The Duhamel heat potential, its spatial derivatives and the Hölder estimates of its Hessian #

For a forcing term f : ℝ → E → F on a finite-dimensional real inner product space E, the Duhamel potential (or volume potential) is

w(t, x) = ∫₀ᵗ (K_{t-s} ⋆ f(s, ·))(x) ds,

where K_τ is the heat kernel. By Duhamel's principle it is the candidate solution of the inhomogeneous heat equation ∂ₜw = Δw + f with zero initial datum, complementing the heat convolution K_t ⋆ g of an initial datum g. This file proves the spatial regularity of each time slice w(t, ·), t > 0, and the temporal regularity of its Hessian, with bounds whose constants depend only on the dimension (and on α below):

if ‖f‖ ≤ M almost everywhere, then ‖w(t, x)‖ ≤ M t;
if f is moreover jointly measurable, w(t, ·) is C¹, its derivative is computed under the time integral, and ‖D w(t, x)‖ ≤ C M √t;
if moreover f(s, ·) is α-Hölder with constant H for almost every 0 < s < t, and α > 0, then w(t, ·) is C², its second derivative is computed under the time integral, and ‖D² w(t, x)‖ ≤ C H t^(α / 2);
if moreover α < 1, then D² w(t, ·) is α-Hölder with constant C H, independently of t;
if instead α < 2, then at each point x the Hessian t ↦ D² w(t, ·)(x) is α/2-Hölder in time on [0, T] with constant C H, independently of T.

The time integrand is singular at s = t. For the first derivative, the bound ‖D(K_τ ⋆ g)‖ ≤ ‖DK_τ‖_{L¹} M = O(M τ^(-1/2)) is integrable in τ. For the second derivative, ‖D²K_τ‖_{L¹} is of order τ⁻¹, which is not integrable. Since ∫ D²K_τ = 0, the value of g at the base point can be subtracted, and Hölder continuity improves the bound to O(H τ^(α/2 - 1)) (TauCeti.norm_iteratedFDeriv_heatKernel_convolution_le_of_holderWith), which is integrable for α > 0. Derivatives are then taken under the time integral with TauCeti.iteratedFDeriv_intervalIntegral. Measurability of the derivatives in the time variable comes from the parabolic scaling D^m K_τ(z) = τ^(-(n + m)/2) D^m K_1(τ^(-1/2) z).

Without the Hölder condition the second derivative of w(t, ·) need not exist, so this is the form in which D² w is controlled by the forcing.

For the Hölder estimate of D² w(t, ·) at two points at distance δ, the times t - δ² < s < t are handled by the bound above, which integrates to O(H δ^α). For s < t - δ² the mean value theorem and the bound ‖D³(K_τ ⋆ g)‖ = O(H τ^((α - 3)/2)), obtained by the same cancellation, give a contribution O(H δ ∫_{δ²}^∞ τ^((α - 3)/2) dτ) = O(H δ^α), which requires α < 1. This is the spatial half of the order-two parabolic Hölder estimate of the Duhamel potential.

The temporal half compares D² w at times t' < t, with h = t - t'. If t' ≤ h, both Hessians are O(H h^(α/2)) by the bound above. Otherwise the times s > t' - h contribute O(H h^(α/2)) by the same bound, while for s < t' - h the slices at elapsed times t' - s and t - s differ by O(H h (t' - s)^((α - 4)/2)) (TauCeti.norm_iteratedFDeriv_heatKernel_convolution_sub_le_of_holderWith, one time derivative of the kernel at the cost of a factor (t' - s)⁻¹), which integrates to O(H h^(α/2)) when α < 2. For 0 < α < 1, the two halves together give the parabolic α-Hölder estimate of D² w.

Main declarations #

TauCeti.heatDuhamelPotential: the Duhamel potential ∫₀ᵗ (K_{t-s} ⋆ f(s, ·))(x) ds.
TauCeti.norm_heatDuhamelPotential_le: the bound ‖w(t, x)‖ ≤ M t.
TauCeti.exists_stronglyMeasurable_eq_iteratedFDeriv_heatKernel_convolution: the derivatives D^m (K_τ ⋆ f(s, ·))(x) agree with a jointly measurable function of (s, τ).
TauCeti.intervalIntegrable_heatKernel_convolution, TauCeti.intervalIntegrable_iteratedFDeriv_heatKernel_convolution, TauCeti.intervalIntegrable_iteratedFDeriv_heatKernel_convolution_of_holderWith: the time integrands of the potential and of its spatial derivatives are interval integrable.
TauCeti.iteratedFDeriv_heatDuhamelPotential, TauCeti.contDiff_heatDuhamelPotential: the first spatial derivative under the time integral, for essentially bounded measurable forcing.
TauCeti.exists_norm_fderiv_heatDuhamelPotential_le: the bound ‖D w(t, x)‖ ≤ C M √t.
TauCeti.iteratedFDeriv_heatDuhamelPotential_of_holderWith, TauCeti.contDiff_heatDuhamelPotential_of_holderWith: the spatial derivatives up to order two under the time integral, for forcing Hölder in space.
TauCeti.exists_norm_iteratedFDeriv_two_heatDuhamelPotential_le: the bound ‖D² w(t, x)‖ ≤ C H t^(α / 2).
TauCeti.exists_holderWith_iteratedFDeriv_two_heatDuhamelPotential: for 0 < α < 1, D² w(t, ·) is α-Hölder with constant C H.
TauCeti.exists_holderOnWith_iteratedFDeriv_two_heatDuhamelPotential: for 0 < α < 2, t ↦ D² w(t, ·)(x) is α/2-Hölder on [0, T] with constant C H.

References #

L. C. Evans, Partial Differential Equations, 2nd ed., §2.3.1 (Duhamel's principle for the nonhomogeneous heat equation).
A. Friedman, Partial Differential Equations of Parabolic Type, Chapter 1, Section 3 (the derivatives of the volume potential of a forcing Hölder continuous in space).
A. Lunardi, Linear and nonlinear diffusion problems, lecture notes, Cortona, 2004, Chapter 4 (Hölder estimates for the heat equation).

The Duhamel heat potential, its spatial derivatives and the Hölder estimates of its Hessian #

Main declarations #

References #

The Hölder estimate of the Hessian #

The temporal Hölder estimate of the Hessian #