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Mass conservation and positivity of the heat flow

Statement

Assume Countable Choice, let $n \geq 1$ , and let $H_{t}$ be the heat evolution of The heat evolution $H_{t}$ of initial data. (i) If $f \in L^{1} (R^{n})$ then $\int_{R^{n}} H_{t} f (x) d x = \int_{R^{n}} f (x) d x$ for every $t > 0$ . (ii) If $f \in L^{p} (R^{n})$ , $1 \leq p \leq \infty$ , satisfies $f \geq 0$ almost everywhere, then $H_{t} f \geq 0$ almost everywhere for every $t > 0$ .

Facts & Assumptions

Given: Countable Choice, $n \geq 1$ , $t > 0$ , and data $f$ in the class named in the respective clause.

[A1]

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

[F1]

For $t > 0$ the heat kernel is positive and has $\int_{R^{n}} Γ (x, t) d x = 1$ (Normalisation, parabolic scaling, heat equation and derivative bounds for the heat kernel).

[F2]

For $1 \leq p \leq \infty$ , $t > 0$ and $f \in L^{p} (R^{n})$ , the heat evolution $H_{t} f$ is the $L^{p}$ class of the almost-everywhere defined function $x \mapsto \int_{R^{n}} Γ (x - y, t) f (y) d y$ (The heat evolution $H_{t}$ of initial data).

[F3]

On sigma-finite product spaces Tonelli's theorem gives $\int_{X \times Y} f d (μ \times ν) = \int_{X} \int_{Y} f_{x} d ν d μ$ for nonnegative product-measurable $f$ (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product).

[F4]

On sigma-finite product spaces Fubini's theorem gives the same iterated equality for $f \in L^{1} (μ \times ν)$ , the sections being integrable almost everywhere (Fubini's theorem for L^1 functions on a sigma-finite product).

[F5]

Under $R^{m + n} = R^{m} \times R^{n}$ the Lebesgue measure $λ_{m + n}$ is the completion of the product measure $λ_{m} \times λ_{n}$ (The Euclidean Lebesgue measure is the completion of the product of the factor Lebesgue measures).

[F6]

If a measure-preserving $T$ and an integrable $f$ are given, then $\int f \circ T d μ = \int f d μ$ (Integral invariance under measure-preserving maps); for each fixed $y$ the translation $x \mapsto x + y$ preserves Lebesgue measure.

[F7]

$\int_{R^{n}} ∣ f ∣ d λ_{n}$ denotes the $L^{1}$ norm of The class $L^{1} (μ)$ of integrable functions.

Proof

technique · direct

1.1A1F1F2given

Work under [A1] and fix $t > 0$ . By [F2] the evolution $H_{t} f$ is the $L^{p}$ class of the representative $u (x) = \int_{R^{n}} Γ (x - y, t) f (y) d y$ , defined for almost every $x$ ; by [F1] the kernel is positive with unit mass.

2.1step 1.1F1F3F4F5F6F7given

Mass conservation: assume $f \in L^{1} (R^{n})$ , so that by [F2] and [F1] the function $(x, y) \mapsto ∣ f (y) ∣ Γ (x - y, t)$ is nonnegative and measurable. The identification [F5] makes the integral over $R^{2 n}$ the completed product integral, so Tonelli [F3] gives $\int_{R^{n}} \int_{R^{n}} ∣ f (y) ∣ Γ (x - y, t) d y d x = \int_{R^{n}} ∣ f (y) ∣ (\int_{R^{n}} Γ (x - y, t) d x) d y = \int_{R^{n}} ∣ f (y) ∣ d y < \infty$ , the inner integral being $1$ for every $y$ by the translation invariance [F6] and the unit mass of [F1]; hence $(x, y) \mapsto f (y) Γ (x - y, t)$ lies in $L^{1} (λ_{2 n})$ and Fubini [F4] gives $\int_{R^{n}} u (x) d x = \int_{R^{n}} f (y) (\int_{R^{n}} Γ (x - y, t) d x) d y = \int_{R^{n}} f (y) d y$ , which is (i), the integral of the class $H_{t} f$ being computed from its representative $u$ .

2.2step 1.1F1F2given

Positivity: assume $f \in L^{p} (R^{n})$ satisfies $f \geq 0$ almost everywhere and let $N$ be the null set where $f < 0$ . For every $x$ at which the defining integral converges, the function $y \mapsto Γ (x - y, t) f (y)$ is $\geq 0$ for every $y \notin N$ , because $Γ > 0$ by [F1]; a function that is nonnegative almost everywhere has nonnegative integral, so $u (x) \geq 0$ wherever $u$ is defined, and $u$ is defined almost everywhere by [F2]; hence the class $H_{t} f$ is $\geq 0$ almost everywhere, which is (ii).

3.1step 2.1step 2.2given∎

Steps 2.1 and 2.2 prove the mass-conservation clause (i) for $L^{1}$ data and the positivity clause (ii) for nonnegative $L^{p}$ data, so the corollary holds.

Depends on

Used by

Infinite propagation speed for nonnegative heat data Corollary
Monotonicity and Lᵖ contractivity of the heat flow Corollary

Dependency tree · two levels

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Sources

Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript, AMS Graduate Studies in Mathematics) (standard reference, not scraped)
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)