The Cauchy heat potential, its derivatives and its parabolic Hölder estimate #
For a suitable initial datum g, for instance a bounded continuous one, the classical solution of
the heat equation ∂ₜu = Δu on E × [0, ∞) with initial datum g is the Cauchy potential
u(t, x) = (K_t ⋆ g)(x), where K_t is the heat kernel. Substituting z = √t • y in the
convolution integral writes it as
u(t, x) = ∫ K_1(y) g(x - √t • y) dy,
an average of translates of g against the fixed probability density K_1. This formula is the
definition TauCeti.heatCauchyPotential used here. It makes sense for every real t:
TauCeti.heatCauchyPotential_eq_convolution identifies it with K_t ⋆ g for t > 0, and when F
is complete TauCeti.heatCauchyPotential_of_nonpos gives u(t, ·) = g for t ≤ 0, because K_1
has mass one. If the translates of g are not integrable against K_1, the Bochner integral takes
the junk value 0, so the potential is meaningful only under integrability assumptions, such as
the continuity and Hölder hypotheses of the estimates below
(TauCeti.integrable_heatKernel_one_smul_sub_smul).
In this form the Hölder estimates are immediate. If g is α-Hölder with constant H, then for
all t, s and x, x'
‖g(x - √t • y) - g(x' - √s • y)‖ ≤ H (‖x - x'‖ + |√t - √s| ‖y‖)^α,
and |√t - √s| ≤ √|t - s|. Integrating against K_1 shows that u is α-Hölder with respect to
the parabolic distance ‖x - x'‖ + √|t - s|, with constant H (1 + M_α), where
M_α = ∫ ‖y‖^α K_1(y) dy is TauCeti.heatKernelMoment. The constant depends only on H, α and
the dimension, and in particular not on the time. Separately, every time slice u(t, ·) is
α-Hölder with the constant H of the datum, and u is α / 2-Hölder in time with constant
H M_α; the latter bounds the rate ‖u(t, ·) - g‖ ≤ H M_α t^(α/2) of attainment of the initial
datum.
These are the estimates behind the top-order seminorms of the Cauchy potential in the parabolic
space C^{2+α,1+α/2}, where they are applied to the second spatial derivatives and the Laplacian
of the datum. For this the derivatives pass under the integral sign: if g has a bounded
continuous derivative, then D_x u is the Cauchy potential of Dg, so for a C² datum with
bounded derivatives Δu(t, ·) is the Cauchy potential of Δg. If g is moreover bounded, then
∂ₜu = Δu for t > 0, and the one-sided time derivative at t = 0 is Δg.
Main declarations #
TauCeti.heatCauchyPotential: the Cauchy potential∫ K_1(y) g(x - √t • y) dy.TauCeti.heatCauchyPotential_of_nonpos,TauCeti.heatCauchyPotential_zero: the initial datum.TauCeti.heatCauchyPotential_eq_convolution,TauCeti.heatCauchyPotential_eq_convolution_fun: agreement withK_t ⋆ gfort > 0, pointwise and as functions.TauCeti.norm_heatCauchyPotential_le: the maximum principle‖u‖ ≤ sup ‖g‖.TauCeti.holderWith_heatCauchyPotential: the parabolic Hölder estimate.HolderWith.heatCauchyPotential: the spatial Hölder estimate of each time slice.TauCeti.norm_heatCauchyPotential_sub_heatCauchyPotential_le: the temporal Hölder estimate.TauCeti.norm_heatCauchyPotential_sub_le: the rate of attainment of the initial datum.TauCeti.heatCauchyPotential_add,TauCeti.heatCauchyPotential_smul: linearity in the datum.TauCeti.continuous_heatCauchyPotential: joint continuity in time and space for bounded continuous data.TauCeti.hasFDerivAt_heatCauchyPotential,TauCeti.fderiv_heatCauchyPotential: the spatial derivative is the Cauchy potential of the derivative.TauCeti.laplacian_heatCauchyPotential: the Laplacian is the Cauchy potential of the Laplacian.TauCeti.hasDerivAt_heatCauchyPotential: the heat equation∂ₜu = Δufort > 0, andTauCeti.hasDerivAt_heatCauchyPotential_of_aestronglyMeasurable, its form for merely measurable and essentially bounded data.TauCeti.hasDerivWithinAt_heatCauchyPotential: the time derivative within[0, ∞), including the one-sided derivativeΔgat the initial time.TauCeti.exists_norm_laplacian_heatCauchyPotential_le: for Hölder data, the Laplacian at timet > 0, hence the time derivative, isO(H t^(α/2 - 1)).TauCeti.exists_norm_heatCauchyPotential_sub_heatCauchyPotential_le_mul: for Hölder data, the potential is Lipschitz in time away fromt = 0, with constantO(|t|^(α/2 - 1)).
References #
- A. Friedman, Partial Differential Equations of Parabolic Type, Chapter 1 (the Cauchy problem for the heat equation).
- A. Lunardi, Linear and nonlinear diffusion problems, lecture notes, Cortona, 2004, Chapter 4 (the heat equation in parabolic Hölder spaces).
