The Fourier transform and the heat kernel

On the whole line there are no boundary conditions to pick out a discrete set of modes, and the Fourier series becomes the Fourier transform \[ \hat f(\xi) = \int_{-\infty}^\infty f(x)\,e^{-ix\xi}\,dx, \qquad f(x) = \frac1{2\pi}\int_{-\infty}^\infty\hat f(\xi)\,e^{ix\xi}\,d\xi. \] Its key property: integration by parts gives \( \widehat{f'}(\xi) = i\xi\,\hat f(\xi) \). Differentiation becomes multiplication, so constant-coefficient PDEs become ODEs in \( t \), one for each frequency \( \xi \).

Transform \( u_t = u_{xx} \) in \( x \): \( \hat u_t = -\xi^2\hat u \), so \( \hat u(\xi, t) = e^{-\xi^2t}\hat f(\xi) \). A product of transforms is the transform of a convolution, and \( e^{-\xi^2t} \) is the transform of a Gaussian, so \[ u(x,t) = \int_{-\infty}^\infty K_t(x - y)f(y)\,dy, \qquad K_t(x) = \frac{1}{\sqrt{4\pi t}}\,e^{-x^2/(4t)}. \] The heat kernel \( K_t \) is positive, has total mass 1 for every \( t \) (the example checks \( t = 1 \)), has variance \( \int x^2K_t\,dx = 2t \), and concentrates at the origin as \( t \to 0^+ \). In \( \mathbb R^n \) it is \( (4\pi t)^{-n/2}e^{-|x|^2/(4t)} \).

Plancherel's theorem says \( \int|\hat f|^2\,d\xi = 2\pi\int|f|^2\,dx \): the transform preserves energy up to a constant. Combined with the formula for \( \hat u \) it shows at once that \( \int|u(x,t)|^2\,dx \) decreases, and that high frequencies are damped by \( e^{-\xi^2t} \), which is the smoothing effect in its cleanest form.

Picture it: put a unit of heat at one point. At time \( t \) it has become a bell curve of width about \( \sqrt{2t} \) and height \( 1/\sqrt{4\pi t} \): half as tall and twice as wide after four times as long, with the area underneath fixed at 1.

Think it: the kernel obeys the scaling \( K_t(x) = t^{-1/2}K_1(x/\sqrt t) \), the same parabolic scaling as the equation. The Navier-Stokes equations can be rewritten as \( u(t) = e^{t\nu\Delta}u_0 - \int_0^te^{(t-s)\nu\Delta}\mathbb P\nabla\cdot(u\otimes u)(s)\,ds \) using this kernel (with \( \mathbb P \) the projection onto divergence-free fields), and local existence theory for Navier-Stokes is a fixed-point argument for that integral equation.