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Sine modes decay under Dirichlet heat flow

Example

Assume Countable Choice. For every integer $k \geq 1$ and every $T > 0$ , the function $u_{k} (x, t) : = e^{- k^{2} t} \sin (k x)$ is a classical solution of the heat equation $u_{t} = Δ u$ on $(0, π) \times (0, T]$ with Dirichlet data $u_{k} (0, t) = u_{k} (π, t) = 0$ and initial data $u_{k} (x, 0) = \sin (k x)$ ; it is smooth up to $t = 0$ in this one-dimensional setting. Its $L^{2} (0, π)$ norm decays at the rate of the $k$ -th Dirichlet eigenvalue, $∥ u_{k} (\cdot, t) ∥_{2} = e^{- k^{2} t} \sqrt{π / 2} (t \geq 0),$ so higher modes decay faster, and the nodal set of $u_{k} (\cdot, t)$ does not depend on $t$ .

Facts & Assumptions

Given: Countable Choice, an integer $k \geq 1$ , $T > 0$ , and the function $u_{k} (x, t) = e^{- k^{2} t} \sin (k x)$ on $[0, π] \times [0, T]$ .

[A1]

Countable Choice is the ambient hypothesis, inherited through the $L^{2}$ dictionary in step 3.1 (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

The heat operator is $\partial_{t} - Δ$ , with $Δ = \partial_{x}^{2}$ in one space dimension (The heat operator, the heat equation, and the Cauchy problem, The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field).

[F2]

$(\sin x)^{'} = \cos x$ and $(\cos x)^{'} = - \sin x$ (The derivatives of sine and cosine are cosine and minus sine), hence $\partial_{x}^{2} \sin (k x) = - k^{2} \sin (k x)$ , and $\sin 0 = \sin (k π) = 0$ (The zero sets of sine and cosine and the least positive common period 2 pi); the exponential factor has time derivative $- k^{2} e^{- k^{2} t}$ by The exponential function is smooth and $(\exp)^{'} = \exp$ and The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ .

[F3]

The $L^{2} (0, π)$ inner product is $⟨ f, g ⟩ = \int_{0}^{π} f g$ on the quotient space of The space $L^{p} (μ)$ as the quotient by null functions ( $L^{2}$ with the integral pairing is a Hilbert space), and $\int_{0}^{π} \sin (k x)^{2} d x = π / 2$ (L2 normalisation of the sine modes on an interval).

Verification

Given: Countable Choice, $k \geq 1$ , $T > 0$ , and $u_{k} (x, t) = e^{- k^{2} t} \sin (k x)$ .

1.1F1F2given

The function $u_{k}$ is smooth on the closed rectangle (a product of a smooth exponential and a smooth sine), and [F2] gives $\partial_{t} u_{k} = - k^{2} e^{- k^{2} t} \sin (k x) = - k^{2} u_{k}$ together with $\partial_{x}^{2} u_{k} = - k^{2} e^{- k^{2} t} \sin (k x) = - k^{2} u_{k}$ ; hence $u_{k, t} = \partial_{x}^{2} u_{k} = Δ u_{k}$ on the open rectangle by [F1].

1.2F2given

The boundary values are $u_{k} (0, t) = e^{- k^{2} t} \sin 0 = 0$ and $u_{k} (π, t) = e^{- k^{2} t} \sin (k π) = 0$ for every $t$ , while $u_{k} (x, 0) = e^{0} \sin (k x) = \sin (k x)$ ; the nodal set at time $t$ is ${x \in (0, π) : \sin (k x) = 0} = {m π / k : 1 \leq m \leq k - 1}$ , independent of $t$ because the positive factor $e^{- k^{2} t}$ never vanishes.

2.1A1F3given

By [F3] the squared norm of step 1.1's function is $∥ u_{k} (\cdot, t) ∥_{2}^{2} = e^{- 2 k^{2} t} \int_{0}^{π} \sin (k x)^{2} d x = e^{- 2 k^{2} t} π / 2$ , so $∥ u_{k} (\cdot, t) ∥_{2} = e^{- k^{2} t} \sqrt{π / 2}$ for every $t \geq 0$ .

3.1step 2.1given

Since $k \mapsto e^{- k^{2} t}$ is strictly decreasing in $k \geq 1$ for every fixed $t > 0$ , higher modes decay faster at each positive time, with the ratio $e^{- (k^{2} - l^{2}) t}$ between the $k$ -th and $l$ -th modes for $k > l$ .

4.1step 1.1step 1.2step 2.1step 3.1given∎

Steps 1.1, 1.2, 2.1 and 3.1 verify that $u_{k}$ is a classical solution smooth up to $t = 0$ with Dirichlet data, decay rate $e^{- k^{2} t} \sqrt{π / 2}$ , faster decay for higher modes, and a time-independent nodal set.

Depends on

Used by

Backward heat amplifies small high-frequency errors Counterexample

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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)