© 2026 NeckPinch · www.neckpinch.com · All rights reserved.

Book 6A

The Heat Equation and Its Relatives

The analytic engine of Ricci flow

Laplace and heat equations, maximum principles, weak solutions and regularity, nonlinear parabolic equations and blow-up, curve shortening, the calculus of variations, and entropy.

  • Main companionPartial Differential Equations · Lawrence C. Evans
  • ReferenceSecond Order Parabolic Differential Equations · Gary Lieberman
In preparation

Contents

  1. 1
    What a PDE Is

    Well-posedness, classification, parabolic scaling, and why the backward heat equation is ill-posed.

    In preparation

  2. 2
    Harmonic Functions

    The mean value property, the maximum principle, Harnack and Liouville.

    In preparation

  3. 3
    The Heat Equation on ℝⁿ

    The heat kernel, smoothing, uniqueness, Brownian motion, and a Gaussian that runs to Perelman.

    In preparation

  4. 4
    Maximum Principles

    Weak and strong maximum principles, comparison, and invariant regions for systems.

    In preparation

  5. 5
    Weak Solutions and Elliptic Regularity

    Lax–Milgram in action, energy estimates and bootstrapping.

    In preparation

  6. 6
    Parabolic Regularity

    Parabolic Hölder spaces, Schauder estimates and Bernstein’s gradient method.

    In preparation

  7. 7
    Nonlinear Parabolic Equations

    Short-time existence by linearisation, continuation criteria and finite-time blow-up.

    In preparation

  8. 8
    Curve Shortening and the First Geometric Flows

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.

    In preparation

  9. 9
    Calculus of Variations and Gradient Flows

    Euler–Lagrange equations, minimal surfaces, soap films and gradient flows.

    In preparation

  10. 10
    Entropy, Information and Diffusion

    Entropy along the heat flow, the log-Sobolev inequality and Li–Yau: the ancestors of Perelman’s entropy.

    In preparation

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