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f18e7de | 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 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 | import OAI.MathematicalPhysics.Boltzmann.Approximation.PositiveNonlinearLimit
import OAI.MathematicalPhysics.Boltzmann.Branching.CompactCapFamily
import OAI.MathematicalPhysics.Boltzmann.Collision.CollisionDissipation
import OAI.MathematicalPhysics.Boltzmann.Compactness.StrongMassClip
import OAI.MathematicalPhysics.Boltzmann.Compactness.WeakDuhamelLimit
import OAI.MathematicalPhysics.Boltzmann.Compactness.WeakEntropyMarginal
import OAI.MathematicalPhysics.Boltzmann.Compactness.WeakL1Operations
import OAI.MathematicalPhysics.Boltzmann.Jets.TorusVelocityWindowAE
namespace OAI
section
/-! # Convergence of the actual right-hand sides of the cap equations -/
noncomputable section
open MeasureTheory Filter
namespace BoltzmannNonuniqueness.JetCopyDatum
theorem TightCapFamily.exists_cutoff_equation_limit_with_pair
(hcarleman : Published.GPV2009CarlemanInput) (hGLPS : Published.GLPS1988Proposition3)
(hDP : Published.DunfordPettis1940Theorem321)
{s₀ η K α γ d₀ q : ℝ} (F : TightCapFamily s₀ η K α γ d₀ q)
(hq₁ : 1 < q) (hq₂ : q < 2) :
∃ (φ : ℕ → ℕ) (g : (ℝ × Position) × Velocity → ℝ), StrictMono φ ∧
Published.WeakL1Converges ((AnnularEnvelope.timePositionVolume F.time).prod volume)
(fun n => torusPhysical q (F.approximation (φ n)).solution.value) g ∧
Published.WeakL1Converges ((AnnularEnvelope.timePositionVolume F.time).prod volume)
(fun n => CollisionCompactness.collisionSource (φ n+1)
(torusPhysical q (F.approximation (φ n)).solution.value))
(fun p => CollisionCompactness.gainDensity g p-CollisionCompactness.lossDensity g p) ∧
Integrable (CollisionCompactness.pairDensity g)
((AnnularEnvelope.timePositionVolume F.time).prod volume) ∧ (∀ z, 0 ≤ g z) ∧
Published.WeakL1Converges ((AnnularEnvelope.timePositionVolume F.time).prod volume)
(fun n => CollisionCompactness.pairDensity
(torusPhysical q (F.approximation (φ n)).solution.value))
(CollisionCompactness.pairDensity g) := by
obtain ⟨φ,g,hφ,hg,hpair,hgn⟩ := F.exists_positive_nonlinear_limit hcarleman hGLPS hDP hq₁ hq₂
have hfm n := torusPhysical_measurable (F.approximation (φ n)).solution.measurable_value q
have hfn n := torusPhysical_nonneg
(fun z => ((F.approximation (φ n)).solution.value_bounds z).1) q
have hf n := memLp_one_iff_integrable.mp (F.pairs_ui.memLp (φ n))
have hb : Tendsto (fun n => (φ n : ℝ)+1) atTop atTop :=
tendsto_atTop_mono (fun n => by linarith : ∀ n, (φ n : ℝ) ≤ (φ n : ℝ)+1)
(tendsto_natCast_atTop_atTop.comp hφ.tendsto_atTop)
have hterms := CollisionCompactness.weak_cutoff_gainLoss hfm hfn hg.1 hpair hf hb
have hdiff := hterms.1.sub hterms.2
(fun n => memLp_one_iff_integrable.mp (F.gain_ui.memLp (φ n)))
(fun n => memLp_one_iff_integrable.mp (F.loss_ui.memLp (φ n)))
have he n : (fun p => KineticCollisionNullSets.gain (φ n+1)
(torusPhysical q (F.approximation (φ n)).solution.value) p-
CollisionCompactness.cutoffLossDensity (φ n+1)
(torusPhysical q (F.approximation (φ n)).solution.value) p) =ᵐ[
