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The heat Cauchy problem for bounded uniformly continuous data

Statement

Assume Countable Choice and let $n \geq 1$ , and let $u_{0} : R^{n} \to C$ be bounded and uniformly continuous. Define $u (x, t) : = \int_{R^{n}} Γ (x - y, t) u_{0} (y) d y$ for $t > 0$ and $u (x, 0) : = u_{0} (x)$ . Then $u$ is $C^{\infty}$ on $R^{n} \times (0, \infty)$ , satisfies $\partial_{t} u = Δ_{x} u$ there, is bounded with $∥ u (\cdot, t) ∥_{\infty} \leq ∥ u_{0} ∥_{\infty}$ , and converges locally uniformly at the initial time: $\sup_{x \in K} ∣ u (x, t) - u_{0} (x) ∣ \to 0$ as $t ↓ 0^{+}$ for every compact $K \subseteq R^{n}$ .

Facts & Assumptions

Given: Countable Choice, $n \geq 1$ , a bounded uniformly continuous $u_{0} : R^{n} \to C$ , a multi-index $α$ , an integer $k \geq 0$ , and $0 < τ < T$ with $K \subseteq R^{n}$ compact.

[A1]

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

[F1]

For $p = \infty$ the heat evolution $H_{t}$ of The heat evolution $H_{t}$ of initial data is defined on bounded measurable data $g$ by the everywhere absolutely convergent integral $(H_{t} g) (x) = \int_{R^{n}} Γ (x - y, t) g (y) d y$ , with $∥ H_{t} g ∥_{\infty} \leq ∥ g ∥_{\infty}$ , and $H_{0} g = g$ .

[F2]

For every $t > 0$ the kernel is $C^{\infty}$ with $\partial_{t} Γ = Δ_{x} Γ$ , for every multi-index $β$ there is $C_{n, β} < \infty$ with $∣ D^{β} Γ (z, t) ∣ \leq C_{n, β} t^{- (n + ∣ β ∣) / 2} e^{- ∣ z ∣^{2} / (8 t)}$ , the unit-mass identity $\int Γ (z, t) d z = 1$ holds, and for every $δ > 0$ $\int_{∣ z ∣ > δ} Γ (z, t) d z \to 0$ as $t ↓ 0^{+}$ (Normalisation, parabolic scaling, heat equation and derivative bounds for the heat kernel).

[F3]

For $f \in L^{p}$ , $1 \leq p \leq \infty$ , the absolutely convergent heat convolution is $C^{\infty}$ in space and time for $t > 0$ , every derivative passes through the integral, and $u_{t} = Δ_{x} u$ (Spatial and time derivatives pass through heat convolution for positive time).

[F4]

If $(K_{ε})$ is an $L^{1}$ approximate identity and $g$ is bounded and continuous, then $(g * K_{ε}) (x) \to g (x)$ uniformly on compact sets ( $L^{1}$ approximate identities converge uniformly on compacta for bounded continuous functions).

Proof

technique · direct

1.1A1F1F2given

Work under [A1] and put $M : = ∥ u_{0} ∥_{\infty} < \infty$ . For every $x$ and $t > 0$ the integral defining $u (x, t)$ is absolutely convergent with $∣ u (x, t) ∣ \leq M \int Γ (x - y, t) d y = M$ by unit mass [F2], so $u$ is a bounded function with $∥ u (\cdot, t) ∥_{\infty} \leq M = ∥ u_{0} ∥_{\infty}$ and $u (\cdot, 0) = u_{0}$ by the definition of $H_{0}$ in [F1].

2.1step 1.1F3given

Since $u_{0}$ is bounded and measurable, it represents an $L^{\infty}$ class. Applying [F3] with $p = \infty$ gives the $C^{\infty}$ representative $u$ on $R^{n} \times (0, \infty)$ and the heat equation $\partial_{t} u = Δ_{x} u$ .

3.1step 2.1F2F4given

Initial convergence: by the evenness of $Γ (\cdot, t)$ in its first argument, $u (x, t) = \int_{R^{n}} Γ (x - y, t) u_{0} (y) d y = \int_{R^{n}} Γ (y - x, t) u_{0} (y) d y = (u_{0} * Γ_{t}) (x)$ for every $t > 0$ ; the datum $u_{0}$ is bounded and continuous and the family $(Γ_{t})_{t > 0}$ is an $L^{1}$ approximate identity by [F2], so the published approximate-identity corollary [F4] gives $\sup_{x \in K} ∣ u (x, t) - u_{0} (x) ∣ \to 0$ as $t ↓ 0^{+}$ for every compact $K \subseteq R^{n}$ , which is the asserted local uniform convergence.

4.1step 1.1step 2.1step 3.1given∎

Steps 1.1, 2.1 and 3.1 prove boundedness with the stated sup-norm bound, $C^{\infty}$ smoothness and the heat equation at positive time, and locally uniform recovery of the initial datum, which is the whole statement.

Depends on

Used by

The heat flow need not converge in supremum norm Counterexample
Heat generator at zero on compactly supported smooth data Lemma

Dependency tree · two levels

40 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John K. Hunter, Notes on Partial Differential Equations (revised 18 June 2014, UC Davis) (standard reference, not scraped)
Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript, AMS Graduate Studies in Mathematics) (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)