The Navier–Stokes scaling transports a classical solution #
The scaling u ↦ R u(R·, R²·), π ↦ R² π(R·, R²·), f ↦ R³ f(R·, R²·) — the rescaled
fields CKN.rescaleVelocity, CKN.rescalePressure, CKN.rescaleForce of the parabolic
dilation CKN.scalingParabolic — carries a classical solution of eq:nse:forced on a
space-time set S to a classical solution on the preimage of S.
The three derivative identities behind it are unconditional: for every scalar c and every
dilation parameter μ,
∂_i (c · g ∘ T) = c μ (∂_i g) ∘ T,∂_t (c · g ∘ T) = c μ² (∂_t g) ∘ T,∂_j ∂_i (c · g ∘ T) = c μ² (∂_j ∂_i g) ∘ T,
where T = CKN.scalingParabolic μ z₀. No smoothness is needed for them: the spatial and
temporal slices of g ∘ T are reparametrisations of the slices of g by an affine map, and
the composition rules fderiv_comp_smul, fderiv_comp_add_left and fderiv_const_smul_field
that they use hold with no differentiability hypothesis. Every term of the momentum equation
therefore picks up the same factor μ³, and the divergence picks up μ².
The same rule iterated gives the meridional derivatives of eq:interior:mean:bounds: a
derivative of order (a, b) of the rescaled velocity carries the factor μ^(1+a+b), which is
exactly the prefactor that CIV.rpow_scaled_le_of_radialAnisoBound and
CIV.rpow_scaled_le_of_swirlVerticalAnisoBound convert into the rescaled constants.
The last theorem is the form used in cor:interior:nonanalytic: for 0 < R ≤ R' and
R² ≤ δ', the scaling carries a classical solution on B(R') × (-δ', 0) to one on the unit
parabolic cylinder CIV.unitCylinder, which is the cylinder the hypotheses of thm:main
are read on.
Derivatives under an affine reparametrisation #
Derivatives under the parabolic rescaling #
A spatial derivative of c · g ∘ T, for T the parabolic dilation of ratio μ at z₀,
is c μ times the same derivative of g at the dilated point.
The time derivative of c · g ∘ T, for T the parabolic dilation of ratio μ at z₀,
is c μ² times the time derivative of g at the dilated point.
A second spatial derivative of c · g ∘ T, for T the parabolic dilation of ratio μ
at z₀, is c μ² times the same derivative of g at the dilated point: the first
derivative already has the shape to which the first-order rule applies again.
Meridional derivatives under the parabolic rescaling #
The meridional derivative ∂_r^a ∂_z^b of c · g ∘ T is c μ^(a+b) times the same
derivative of g at the dilated point, for all orders a and b.
The prefactor of the rescaled anisotropic bounds eq:interior:mean:bounds: a meridional
derivative of order (a, b) of the rescaled velocity is μ^(1+a+b) times the same derivative
of the velocity at the dilated point. This is the left-hand factor R^(1+a+b) of
CIV.rpow_scaled_le_of_radialAnisoBound and of
CIV.rpow_scaled_le_of_swirlVerticalAnisoBound, which convert it into the rescaled
constants.
The transport of eq:nse:forced #
The Navier–Stokes scaling of the footnote to eq:nse:forced, u ↦ R u(R·, R²·),
π ↦ R² π(R·, R²·), f ↦ R³ f(R·, R²·) based at z₀, carries a classical solution on S
to a classical solution on the preimage of S under the dilation. Every term of the momentum
equation is multiplied by μ³ and the divergence by μ², so both equations are preserved
exactly; the identity therefore needs no restriction on μ, and the case used in
cor:interior:nonanalytic is 0 < R ≤ 1.
The form of the scaling used in cor:interior:nonanalytic: a classical solution of
eq:nse:forced on B(R') × (-δ', 0) is carried to a classical solution on the unit parabolic
cylinder by the dilation of ratio R, for every 0 < R ≤ R' with R² ≤ δ'.