(AnnularEnvelope.timePositionVolume F.time).prod volume]
CollisionCompactness.collisionSource (φ n+1)
(torusPhysical q (F.approximation (φ n)).solution.value) := by
filter_upwards [CollisionCompactness.cutoff_loss_ae_eq_frequency (hfm n) (hfn n) (hf n)
(φ n+1)] with p hp
simp only [CollisionCompactness.collisionSource,hp]
exact ⟨φ,g,hφ,hg,hdiff.congr_sequence he,hpair.2.1,hgn,hpair⟩
theorem TightCapFamily.exists_cutoff_equation_limit
(hcarleman : Published.GPV2009CarlemanInput) (hGLPS : Published.GLPS1988Proposition3)
(hDP : Published.DunfordPettis1940Theorem321)
{s₀ η K α γ d₀ q : ℝ} (F : TightCapFamily s₀ η K α γ d₀ q)
(hq₁ : 1 < q) (hq₂ : q < 2) :
∃ (φ : ℕ → ℕ) (g : (ℝ × Position) × Velocity → ℝ), StrictMono φ ∧
Published.WeakL1Converges ((AnnularEnvelope.timePositionVolume F.time).prod volume)
(fun n => torusPhysical q (F.approximation (φ n)).solution.value) g ∧
Published.WeakL1Converges ((AnnularEnvelope.timePositionVolume F.time).prod volume)
(fun n => CollisionCompactness.collisionSource (φ n+1)
(torusPhysical q (F.approximation (φ n)).solution.value))
(fun p => CollisionCompactness.gainDensity g p-CollisionCompactness.lossDensity g p) ∧
Integrable (CollisionCompactness.pairDensity g)
((AnnularEnvelope.timePositionVolume F.time).prod volume) ∧ (∀ z, 0 ≤ g z) := by
obtain ⟨φ,g,hφ,hg,hr,hpi,hgn,_hpair⟩ :=
F.exists_cutoff_equation_limit_with_pair hcarleman hGLPS hDP hq₁ hq₂
exact ⟨φ,g,hφ,hg,hr,hpi,hgn⟩
end BoltzmannNonuniqueness.JetCopyDatum
end
end
section
/-! # The cap approximations satisfy the actual torus initial-value identity -/
noncomputable section
open MeasureTheory Set
namespace BoltzmannNonuniqueness.JetCopyDatum
theorem periodicInitial_measurable (s₀ η K α γ d₀ q : ℝ) (N : ℕ) (ell : ℝ) :
Measurable (periodicInitial s₀ η K α γ d₀ q N ell) := by
have hm := (measurable_seed s₀ η K α γ d₀ q N).add
((HotSeedProfile.normalized_measurable q (DatumBounds.scale s₀ η (N+1))).const_mul ell)
have he : periodicInitial s₀ η K α γ d₀ q N ell =
TorusFlight.initialDensity q (initial s₀ η K α γ d₀ q N
(fun p => ell*HotSeedProfile.normalized q (DatumBounds.scale s₀ η (N+1)) p)) := by
funext p
exact mul_comm _ _
rw [he]
exact TorusFlight.initialDensity_measurable hm q
theorem CapApproximation.spacetime_mild (hcarleman : Published.GPV2009CarlemanInput)
{s₀ η K α γ d₀ q T : ℝ} {N : ℕ} (F : CapApproximation s₀ η K α γ d₀ q T N)
(hq₁ : 1 < q) (hq₂ : q < 2) (hT : 0 ≤ T) :
TorusFlight.along (TorusFlight.regroup (torusPhysical q F.solution.value)) =ᵐ[
(volume.restrict (Icc 0 T)).prod (volume : Measure Phase)]
(fun z => periodicInitial s₀ η K α γ d₀ q N F.coefficient z.2 +
Published.timePrimitive (volume.restrict (Icc 0 T))
(TorusFlight.along (TorusFlight.regroup
(CollisionCompactness.collisionSource (N+1) (torusPhysical q F.solution.value)))) z) := by
have hm := (measurable_seed s₀ η K α γ d₀ q N).add
((HotSeedProfile.normalized_measurable q (DatumBounds.scale s₀ η (N+1))).const_mul F.coefficient)
have h := TorusFlight.spacetime_mild hcarleman F.solution hq₁ hq₂
