ⟨projIcc 0 1 zero_le_one, continuous_projIcc⟩
omit [NormedSpace ℝ E] [CompleteSpace E] in
@[simp] theorem extendPath_apply (u : C(UnitTime, E)) (t : ℝ) :
extendPath u t = u (projIcc 0 1 zero_le_one t) := rfl
omit [NormedSpace ℝ E] [CompleteSpace E] in
@[simp] theorem extendPath_coe (u : C(UnitTime, E)) (t : UnitTime) :
extendPath u t = u t := by simp [extendPath]
def primitivePath (u : C(UnitTime, E)) : C(UnitTime, E) :=
⟨fun t => ∫ s in (0 : ℝ)..t.1, extendPath u s,
(continuous_iff_continuousAt.mpr fun t =>
((extendPath u).continuous.integral_hasStrictDerivAt 0 t).hasDerivAt.continuousAt).comp
continuous_subtype_val⟩
@[simp] theorem primitivePath_apply (u : C(UnitTime, E)) (t : UnitTime) :
primitivePath u t = ∫ s in (0 : ℝ)..t.1, extendPath u s := rfl
theorem primitivePath_norm (u : C(UnitTime, E)) : ‖primitivePath u‖ ≤ ‖u‖ := by
apply (ContinuousMap.norm_le _ (norm_nonneg u)).mpr
intro t
calc
‖primitivePath u t‖ ≤ ‖u‖ * |t.1 - 0| :=
intervalIntegral.norm_integral_le_of_norm_le_const fun s _ =>
u.norm_coe_le_norm _
_ ≤ ‖u‖ := by
rw [sub_zero, abs_of_nonneg t.2.1]
exact mul_le_of_le_one_right (norm_nonneg u) t.2.2
def primitivePathL : C(UnitTime, E) →L[ℝ] C(UnitTime, E) :=
LinearMap.mkContinuous
{ toFun := primitivePath
map_add' := by
intro u v
ext t
change (∫ s in (0 : ℝ)..t.1, extendPath u s + extendPath v s) = _
exact intervalIntegral.integral_add
((extendPath u).continuous.intervalIntegrable _ _)
((extendPath v).continuous.intervalIntegrable _ _)
map_smul' := by
intro c u
ext t
change (∫ s in (0 : ℝ)..t.1, c • extendPath u s) = _
exact intervalIntegral.integral_smul c (extendPath u) }
1 (by
intro u
change ‖primitivePath u‖ ≤ (1 : ℝ) * ‖u‖
rw [one_mul]
exact primitivePath_norm u)
@[simp] theorem primitivePathL_apply (u : C(UnitTime, E)) :
primitivePathL u = primitivePath u := rfl
theorem primitivePathL_norm : ‖primitivePathL (E := E)‖ ≤ 1 :=
LinearMap.mkContinuous_norm_le _ zero_le_one _
end ForcedComputation.Flow
end
end OAI
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