The tensor U built from a mean-free velocity field #
This file records the tensor that paper/ckn.tex attaches to the local energy
class in the proof of Lemma lem:U-bounds (equation eq:Uij): at a point
y of the spatial ball B_ρ = vec3Ball x₀ ρ and at time t,
U_ij(y,t) = -u_i(y,t) * (u_j(y,t) - ⨍_{B_ρ} u_j(·,t)),
where the second factor is the mean-free part of the velocity field, averaged
in space over B_ρ with MeasureTheory.average. Writing
|U| = (∑_{i,j} U_ij²)^{1/2},
the main result U_bounds_of_sobolevPoincare is the two displays
eq:U-bounds of the paper:
(∫_{B_ρ} |U|^{3/2})^{2/3} ≤ C₅ · 𝔲(t) · 𝔤(t);∫_{B_ρ} |U| ≤ (4π/3)^{1/3} C₅ ρ · 𝔲(t) · 𝔤(t),
with 𝔲(t) = (∫_{B_ρ} |u(·,t)|²)^{1/2} and
𝔤(t) = (∫_{B_ρ} |∇u(·,t)|²)^{1/2}. The only analytic input is the
same-ball L⁶ Sobolev–Poincaré inequality (equation
eq:sobolev-poincare-6), taken as an explicit hypothesis on the mean-free
field; everything else is the pointwise rank-one identity |U| = |u| |v|,
Hölder's inequality, and the volume of B_ρ.
The j-th component of u(·,t) with its spatial average over
B_ρ = vec3Ball x₀ ρ subtracted, the mean-free component appearing in
equation eq:Uij of paper/ckn.tex.
Equations
- CKN.meanFreeComponent u x₀ ρ t j y = u (y, t) j - ⨍ (z : CKN.Foundation.Parabolic.Vec3) in CKN.Foundation.Parabolic.vec3Ball x₀ ρ, u (z, t) j
Instances For
The mean-free velocity v = u(·,t) - ⨍_{B_ρ} u(·,t) of equation
eq:Uij in paper/ckn.tex.
Equations
- CKN.meanFreeVec u x₀ ρ t y j = CKN.meanFreeComponent u x₀ ρ t j y
Instances For
The rank-one tensor U_ij = -u_i (u_j - ⨍_{B_ρ} u_j) of equation
eq:Uij in paper/ckn.tex.
Equations
- CKN.utensor u x₀ ρ t i j y = -u (y, t) i * CKN.meanFreeComponent u x₀ ρ t j y
Instances For
The pointwise norm |U| = (∑_{i,j} U_ij²)^{1/2} of the tensor of
equation eq:Uij in paper/ckn.tex.
Equations
- CKN.utensorNorm u x₀ ρ t y = √(∑ i : Fin 3, ∑ j : Fin 3, CKN.utensor u x₀ ρ t i j y ^ 2)
Instances For
U is rank one: |U| is the product of the norms of u(·,t) and of its
mean-free part. This is the pointwise identity used in eq:U-bounds.
Lemma lem:U-bounds of paper/ckn.tex (equation eq:U-bounds). For a
velocity field u whose time slice is in L² and L⁶ on the spatial ball
B_ρ = vec3Ball x₀ ρ, and assuming the same-ball L⁶ Sobolev–Poincaré
inequality (equation eq:sobolev-poincare-6) for the mean-free field, the
tensor U of eq:Uij satisfies
(∫_{B_ρ} |U|^{3/2})^{2/3} ≤ C₅ 𝔲(t) 𝔤(t) and
∫_{B_ρ} |U| ≤ (4π/3)^{1/3} C₅ ρ 𝔲(t) 𝔤(t),
where 𝔲(t) = (∫_{B_ρ} |u(·,t)|²)^{1/2} and
𝔤(t) = (∫_{B_ρ} |∇u(·,t)|²)^{1/2}.
The paper's eq:U-bounds holds for almost every time slice of a suitable
weak solution on a cylinder whose closure lies in its open carrier.