(by positivity) (by have := F.amplitude_pos; positivity) hT le_rfl le_rfl hm
filter_upwards [h] with z hz
convert hz using 1
congr 1
exact mul_comm _ _
end BoltzmannNonuniqueness.JetCopyDatum
end
end
section
/-! # Every-time characteristic traces of the finite cap approximations -/
noncomputable section
open MeasureTheory Set
namespace BoltzmannNonuniqueness.JetCopyDatum
theorem CapApproximation.characteristic_slice (hcarleman : Published.GPV2009CarlemanInput)
{s₀ η K α γ d₀ q T : ℝ} {N : ℕ} (F : CapApproximation s₀ η K α γ d₀ q T N)
(hq₁ : 1 < q) (hq₂ : q < 2) {t : ℝ} (ht : t ∈ Icc 0 T) :
(fun p => torusPhysical q F.solution.value
((t,(TorusFlight.flight t p).1),(TorusFlight.flight t p).2)) =ᵐ[volume]
TorusFlight.characteristicProfile (volume.restrict (Icc 0 T))
(periodicInitial s₀ η K α γ d₀ q N F.coefficient)
(TorusFlight.along (TorusFlight.regroup
(CollisionCompactness.collisionSource (N+1) (torusPhysical q F.solution.value)))) t := by
have hm := (measurable_seed s₀ η K α γ d₀ q N).add
((HotSeedProfile.normalized_measurable q (DatumBounds.scale s₀ η (N+1))).const_mul F.coefficient)
have h := TorusFlight.characteristic hcarleman F.solution hq₁ hq₂
(by positivity) (by have := F.amplitude_pos; positivity) ht.1 ht.2 le_rfl hm
filter_upwards [h] with p hp
change _ = periodicInitial s₀ η K α γ d₀ q N F.coefficient p+
Published.timePrimitive (volume.restrict (Icc 0 T)) _ (t,p)
rw [Published.timePrimitive_eq_interval ht]
convert hp using 1
congr 1
exact mul_comm _ _
end BoltzmannNonuniqueness.JetCopyDatum
end
end
section
/-! # The actual cap initial data converge weakly to the prescribed cold datum -/
noncomputable section
open MeasureTheory
namespace BoltzmannNonuniqueness.JetCopyDatum
theorem periodicInitial_integrable {s₀ η K γ d₀ q : ℝ}
(hs₀ : 0 < s₀) (hsmax : s₀ ≤ 1/8) (hη : 0 < η) (hηmax : η ≤ Real.log 2)
(hK : 0 ≤ K) (hγ : 0 < γ) (hd₀ : 0 < d₀) (hq : 0 ≤ q) (hqmax : q < 2)
(α : ℝ) (N : ℕ) (ell : ℝ) :
Integrable (periodicInitial s₀ η K α γ d₀ q N ell) := by
have hδ : 0 < DatumBounds.scale s₀ η (N+1) := DatumBounds.scale_pos hs₀ _
have hδmax : DatumBounds.scale s₀ η (N+1) ≤ 1 := by
have h := JetGeometry.scale_antitone hs₀.le hη.le (Nat.zero_le (N+1))
rw [show DatumBounds.scale s₀ η 0 = s₀ by simp [DatumBounds.scale]] at h
linarith
have he : periodicInitial s₀ η K α γ d₀ q N ell =
fun p => JetPeriodicDatum.truncated s₀ η K α γ d₀ N p+
ell*HotSeedProfile.periodicDensity q (DatumBounds.scale s₀ η (N+1)) p :=
funext (periodicInitial_eq hs₀ hsmax hη hηmax K α γ d₀ q N ell)
rw [he]
exact (JetPeriodicDatum.truncated_integrable hs₀ hsmax hη hηmax hK hγ hd₀ α N).add
((HotSeedProfile.periodicDensity_integrable hq hqmax hδ hδmax).const_mul ell)
theorem CompactCapFamily.initial_weak_limit {s₀ η K α γ d₀ q : ℝ}
(F : CompactCapFamily s₀ η K α γ d₀ q)
(hs₀ : 0 < s₀) (hsmax : s₀ ≤ 1/8) (hη : 0 < η) (hηmax : η ≤ Real.log 2)
(hK : 0 ≤ K) (hγ : 0 < γ) (hd₀ : 0 < d₀) (hq : 0 ≤ q) (hqmax : q < 2) :
Published.WeakL1Converges volume
(fun n => periodicInitial s₀ η K α γ d₀ q n (F.approximation n).coefficient)
(JetPeriodicDatum.datum s₀ η K α γ d₀) := by
