The Navier–Stokes forcing of the short-time construction #
For a velocity U on the slab ℝ³ × (0, σ) the construction of
prop:pv-local-solution uses the quadratic tensor U ⊗ U and its canonical
Riesz pressure, both cut off to the slab, and the forcing tensor
-(U ⊗ U + p I). The Duhamel map adds the forced heat response of this tensor
to the heat orbit of the datum. This file sets up these objects and their
integrability.
The quadratic tensor U ⊗ V, cut off to the slab ℝ³ × (0, σ).
Equations
- ESS.pvSlabTensor σ U V i j = (CKN.spaceTimeSet Set.univ (Set.Ioo 0 σ)).indicator fun (z : CKN.Foundation.Parabolic.ParabolicPoint) => U z i * V z j
Instances For
The canonical pressure of U ⊗ U (the double Riesz transform at exponent
two), cut off to the slab.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The Navier–Stokes forcing tensor -(U ⊗ U + p I) on the slab.
Equations
- ESS.pvSlabForce σ U i j z = -(ESS.pvSlabTensor σ U U i j z + if i = j then ESS.pvSlabPressure σ U z else 0)
Instances For
The Duhamel map of prop:pv-local-solution: heat orbit of the datum plus
the forced heat response of the Navier–Stokes forcing.
Equations
- ESS.pvLocalMap a σ U z = ESS.heatOrbit a z + ESS.forcedHeat (ESS.pvSlabForce σ U) z
Instances For
On the slab, the pressure is the canonical Riesz pressure of U ⊗ U.
Hölder's inequality for one entry of the cut-off quadratic tensor.
Each entry of the cut-off quadratic tensor lies in L^r for velocities in
L^p of the slab, 2/p = 1/r.
The difference of two quadratic tensors splits into two bilinear terms.