Iterates Gradient Material Matrix

Support for the Armstrong–Vicol anomalous-diffusion formalization.

theorem AVenhance.Infra.Section4.iterate_gradient_material_matrix_pairing_integrable {b : ℝ → Homogenization.Vec 2 → Homogenization.Vec 2} {u v : ℝ → Homogenization.Vec 2 → ℝ} {Q : ℝ → Matrix (Fin 2) (Fin 2) ℝ} (hb : ContDiffOn ℝ (↑⊤) (fun (z : AmnrSpace) => b z.1 z.2) (Set.Ici 0 ×ˢ Set.univ)) (hu : ContDiffOn ℝ (↑⊤) (fun (z : AmnrSpace) => u z.1 z.2) (Set.Ici 0 ×ˢ Set.univ)) (hv : ContDiffOn ℝ (↑⊤) (fun (z : AmnrSpace) => v z.1 z.2) (Set.Ici 0 ×ˢ Set.univ)) (hQ : ContinuousOn (fun (z : AmnrSpace) => Q z.1) (Set.Ici 0 ×ˢ Set.univ)) : MeasureTheory.IntegrableOn (fun (z : AmnrSpace) => Homogenization.vecDot (spaceGrad (u z.1) z.2) ((Q z.1).mulVec (spaceGrad (amnrMaterial b v z.1) z.2))) timeCube MeasureTheory.volume

Natural integrability of the vector material-gradient pairing follows componentwise from the exact gradient commutator.