have hi := JetPeriodicDatum.integrable_pullback
(JetDatum.datum_integrable hs₀ hsmax hη hηmax hK hγ hd₀ α)
exact Published.weakL1_of_integral_norm_sub
(fun n => periodicInitial_integrable hs₀ hsmax hη hηmax hK hγ hd₀ hq hqmax α n _)
(JetPeriodicDatum.measurable_pullback (Measurable.tsum
(fun i => JetDatum.term_measurable s₀ η K α γ d₀ i))) hi F.initial_l1
end BoltzmannNonuniqueness.JetCopyDatum
end
end
section
/-! # A genuine local mild solution obtained from the actual positive cap family -/
noncomputable section
open MeasureTheory Set
namespace BoltzmannNonuniqueness
/-- The characteristic identity uses the full physical collision terms and the
prescribed initial density. The spacetime identity determines a continuous-time
representative through the Bochner integral of the collision source. -/
structure MildCollisionSolution (T : ℝ) (f₀ : Density)
(g : (ℝ × Position) × Velocity → ℝ) : Prop where
measurable : Measurable g
nonnegative : ∀ z, 0 ≤ g z
integrable : Integrable g ((AnnularEnvelope.timePositionVolume T).prod volume)
pair_integrable : Integrable (CollisionCompactness.pairDensity g)
((AnnularEnvelope.timePositionVolume T).prod volume)
characteristic : TorusFlight.along (TorusFlight.regroup g) =ᵐ[
(volume.restrict (Icc 0 T)).prod (volume : Measure Phase)]
(fun z => f₀ z.2+Published.timePrimitive (volume.restrict (Icc 0 T))
(TorusFlight.along (TorusFlight.regroup
(fun p => CollisionCompactness.gainDensity g p-CollisionCompactness.lossDensity g p))) z)
namespace JetCopyDatum
theorem TightCapFamily.exists_mild_limit_with_convergence
(hcarleman : Published.GPV2009CarlemanInput) (hGLPS : Published.GLPS1988Proposition3)
(hDP : Published.DunfordPettis1940Theorem321)
{s₀ η K α γ d₀ q : ℝ} (F : TightCapFamily s₀ η K α γ d₀ q)
(hs₀ : 0 < s₀) (hsmax : s₀ ≤ 1/8) (hη : 0 < η) (hηmax : η ≤ Real.log 2)
(hK : 0 ≤ K) (hγ : 0 < γ) (hd₀ : 0 < d₀) (hq₁ : 1 < q) (hq₂ : q < 2) :
∃ (φ : ℕ → ℕ) (g : (ℝ × Position) × Velocity → ℝ), StrictMono φ ∧
Published.WeakL1Converges ((AnnularEnvelope.timePositionVolume F.time).prod volume)
(fun n => torusPhysical q (F.approximation (φ n)).solution.value) g ∧
MildCollisionSolution F.time (JetPeriodicDatum.datum s₀ η K α γ d₀) g ∧
Published.WeakL1Converges ((AnnularEnvelope.timePositionVolume F.time).prod volume)
(fun n => CollisionCompactness.pairDensity
(torusPhysical q (F.approximation (φ n)).solution.value))
(CollisionCompactness.pairDensity g) ∧
Published.WeakL1Converges ((AnnularEnvelope.timePositionVolume F.time).prod volume)
(fun n => CollisionCompactness.collisionSource (φ n+1)
(torusPhysical q (F.approximation (φ n)).solution.value))
(fun p => CollisionCompactness.gainDensity g p-CollisionCompactness.lossDensity g p) := by
obtain ⟨φ,g,hφ,hg,hr,hpair,hgn,hpairweak⟩ := F.exists_cutoff_equation_limit_with_pair hcarleman hGLPS hDP hq₁ hq₂
let μ : Measure ℝ := volume.restrict (Icc 0 F.time)
let f : ℕ → (ℝ × Position) × Velocity → ℝ :=
fun n => torusPhysical q (F.approximation (φ n)).solution.value
let r : ℕ → (ℝ × Position) × Velocity → ℝ :=
