Source facts on a finite slab for lem:pv-positive-bound #
For a Leray–Hopf solution on ℝ³ × (0, T) with a pressure p whose spatial
L^{3/2} norms are essentially bounded in time, this file records the facts on
the whole slab ℝ³ × (0, T) that feed the rescaling step of
lem:pv-positive-bound: a Tonelli bound for slab integrals of functions with
uniformly bounded slices, the global L^{3/2} membership of the pressure, the
CKN data clauses on every compact box of the slab, and the space-time form of
the divergence-free condition.
Tonelli bound on a slab: if almost every time slice of a nonnegative integrand has spatial integral at most K, the slab integral is at most |J| K.
A pressure with essentially bounded spatial L^{3/2} norms lies in L^{3/2} of the finite slab ℝ³ × (0, T).
A Leray–Hopf solution together with a slab L^{3/2} pressure satisfies the CKN data clauses on every compact box of ℝ³ × (0, T).
The time slice of a whole-space test function, as a spatial test function.
Equations
- ESS.pvSliceTest hψ t = { toFun := fun (x : CKN.Foundation.Parabolic.Vec3) => ψ (x, t), contDiff := ⋯, hasCompactSupport := ⋯, tsupport_subset := ⋯ }
Instances For
The divergence-free clause of a Leray–Hopf solution in space-time form on the slab ℝ³ × (0, T), with the integrability of the tested integrand on the support of the test function.