The Duhamel heat potential solves the inhomogeneous heat equation #
For a forcing term f : ℝ → E → F on a finite-dimensional real inner product space E, the
Duhamel potential w(t, x) = ∫₀ᵗ (K_{t-s} ⋆ f(s, ·))(x) ds (TauCeti.heatDuhamelPotential) is
the candidate solution of ∂ₜw = Δw + f with zero initial datum. This file proves that it is a
solution at positive times. Let f be jointly measurable and essentially bounded on (0, T) × E,
with f(s, ·) α-Hölder with constant H for almost every 0 < s < T, where α > 0.
TauCeti.heatDuhamelPotential_eq_integral: Duhamel's principle in integrated form,w(t, x) = ∫₀ᵗ (Δw(σ, x) + f(σ, x)) dσfor0 ≤ t ≤ T.TauCeti.laplacian_heatDuhamelPotential_of_holderWith: the Laplacian under the time integral,Δw(t, x) = ∫₀ᵗ Δ(K_{t-s} ⋆ f(s, ·))(x) ds.TauCeti.continuousOn_laplacian_heatDuhamelPotential: forα < 2,σ ↦ Δw(σ, x)is continuous on[0, T].TauCeti.hasDerivWithinAt_heatDuhamelPotential,TauCeti.hasDerivAt_heatDuhamelPotential: the inhomogeneous heat equation. If moreoverα < 2ands ↦ f(s, x)is continuous att, then∂ₜw(t, x) = Δw(t, x) + f(t, x), within[0, T]fort ∈ [0, T]and two-sided fort ∈ (0, T).TauCeti.hasDerivWithinAt_heatDuhamelPotential_of_holderOnWith: the same on the closed slabE × [0, T]for forcing parabolically Hölder there, which supplies both hypotheses.
Together with the Hölder estimates of the Hessian in
TauCeti.Analysis.PDE.HeatKernel.DuhamelPotential.Basic, this is the time regularity of the
Duhamel potential needed to place it in the parabolic Hölder spaces of a closed slab.
The argument #
The time integrand of the potential is singular at s = t, so the derivative is not computed by
differentiating under the integral. Instead, write v_s(σ) = u_s(σ - s, x), where u_s is the
Cauchy heat potential of the slice f(s, ·) (TauCeti.heatCauchyPotential), so that
v_s(σ) = (K_{σ-s} ⋆ f(s, ·))(x) for σ > s and v_s(σ) = f(s, x) for σ ≤ s. The function
v_s is continuous, by the temporal Hölder estimate of the Cauchy potential; it is constant for
σ < s; and for σ > s its derivative is Δ(K_{σ-s} ⋆ f(s, ·))(x) by the heat equation for the
heat convolution. This derivative is O(H (σ - s)^((α - 2)/2)), which is integrable for α > 0,
so the fundamental theorem of calculus with one exceptional point gives
(K_{t-s} ⋆ f(s, ·))(x) - f(s, x) = ∫₀ᵗ ∂_σ v_s(σ) dσ. Integrating in s and exchanging the
order of integration by Fubini's theorem, the inner integral ∫₀^σ Δ(K_{σ-s} ⋆ f(s, ·))(x) ds is
Δw(σ, x), because the Hessian of w(σ, ·) is computed under the time integral
(TauCeti.iteratedFDeriv_heatDuhamelPotential_of_holderWith). This gives the integrated form.
For the derivative at t, the integrand Δw(·, x) + f(·, x) is continuous at t: the Hessian
of w is Hölder in time (TauCeti.exists_holderOnWith_iteratedFDeriv_two_heatDuhamelPotential),
and the Laplacian is a sum of its values along an orthonormal basis.
References #
- L. C. Evans, Partial Differential Equations, 2nd ed., §2.3.1, Theorem 2 (Duhamel's principle for the nonhomogeneous heat equation).
- A. Friedman, Partial Differential Equations of Parabolic Type, Chapter 1, Section 3 (the volume potential of a forcing Hölder continuous in space).
The Laplacian of the Duhamel potential. Let f be jointly measurable and essentially
bounded on (0, t) × E, with f(s, ·) α-Hölder for almost every 0 < s < t, where α > 0.
Then the Laplacian of w(t, ·) is computed under the time integral:
Δw(t, x) = ∫₀ᵗ Δ(K_{t-s} ⋆ f(s, ·))(x) ds.
Duhamel's principle in integrated form. Let f be jointly measurable and essentially
bounded on (0, t) × E, with f(s, ·) α-Hölder for almost every 0 < s < t, where α > 0.
Then the Duhamel potential w satisfies w(t, x) = ∫₀ᵗ (Δw(σ, x) + f(σ, x)) dσ.
Time continuity of the Laplacian of the Duhamel potential. Let f be jointly measurable
and essentially bounded on (0, T) × E, with f(s, ·) α-Hölder for almost every 0 < s < T,
where 0 < α < 2. Then σ ↦ Δw(σ, x) is continuous on [0, T].
The inhomogeneous heat equation on [0, T]. Let f be jointly measurable and
essentially bounded on (0, T) × E, with f(s, ·) α-Hölder for almost every 0 < s < T,
where 0 < α < 2. If t ∈ [0, T] and s ↦ f(s, x) is continuous at t within [0, T], then
the Duhamel potential w has time derivative Δw(t, x) + f(t, x) at t within [0, T].
The Duhamel potential solves the inhomogeneous heat equation. Let f be jointly
measurable and essentially bounded on (0, T) × E, with f(s, ·) α-Hölder for almost every
0 < s < T, where 0 < α < 2. If s ↦ f(s, x) is continuous at t ∈ (0, T), then the Duhamel
potential w satisfies ∂ₜw(t, x) = Δw(t, x) + f(t, x).
The inhomogeneous heat equation on a closed slab. Let f be jointly measurable and
essentially bounded on (0, T) × E, and parabolically α-Hölder on the slab E × [0, T], where
0 < α < 2. Then at every time t ∈ [0, T] the Duhamel potential w has time derivative
Δw(t, x) + f(t, x) within [0, T], one-sided at the endpoints.