Euler iff convective Euler
FluidDynamics.euler_iff_convectiveEuler
Plain-language statement
The conservative and convective Euler forms are equivalent when the fields are differentiable enough for the product rules.
Exact Lean statement
theorem euler_iff_convectiveEuler
(d : ℕ) (flow : CauchyFlow d) (pressure : ScalarField d)
(hRhoTime : ∀ t x, DifferentiableAt ℝ (flow.rho · x) t)
(hVelocityTime : ∀ t x, DifferentiableAt ℝ (flow.velocity · x) t)
(hMomentumDensity : ∀ t,
Differentiable ℝ (FluidFlow.momentumDensity d flow.toFluidFlow t))
(hVelocitySpace : ∀ t, Differentiable ℝ (flow.velocity t)) :
Euler d flow pressure ↔ ConvectiveEuler d flow pressureFormal artifact
Lean source
theorem euler_iff_convectiveEuler (d : ℕ) (flow : CauchyFlow d) (pressure : ScalarField d) (hRhoTime : ∀ t x, DifferentiableAt ℝ (flow.rho · x) t) (hVelocityTime : ∀ t x, DifferentiableAt ℝ (flow.velocity · x) t) (hMomentumDensity : ∀ t, Differentiable ℝ (FluidFlow.momentumDensity d flow.toFluidFlow t)) (hVelocitySpace : ∀ t, Differentiable ℝ (flow.velocity t)) : Euler d flow pressure ↔ ConvectiveEuler d flow pressure := by constructor · intro hConservative refine ⟨hConservative.1, ?_, hConservative.2.2⟩ exact (CauchyFlow.cauchyMomentumEquation_iff_convectiveCauchyMomentumEquation d flow hConservative.1 hRhoTime hVelocityTime hMomentumDensity hVelocitySpace).mp hConservative.2.1 · intro hConvective refine ⟨hConvective.1, ?_, hConvective.2.2⟩ exact (CauchyFlow.cauchyMomentumEquation_iff_convectiveCauchyMomentumEquation d flow hConvective.1 hRhoTime hVelocityTime hMomentumDensity hVelocitySpace).mpr hConvective.2.1- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/FluidDynamics/Euler/Basic.lean:65-85
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