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Heat generator at zero on compactly supported smooth data

Statement

Assume Countable Choice. For $n \geq 1$ , $φ \in C_{c}^{\infty} (R^{n})$ and $1 \leq p < \infty$ , $∥ (H_{t} φ - φ) / t - Δ φ ∥_{p} \to 0$ as $t ↓ 0$ .

Facts & Assumptions

Given: Countable Choice, $n \geq 1$ , $φ \in C_{c}^{\infty} (R^{n})$ , $1 \leq p < \infty$ , and $0 < ε < t$ .

[A1]

Countable Choice is the hypothesis carried by the differentiation, integration and evolution suppliers below (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

$C_{c}^{\infty} (R^{n})$ is the test-function space of Test function space d of an open set; $φ$ and every derivative of it is smooth with compact support, so $Δ φ \in C_{c}^{\infty} (R^{n})$ and in particular $Δ φ \in L^{p} (R^{n})$ . In this item $H_{s} ψ$ denotes the evolution of the class $ψ$ as in The heat evolution $H_{t}$ of initial data.

[F2]

For $s > 0$ the function $H_{s} φ$ is $C^{\infty}$ on $R^{n}$ and every spatial and time derivative passes through the convolution, $D^{α} \partial_{s}^{k} H_{s} φ = (D^{α} \partial_{s}^{k} Γ_{s}) * φ$ (Spatial and time derivatives pass through heat convolution for positive time).

[F3]

The kernel satisfies $\partial_{s} Γ = Δ_{x} Γ$ on $R^{n} \times (0, \infty)$ (Normalisation, parabolic scaling, heat equation and derivative bounds for the heat kernel).

[F4]

For continuous $F, G$ on $[a, b]$ differentiable on $(a, b)$ with $F^{'} = f$ , $G^{'} = g$ Riemann integrable there, $\int_{a}^{b} F g + \int_{a}^{b} f G = F (b) G (b) - F (a) G (a)$ (Integration by parts for continuous factors with Riemann-integrable extensions of their interior derivatives).

[F5]

If $G$ is continuous on $[a, b]$ , differentiable on $(a, b)$ and $f = G^{'}$ is Riemann integrable, then $\int_{a}^{b} f = G (b) - G (a)$ (Newton–Leibniz needs only continuity on $[a, b]$ , differentiability on $(a, b)$ , and a Riemann-integrable extension of the interior derivative).

[F6]

For $1 \leq p < \infty$ and $ψ \in L^{p} (R^{n})$ , $∥ H_{s} ψ - ψ ∥_{p} \to 0$ as $s ↓ 0^{+}$ , and $∥ H_{s} ψ ∥_{p} \leq ∥ ψ ∥_{p}$ (The heat Cauchy problem for $L^{p}$ data).

[F7]

Let $1 \leq p < \infty$ and let $F (x, s)$ be measurable with $\int_{0}^{t} ∥ F (\cdot, s) ∥_{L^{p}} d s < \infty$ ; then $∥ \int_{0}^{t} ∣ F (x, s) ∣ d s ∥_{L^{p}} \leq \int_{0}^{t} ∥ F (\cdot, s) ∥_{L^{p}} d s$ (Minkowski's integral inequality).

[F8]

Bounded uniformly continuous data are recovered locally uniformly at $t = 0$ by their heat convolution (The heat Cauchy problem for bounded uniformly continuous data). This applies to both $φ$ and $Δ φ$ , which are smooth with compact support.

Proof

technique · direct

1.1A1F1F2F3F4given

Derivative identity: fix $x \in R^{n}$ and $s > 0$ . By [F2] with $α = 0$ , $k = 1$ and $f = φ$ the function $s \mapsto H_{s} φ (x)$ is differentiable with $\partial_{s} H_{s} φ (x) = \int \partial_{s} Γ (x - y, s) φ (y) d y$ ; by [F3] and $\partial_{x_{i}} \partial_{x_{i}} Γ (x - y, s) = \partial_{y_{i}} \partial_{y_{i}} Γ (x - y, s)$ , applying the scalar integration by parts [F4] twice in each coordinate (the boundary terms vanish because $φ$ has compact support, and this works for every $n \geq 1$ including $n = 1$ ) turns the last integral into $\int Γ (x - y, s) Δ φ (y) d y = H_{s} Δ φ (x)$ ; hence $\partial_{s} H_{s} φ (x) = H_{s} Δ φ (x)$ for every $x$ and $s > 0$ .

2.1step 1.1F5given

Newton–Leibniz: by step 1.1 the map $s \mapsto H_{s} φ (x)$ is continuous on $[ε, t]$ with derivative $H_{s} Δ φ (x)$ on $(ε, t)$ , for its real and imaginary parts separately; the fundamental theorem [F5] applied on $[ε, t]$ gives $H_{t} φ (x) - H_{ε} φ (x) = \int_{ε}^{t} H_{s} Δ φ (x) d s$ for every $x$ .

3.1step 2.1F1F2F7F8given

Both $H_{ε} φ (x) \to φ (x)$ and $H_{s} Δ φ (x) \to Δ φ (x)$ hold at every $x$ by [F8]. Thus the integrand in step 2.1 extends continuously to $s = 0$ , and letting $ε ↓ 0$ gives $H_{t} φ (x) - φ (x) = \int_{0}^{t} H_{s} Δ φ (x) d s$ . Dividing by $t$ and subtracting $Δ φ (x)$ yields $\frac{H_{t} φ (x) - φ (x)}{t} - Δ φ (x) = \frac{1}{t} \int_{0}^{t} (H_{s} Δ φ (x) - Δ φ (x)) d s$ . The integrand is jointly continuous for $s > 0$ by [F2] applied to $Δ φ$ , hence measurable as required for [F7].

4.1step 3.1F6F7given

$L^{p}$ bound and limit: applying the Minkowski integral inequality [F7] to $F (x, s) : = H_{s} Δ φ (x) - Δ φ (x)$ on $R^{n} \times (0, t)$ — whose hypothesis holds because $∥ F (\cdot, s) ∥_{p} \leq 2 ∥ Δ φ ∥_{p}$ by the contraction clause of [F6] — gives $∥ \frac{H_{t} φ - φ}{t} - Δ φ ∥_{p} \leq \frac{1}{t} \int_{0}^{t} ∥ H_{s} Δ φ - Δ φ ∥_{p} d s$ . Given $η > 0$ , the strong convergence clause of [F6] applied to $Δ φ \in L^{p}$ supplies $δ > 0$ with $∥ H_{s} Δ φ - Δ φ ∥_{p} < η$ for every $0 < s < δ$ ; for every $0 < t < δ$ the right-hand side is then at most $η$ . Hence $∥ (H_{t} φ - φ) / t - Δ φ ∥_{p} \to 0$ as $t ↓ 0^{+}$ .

5.1step 1.1step 4.1given∎

Steps 1.1, 2.1, 3.1 and 4.1 prove the stated generator limit for compactly supported smooth data in every $L^{p}$ , $1 \leq p < \infty$ .

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Sources

John K. Hunter, Notes on Partial Differential Equations (revised 18 June 2014, UC Davis) (standard reference, not scraped)
Jared Speck, MIT 18.152 Introduction to Partial Differential Equations, Class Meeting #5: The Fundamental Solution for the Heat Equation (Fall 2011) (standard reference, not scraped)