Fast diffusion to self-similarity: Complete spectrum, long-time asymptotics, and numerology

The complete spectrum is determined for the operator H = -mp(m-1) Delta + x (.) del on the Sobolev space W-p(1,2) (R-n) formed by closing the smooth functions of compact support with respect to the norm parallel topsiparallel toW(p1,2)(2) ((Rn)) := integral(Rn) \delpsi\(2) rho dx. Here the Barenblatt profile rho is the stationary attractor of the rescaled diffusion equation. partial derivativeu/partial derivativet = Delta(u(m)) + div(x u) in the fast, supercritical regime m is an element of ]n-2/n, 1[. For m greater than or equal to n/n+2, the same diffusion dynamics represent the steepest descent down an entropy E( u) on probability measures with respect to the Wasserstein distance d(2). Formally, the operator H = Hess(rho)E is the Hessian of this entropy at its minimum rho, so the spectral gap H greater than or equal to alpha := 2- n(1 - m) found below suggests the sharp rate of asymptotic convergence: lim(t-->infinity) log d(2) (u(t), p)/t less than or equal to -alpha < 0 from any centered initial data 0 less than or equal to u(0, x) is an element of L-1(R-n) with second moments. This bound improves various results in the literature, and suggests the conjecture that the self-similar solution u(t, x) = R(t)(-n) rho(x/R(t)) is always slowest to converge. The higher eigenfunctions - which are polynomials with hypergeometric radial parts - and the presence of continuous spectrum yield additional insight into the relations between symmetries of R-n and the flow. Thus the rate of convergence can be improved if we are willing to replace the distance to rho with the distance to its nearest mass-preserving dilation (or still better, affine image). The strange numerology of the spectrum is explained in terms of the number of moments of rho.