openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Programs /BalancedEuclidean.lean
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Flow.CompactEuclideanEnergy | |
| import OAI.MathematicalPhysics.AlternatingFlow.DerivativeBounds | |
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.ResidualScaling | |
| namespace OAI | |
| /-! The compact Euclidean solution uses the prescribed coordinate residual. | |
| The coordinate change commutes with all spatial derivatives and with the | |
| right time derivative on the nonnegative time axis. -/ | |
| noncomputable section | |
| namespace ForcedComputation.CompactEuclidean | |
| open ShearFlows Set RapidForcing.CompactEmbedding | |
| open scoped ContDiff | |
| theorem velocity_residual {V : ShearFlows.Velocity} (hV : ContDiff ℝ ∞ V) | |
| (ν : ℝ) {t : ℝ} (ht : 0 ≤ t) (x : AlternatingNS.Space) : | |
| velocity (ForcedComputation.residual ν V) t x = | |
| AlternatingNS.residual ν (velocity V) t x := by | |
| let U := velocity V | |
| have hU := velocity_smooth hV | |
| have ht' := piField_timeDerivative hU ht (piCoordinates x) | |
| have ha := piField_advection U t (piCoordinates x) | |
| have hl := piField_laplacian U t (piCoordinates x) | |
| rw [piField_velocity] at ht' ha hl | |
| simp only [ContinuousLinearEquiv.symm_apply_apply] at ht' ha hl | |
| have ha' : ShearFlows.advection (fun y => V (t,y)) (piCoordinates x) = | |
| piCoordinates (AlternatingNS.advection U t x) := by | |
| rw [AlternatingNS.Bounds.advection_eq_sum] | |
| simpa only [RapidForcing.advection, RapidForcing.spatialD, RapidForcing.basis, | |
| AlternatingNS.dx, AlternatingNS.e] using ha | |
| apply piCoordinates.injective | |
| change ForcedComputation.residual ν V (t,piCoordinates x) = | |
| piCoordinates (AlternatingNS.residual ν U t x) | |
| rw [ForcedComputation.residual, AlternatingNS.residual, map_sub, map_add, map_smul, | |
| ht', ha'] | |
| congr 2 | |
| theorem compact_prescribed_solution {ν : ℝ} (hν : 0 < ν) | |
| {V : ShearFlows.Velocity} (hV : ContDiff ℝ ∞ V) | |
| {K : Set ShearFlows.Space} (hK : IsCompact K) | |
| (hs : ∀ t, Function.support (fun x => V (t,x)) ⊆ K) | |
| (h0 : ∀ x, V (0,x) = 0) (hd : Solenoidal V) : | |
| let U := velocity V | |
| let f := velocity (ForcedComputation.residual ν V) | |
| AlternatingNS.NavierStokes ν f U (fun _ _ => 0) ∧ | |
| AlternatingNS.EnergyClass U (fun _ _ => 0) ∧ | |
| ∀ v p, IsEnergySolutionModuloConstants ν f v p → | |
| ∃ c : ℝ → ℝ, ∀ t, 0 ≤ t → ∀ x, v t x = U t x ∧ p t x = c t := by | |
| dsimp only | |
| obtain ⟨hsol,hE,_⟩ := compact_velocity_solution hν hV hK hs h0 hd | |
| have hsol' : AlternatingNS.NavierStokes ν (velocity (ForcedComputation.residual ν V)) | |
| (velocity V) (fun _ _ => 0) := by | |
| refine ⟨hsol.1,hsol.2.1,?_⟩ | |
| intro t ht x | |
| simpa only [velocity_residual hV ν ht x] using hsol.2.2 t ht x | |
| exact ⟨hsol',hE,fun v p hv => energy_unique_modulo_constants hν.le hsol' hE hv⟩ | |
| end ForcedComputation.CompactEuclidean | |
| end | |
| end OAI | |