Entropy methods and convergence to equilibrium

Take heat flow with κ>0 on a connected bounded smooth insulated domain, or on a connected periodic torus. Its mass M is conserved, so the equilibrium candidate is the mean ū=M/|Ω|. For a positive sufficiently smooth solution bounded away from zero on the time interval of calculation, define relative entropy as the integral of u log(u/ū)−u+ū. The integrand is nonnegative by convexity and vanishes only at u=ū. The linear terms integrate to zero because mass matches the reference.

Differentiate the functional and use the PDE. Integrating by parts removes the boundary term under zero normal flux or periodicity, leaving minus κ times the integral of |∇u|²/u. This is nonnegative dissipation with a negative sign. For solutions that touch zero, the logarithm and division require an approximation or a justified extended formulation; the classical calculation should not be applied pointwise without that step. The reference mean must be positive for this logarithmic functional, so an identically zero mass solution is handled separately.

Knowing H′≤0 says entropy cannot increase, but it does not identify an exponential rate or rule out slow relaxation. If a valid logarithmic Sobolev inequality in the chosen normalization gives H≤C I, then H′≤−(κ/C)H and Gronwall gives exponential decay. The value and meaning of C depend on domain, reference measure, and normalization. An inequality written for probability densities cannot be transplanted to an unnormalized mass convention without checking how all factors transform.

A simpler quadratic functional is half the squared L² distance to the mean. Its dissipation is κ times the squared gradient norm, and a mean-zero Poincaré inequality gives exponential decay with rate 2κλ₁ for this quadratic energy. Logarithmic entropy and squared L² distance are different functionals. They may control related notions of closeness under additional positivity or boundedness assumptions, but their numerical values and sharp constants should not be conflated. Even identifying the correct zero modes is necessary before claiming a positive gap.

On an insulated disconnected domain, mass is conserved separately on each component, so the equilibrium is componentwise constant. One global mean can leave a persistent mismatch and invalidate a claimed global gap. Under Dirichlet conditions, a nonzero prescribed boundary or forcing changes the equilibrium and can create boundary contributions in the entropy balance. On the whole space, a finite-mass heat solution spreads rather than approaching a nonzero spatial constant; self-similar coordinates and a corresponding reference profile are needed for a meaningful equilibrium comparison.

For nonlinear diffusion, a power-type energy may be more natural than the logarithm. For sufficiently regular whole-space porous-medium flow with vanishing boundary flux, differentiating the integral of uᵐ⁺¹/(m+1) gives minus the integral of |∇uᵐ|². Turning such dissipation into a sharp convergence theorem requires mass constraints, scaling, compactness or functional inequalities, and an appropriate stationary or self-similar reference. A formal negative derivative motivates the analysis; it is not by itself a proof of existence, uniqueness, or a universal relaxation rate.