The Cauchy heat potential of g, u(t, x) = ∫ K_1(y) g(x - √t • y) dy. For t > 0 it is
the heat convolution (K_t ⋆ g)(x) (TauCeti.heatCauchyPotential_eq_convolution), which for
suitable data is the classical solution of the heat equation with initial datum g
(TauCeti.hasDerivAt_heatCauchyPotential). When F is complete it is g(x) for t ≤ 0
(TauCeti.heatCauchyPotential_of_nonpos). If the translates of g are not integrable against
K_1, the integral takes the junk value 0.
Instances For
The defining formula of the Cauchy heat potential.
The initial datum. For t ≤ 0 the Cauchy heat potential of g is g.
The initial datum. At time zero the Cauchy heat potential of g is g.
For t > 0 the Cauchy heat potential is the heat convolution K_t ⋆ g.
For t > 0 the time slice u(t, ·) of the Cauchy heat potential is the heat convolution
K_t ⋆ g, as a function.
The maximum principle. If ‖g‖ ≤ M everywhere, then ‖u(t, x)‖ ≤ M for the Cauchy heat
potential u of g, at every time t and point x.
A Hölder function grows at most like ‖y‖^α along each line through x, so its translates
are integrable against K_1.
The parabolic Hölder estimate of the Cauchy heat potential. If g is continuous and
α-Hölder with constant H, its Cauchy potential is α-Hölder on parabolic space-time, whose
distance is ‖x - x'‖ + √|t - s|, with constant H (1 + M_α). Here
M_α = ∫ ‖y‖^α K_1(y) dy is TauCeti.heatKernelMoment, which depends only on α and the
dimension. Continuity is automatic when α > 0 (HolderWith.continuous).
The spatial Hölder estimate of the Cauchy heat potential. If g is continuous and
α-Hölder with constant H, then so is every time slice of its Cauchy potential, with the same
constant.
The temporal Hölder estimate of the Cauchy heat potential. If g is continuous and
α-Hölder with constant H, its Cauchy potential u satisfies
‖u(t, x) - u(s, x)‖ ≤ H M_α |t - s|^(α/2), where M_α = ∫ ‖y‖^α K_1(y) dy is
TauCeti.heatKernelMoment.
Attainment of the initial datum at a rate. If g is continuous and α-Hölder with
constant H, its Cauchy potential u satisfies ‖u(t, x) - g(x)‖ ≤ H M_α t^(α/2) for t ≥ 0,
where M_α = ∫ ‖y‖^α K_1(y) dy is TauCeti.heatKernelMoment.
Linearity, continuity and derivatives #
The translates of a bounded continuous function are integrable against K_1.
The Cauchy heat potential is additive on bounded continuous data.
The Cauchy heat potential commutes with scalar multiplication.
The Cauchy heat potential of a bounded continuous function is jointly continuous in time and
space, including across t = 0.
Spatial derivative of the Cauchy heat potential. If g has a continuous bounded derivative
g', then the spatial derivative of its Cauchy potential is the Cauchy potential of g'.
The spatial derivative of the Cauchy potential of a C¹ function with bounded derivative is
the Cauchy potential of its derivative.
The Laplacian of the Cauchy heat potential. For a C² function with bounded first and
second derivatives, the Laplacian of each time slice of the Cauchy potential is the Cauchy
potential of the Laplacian.
The heat equation for rough data. For g measurable and essentially bounded and t > 0,
the time derivative of the Cauchy potential is the Laplacian of its time slice,
∂ₜu(t, x) = Δ (u(t, ·)) x. No regularity of the datum is needed at positive times.
The Cauchy heat potential solves the heat equation. For a bounded C² function with
bounded first and second derivatives and t > 0, the time derivative of the Cauchy potential is
the Cauchy potential of the Laplacian, which is also its spatial Laplacian
(TauCeti.laplacian_heatCauchyPotential).
The heat equation up to the initial time. For a bounded C² function with bounded first
and second derivatives, the Cauchy potential has time derivative heatCauchyPotential (Δ g) t x
within [0, ∞) at every t ≥ 0. At t = 0 this is the one-sided derivative Δ g x.
The time-Lipschitz bound away from the initial time #
The Laplacian of the Cauchy potential of Hölder data. There is a constant C ≥ 0,
depending only on α and the dimension, with the following property. If g is measurable,
essentially bounded and α-Hölder with constant H, then for t > 0,
‖Δu(t, ·)(x)‖ ≤ C H t^(α/2 - 1). By the heat equation
(TauCeti.hasDerivAt_heatCauchyPotential_of_aestronglyMeasurable) this also bounds the time
derivative. The bound on g does not enter the estimate.
The time-Lipschitz bound of the Cauchy potential. For α ≤ 2 there is a constant C,
depending only on α and the dimension, with the following property. If g is bounded,
continuous and α-Hölder with constant H, then for every a ≠ 0 and every b,
‖u(b, x) - u(a, x)‖ ≤ C H |b - a| |a|^(α/2 - 1).
So the potential is Lipschitz in time near every a ≠ 0, with a constant that blows up at most
like |a|^(α/2 - 1) as a → 0; the time derivative at positive times is O(H t^(α/2 - 1))
(TauCeti.exists_norm_laplacian_heatCauchyPotential_le). The bound on g does not enter the
estimate.