fun n => CollisionCompactness.collisionSource (φ n+1) (f n)
let a : ℕ → Phase → ℝ :=
fun n => periodicInitial s₀ η K α γ d₀ q (φ n) (F.approximation (φ n)).coefficient
have hfm n : Measurable (f n) :=
torusPhysical_measurable (F.approximation (φ n)).solution.measurable_value q
have hrm n : Measurable (r n) := CollisionCompactness.collisionSource_measurable (hfm n) _
have hrint n : Integrable (r n) ((AnnularEnvelope.timePositionVolume F.time).prod volume) :=
memLp_one_iff_integrable.mp (F.toCompactCapFamily.source_ui.memLp (φ n))
have hfr := TorusFlight.weak_regroup hfm hg
have hrr := TorusFlight.weak_regroup hrm hr
have hfa := TorusFlight.weak_along
(fun n => (hfm n).comp MeasurableEquiv.prodAssoc.symm.measurable) hfr
have hra := TorusFlight.weak_along
(fun n => (hrm n).comp MeasurableEquiv.prodAssoc.symm.measurable) hrr
have hrain n : Integrable (TorusFlight.along (TorusFlight.regroup (r n)))
(μ.prod (volume : Measure Phase)) := by
have hrreg : Integrable (TorusFlight.regroup (r n)) (μ.prod (volume : Measure Phase)) := by
have hi := (measurePreserving_prodAssoc μ (volume : Measure Position)
(volume : Measure Velocity)).symm.integrable_comp_of_integrable (hrint n)
rw [Measure.volume_eq_prod]
exact hi
exact (TorusFlight.along_measurePreserving μ).integrable_comp_of_integrable hrreg
have ham n : Measurable (a n) := periodicInitial_measurable s₀ η K α γ d₀ q _ _
have hai n : Integrable (a n) := periodicInitial_integrable hs₀ hsmax hη hηmax hK hγ hd₀
(by linarith : 0 ≤ q) hq₂ α _ _
have ha := (F.toCompactCapFamily.initial_weak_limit hs₀ hsmax hη hηmax hK hγ hd₀
(by linarith : 0 ≤ q) hq₂).comp hφ.tendsto_atTop
have heq n : TorusFlight.along (TorusFlight.regroup (f n)) =ᵐ[μ.prod volume]
(fun z => a n z.2+Published.timePrimitive μ
(TorusFlight.along (TorusFlight.regroup (r n))) z) :=
(F.approximation (φ n)).spacetime_mild hcarleman hq₁ hq₂ F.time_pos.le
have hlim := TorusFlight.weak_duhamel_limit hfa hra
(fun n => (hrm n).comp MeasurableEquiv.prodAssoc.symm.measurable |>.comp
(TorusFlight.along_measurePreserving μ).measurable)
hrain ha ham hai heq
exact ⟨φ,g,hφ,hg,⟨hg.1,hgn,hg.2.1,hpair,hlim⟩,hpairweak,hr⟩
theorem TightCapFamily.exists_mild_limit
(hcarleman : Published.GPV2009CarlemanInput) (hGLPS : Published.GLPS1988Proposition3)
(hDP : Published.DunfordPettis1940Theorem321)
{s₀ η K α γ d₀ q : ℝ} (F : TightCapFamily s₀ η K α γ d₀ q)
(hs₀ : 0 < s₀) (hsmax : s₀ ≤ 1/8) (hη : 0 < η) (hηmax : η ≤ Real.log 2)
(hK : 0 ≤ K) (hγ : 0 < γ) (hd₀ : 0 < d₀) (hq₁ : 1 < q) (hq₂ : q < 2) :
∃ (φ : ℕ → ℕ) (g : (ℝ × Position) × Velocity → ℝ), StrictMono φ ∧
Published.WeakL1Converges ((AnnularEnvelope.timePositionVolume F.time).prod volume)
(fun n => torusPhysical q (F.approximation (φ n)).solution.value) g ∧
MildCollisionSolution F.time (JetPeriodicDatum.datum s₀ η K α γ d₀) g := by
obtain ⟨φ,g,hφ,hg,S,_hpair,_hr⟩ :=
F.exists_mild_limit_with_convergence hcarleman hGLPS hDP
hs₀ hsmax hη hηmax hK hγ hd₀ hq₁ hq₂
exact ⟨φ,g,hφ,hg,S⟩
end JetCopyDatum
end BoltzmannNonuniqueness
end
end
end OAI
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