Questions: Parabolic PDE Theory (Heat Kernel and Regularity)

Question 1

The parabolic scaling that relates time and space derivatives is:

• A. One time derivative is equivalent to two spatial derivatives
• B. One time derivative is equivalent to one spatial derivative
• C. Two time derivatives are equivalent to two spatial derivatives
• D. Time and space derivatives are independent

Answer: (A)

The heat equation u_t = Δu shows that ∂/∂t scales like ∂²/∂x². Under the parabolic scaling (x,t) → (λx, λ²t), both sides of the equation scale the same way. This is why parabolic Sobolev spaces W^{1,0;2,1} count one time derivative as worth two spatial derivatives.

Question 2

A solution of the heat equation with merely L² initial data becomes C^∞ for all t > 0.

• T. True
• F. False

Answer: True

This is the instantaneous smoothing property of parabolic equations. The heat kernel is C^∞ for t > 0, and the solution u(·,t) = K(·,t) * u₀ inherits all the smoothness of K. This is in stark contrast to hyperbolic equations, where initial singularities persist forever.

Question 3

What is the Schauder estimate for the heat equation?

Model answer: If u_t - Δu = f with f ∈ C^{k,α} in space and C^{k/2,α/2} in time, then u ∈ C^{k+2,α} in space and C^{(k+2)/2, α/2} in time

Parabolic Schauder estimates are the analogue of elliptic Schauder estimates, with the anisotropic parabolic scaling. They provide the optimal regularity in Holder spaces and are essential for nonlinear parabolic equations.

Question 4

The heat equation is time-reversible: given the solution at time T, we can uniquely recover the initial data.

• T. True
• F. False

Answer: False

The backward heat equation u_t = -Δu is ill-posed: the smoothing effect of forward evolution destroys information about high-frequency components of the initial data. This irreversibility is connected to the second law of thermodynamics and the arrow of time.