The frequency-wise energy identity of the forced mild equation #
In frequency variables the forced mild equation eq:reg-mild-forced is, at
each frequency ξ, the damped Duhamel formula
Y(ξ,t) = e^{-λt} B(ξ) + ∫₀ᵗ e^{-λ(t-s)} ℙ(ξ) g(ξ,s) ds with λ = 4π²|ξ|²,
source g = -2πi ξ·F̂ + Fourier(H), and the Leray symbol ℙ(ξ). This file defines
the jointly measurable frequency representative Y and proves, frequency by
frequency, the energy identity of the damped formula; the Leray symbol drops
out of the source pairing because Y(ξ,s) lies in its range.
The heat decay rate 4π²|ξ|² at a frequency.
Instances For
The heat symbol is the exponential of minus the decay rate times the time.
The frequency source of the forced mild equation: minus 2πi times the
tensor divergence symbol applied to the tensor, plus the force.
Equations
- CKN.Leray.forcedFourierSource Fh Hh p = -((2 * ↑Real.pi * Complex.I) • (CKN.Leray.tensorDivergenceLinear p.1) (Fh p)) + Hh p
Instances For
The frequency representative of the forced mild equation.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The divergence part of the frequency source is jointly measurable.
The frequency integrand of the representative is jointly measurable.
The frequency representative is jointly strongly measurable.
The pairing of a vector with the divergence part of the source is bounded by half of the weighted squared norms.
At a frequency where the source is integrable in time, the representative satisfies the damped energy identity with the unprojected source.
The energy identity of the forced mild equation in frequency variables:
integrating the frequency-wise identity gives
∫|Y(t)|² + 2∫λ∫₀ᵗ|Y|² = ∫|B|² + 2∫₀ᵗ∫Re⟨Y, -2πi ξ·F̂⟩ + 2∫₀ᵗ∫Re⟨Y, Fourier(H)⟩,
and the weighted dissipation is integrable in frequency and time.