Heat equation and smoothing

Let u denote temperature after taking constant density and heat capacity into the units. Conservation on a region says its rate of change equals negative outward flux plus any source. Fourier's constitutive law sets the flux to −κ∇u with κ>0, so the source-free equation becomes u_t=κΔu. At a smooth local spatial maximum, the Laplacian is nonpositive, giving an intuitive local reason a new higher peak does not grow in an unforced interior. A rigorous maximum principle additionally specifies the domain, boundary data, and regularity assumptions.

Integrating the PDE gives the total-heat balance. On a periodic domain the contributions across matching faces cancel; on an insulated boundary the normal derivative is zero. In both cases the spatial integral stays constant. A boundary held at zero temperature instead exchanges heat with the environment, and its integral need not remain constant. Calling every decreasing heat profile a violation of conservation would omit the boundary reservoir from the bookkeeping. With sources, the source integral joins the flux balance and changes the conclusion again.

For a 2π-periodic profile, the sine and cosine mode of frequency n has second derivative −n² times itself. Its time coefficient therefore solves a'=−κn²a. The zero mode is constant; high frequencies decay faster than low frequencies. With zero Dirichlet values on an interval, the appropriate sine basis has frequencies nπ/L instead. The boundary condition determines the basis, so one cannot select convenient periodic modes and afterward claim they solve arbitrary endpoint values. Initial data expand in the basis in the topology under consideration.

For positive time on the whole space, convolution with a Gaussian heat kernel makes suitable rough initial data smooth. Differentiating the kernel produces finite estimates at each strictly positive time, though those derivative estimates generally deteriorate as time approaches zero. The heat kernel is nonzero at every spatial point for t>0, giving the ideal equation infinite propagation speed. This is a property of the diffusion approximation, not a relativistic signal-speed prediction. Interior smoothing also does not guarantee smoothness at a corner where initial and boundary data disagree.

Multiply the heat equation by u and integrate by parts. With periodic, homogeneous Dirichlet, or homogeneous Neumann conditions, the resulting boundary term vanishes and half the squared L2 norm decreases at rate κ times the squared gradient norm. The same identity applied to the difference of two solutions proves uniqueness and contraction in L2 within the energy class. It is a mathematical energy estimate; the spatial integral of temperature is the separate mass or heat quantity from the flux balance. A constant nonzero Neumann solution has conserved L2 norm because its gradient vanishes.

To inspect spatial scales, increase N in the spectral lab and compare profiles at larger p. More modes introduce finer oscillations; larger p damps their coefficients in this selected finite family. Its terms are sin(nx−t)/nᵖ, so the playback has phase motion with no exponential heat damping. Use the exact worked solution below to calculate thermal decay instead of inferring it from that animation. Seeing which feature a simple model omits is part of connecting a diagram to the PDE rather than mistaking a moving graph for a numerical solution.