The Heat Equation

Temperature diffusion on curves and surfaces — the simplest geometric flow

The Heat Equation

The heat equation ∂u/∂t = Δu is the simplest and most fundamental geometric flow. It describes how a temperature distribution evolves over time, smoothing out hot spots and cold spots until the temperature becomes uniform.

This equation is the prototype for all geometric flows: it takes something irregular and makes it smoother. Understanding the heat equation is the gateway to curve shortening flow, mean curvature flow, and ultimately Ricci flow.

Interactive: Heat Diffusion on a Bar

Click on the bar to place heat sources and watch temperature diffuse over time. The heat equation drives hot regions to cool down and cold regions to warm up, always seeking equilibrium.

The Laplacian Operator

The key ingredient is the Laplacian Δu, which measures how much a function's value at a point differs from its average in a small neighborhood:

Δu = ∂²u/∂x² (in 1D)
Δu = ∂²u/∂x² + ∂²u/∂y² (in 2D)

When Δu > 0, the point is cooler than its neighbors and heats up. When Δu < 0, the point is warmer and cools down. This is why the heat equation always smooths — it pulls every point toward the local average.

The Maximum Principle

A cornerstone result: under the heat equation, the maximum temperature can only decrease and the minimum can only increase.

  • No new extrema: Hot spots cool, cold spots warm
  • Monotone bounds: max(u) is non-increasing in time
  • Uniqueness: Solutions are determined by initial data

Conservation of Energy

The total heat (integral of u) is conserved. Heat is never created or destroyed — it merely redistributes.

d/dt ∫ u(x,t) dx = 0

This is the mathematical expression of the first law of thermodynamics applied to an insulated system.

Interactive: The Heat Kernel

The heat kernel G(x,t) is the fundamental solution — the response to a single point source of heat. Watch a sharp spike spread into ever-wider Gaussians while the total area stays exactly 1.

The Gaussian Heat Kernel

G(x,t) = (1/√(4πt)) · exp(-x²/4t)

Any initial temperature distribution u(x,0) = f(x) can be recovered by convolving with the heat kernel:

u(x,t) = ∫ G(x-y,t) · f(y) dy

This convolution formula shows that the heat equation is a smoothing machine: convolving with a Gaussian blurs sharp features. As t grows, the Gaussian widens and smoothing intensifies.

Interactive: Curve Smoothing via Heat Flow

Apply the heat equation to smooth a closed curve — each vertex moves toward the average of its neighbors. This is a preview of curve shortening flow, the topic of Lesson 2.

Key Takeaways

  • The heat equation ∂u/∂t = Δu is the prototype for all geometric flows — it smooths irregularities
  • The Laplacian measures how a value differs from its local average, driving the flow toward uniformity
  • The maximum principle guarantees that extrema decay — solutions become smoother over time
  • The heat kernel is a spreading Gaussian that encodes the fundamental solution
  • Curve smoothing by averaging neighbors is a discrete version of heat flow on curves — a bridge to curve shortening flow

Next: Curve Shortening Flow — applying heat-like smoothing to evolve curves, with deep connections to topology and the isoperimetric inequality.