File size: 2,746 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 | import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.BalancedFluid
import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.BalancedCoefficientEvaluation
namespace OAI
/-! The compact balanced main theorem includes a terminating evaluator
for every mixed derivative of the prescribed force. -/
noncomputable section
namespace ForcedComputation.Balanced
open ShearFlows Set
open scoped ContDiff
/-- The two finite coefficients prescribe the compact zero-data flow for
every viscosity; named finite queries evaluate every force derivative. -/
theorem balanced_three_stack_realization :
IsCompact BalancedVelocity.support ∧
∀ (I : Alternating.MachineInput) (hI : Alternating.ValidInput I),
(∀ ν : ℝ, force I hI ν = BalancedVelocity.force₀ I hI +
ν • BalancedVelocity.force₁ I hI) ∧
(∀ (α : List (Fin 4)) (b : ℕ → RationalSpaceTime) (y : SpaceTime),
IsFastName b y → ∀ (ε : ℚ) (hε : 0 < ε),
‖mixedDerivative (BalancedVelocity.force₀ I hI) α y -
rationalVector (BalancedExpressions.evaluateCoefficient₀ I hI α b ε hε)‖ ≤ (ε : ℝ) ∧
‖mixedDerivative (BalancedVelocity.force₁ I hI) α y -
rationalVector (BalancedExpressions.evaluateCoefficient₁ I hI α b ε hε)‖ ≤ (ε : ℝ)) ∧
(∀ ν : ℝ, 0 < ν → FluidProperties I hI ν) ∧
(∀ (ν : ℝ) (a : ℕ → ℚ), IsFastRealName a ν →
∀ (α : List (Fin 4)) (b : ℕ → RationalSpaceTime) (y : SpaceTime),
IsFastName b y → ∀ (ε : ℚ) (hε : 0 < ε),
‖mixedDerivative (force I hI ν) α y -
rationalVector (BalancedExpressions.evaluate I hI α a b ε hε)‖ ≤ (ε : ℝ)) ∧
(∀ (ν : ℝ) (a : ℕ → ℚ), IsFastRealName a ν →
∀ (α : List (Fin 4)) (u v M : ℚ) (y : SpaceTime),
y.1 ∈ Ioo (u : ℝ) v →
(∀ j, |timeSpaceCoord j y| ≤ |(M : ℝ)|) →
‖mixedDerivative (force I hI ν) α y‖ ≤
(BalancedExpressions.boxForceBound I hI α u v M a : ℝ)) := by
refine ⟨BalancedVelocity.support_compact,?_⟩
intro I hI
refine ⟨BalancedVelocity.force_affine I hI,?_,
fun ν hν => fluid_properties I hI hν,?_,?_⟩
· intro α b y hb ε hε
exact ⟨BalancedExpressions.evaluateCoefficient₀_spec I hI α hb ε hε,
BalancedExpressions.evaluateCoefficient₁_spec I hI α hb ε hε⟩
· intro ν a ha α b y hb ε hε
exact BalancedExpressions.evaluate_spec I hI α ha hb ε hε
· intro ν a ha α u v M y ht hy
exact BalancedExpressions.boxForceBound_spec I hI α u v M ha y ht hy
end ForcedComputation.Balanced
end
end OAI
|