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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 | 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
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