Long-time asymptotics for the porous medium equation: The spectrum of the linearized operator

Abstract

We compute the complete spectrum of the displacement Hessian operator, which is obtained from the confined porous medium equation by linearization around its stationary attractor, the Barenblatt profile. On a formal level, the operator is conjugate to the Hessian of the entropy via similarity transformation. We show that the displacement Hessian can be understood as a self-adjoint operator and find that its spectrum is purely discrete. The knowledge of the complete spectrum and the explicit information about the corresponding eigenfunctions give new insights on the convergence and higher order asymptotics of solutions to the porous medium equation towards its attractor.More precisely, the inspection of the eigenfunctions allows to identify symmetries in R N with flows whose rates of convergence are faster than the uniform, translation-governed bound. The present work complements the analogous study of Denzler & McCann for the fast-diffusion equation.Solutions to (1) feature different phenomena depending on the degree of the nonlinearity ρ m . In the case m = 1, equation (1) is the ordinary diffusion (or heat) equation. For 0 < m < 1, the diffusion flux mρ m−1 diverges as ρ vanishes and thus, for suitable initial data, the solution spreads over the whole R N immediately. In this situation, equation (1) is often called the fast diffusion equation. In the porous-medium range m > 1, the diffusion flux increases with the density and degenerates where ρ = 0. Consequently, solutions will preserve a compact support and hence this type of propagation goes by the name slow diffusion. In the present paper, we restrict our attention exclusively to the latter case. For a study of the fast diffusion equation and more general evolutions of porous-medium type, we refer to Vázquez [31] and references therein.The Cauchy problem for the porous medium equation is solved in various settings. As the equation is degenerate parabolic, that is, it is parabolic only where the solution is positive, solutions are in general not classical. More precisely, if the initial datum is zero in some open subset of R N , then there is a slowly propagating free boundary that separates the region where the solution is positive $ The author acknowledges support through NSERC grant 217006-08.

The thin-film equation close to self-similarity

“…For this, it is necessary to rigorously linearize the equation, the framework for which is obtained in the current paper. This strategy was recently successfully applied to the porous medium equation near the self-similar Barenblatt solutions [42,43]. The present work parallels in parts [43] as well as the pioneering work by Koch [32] and the further developments by Kienzler [28] and John [27] 1.2.…”

“…Thanks to the Hardy-Poincaré inequality [42,Lemma 3] and because µ σ+1 µ σ , we can drop the term (w − c) 2 in the integrand. To prove the statement of the Lemma, we thus have to establish the estimate…”

The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion

“…Besides, some more information is available: if the initial function is supported in the ball B R (0), then we can write the upper estimate of the regularization time as By scaling and space displacement we can reduce the proof to the case M = 1 and x 0 = 0. The fine asymptotic analysis uses also the results of Seis [214].…”

Well-Posedness and Self-Similar Asymptotics for a Thin-Film Equation

“…also Carlen [9]). We also refer to the work of Bernoff and Witelski [6], explicitly solving the linear evolution for n = 1 in the 1 + 1-dimensional case and providing a numerical analysis for other n. The work of McCann and Seis [26] generalizes the linear analysis to higher dimensions and more general fourth-order equations by exploiting the gradient flow structure of the thin-film equation and relying on the analogous analysis for the porous medium equation [33]. We also refer to the work of Bernoff and Witelski [6], explicitly solving the linear evolution for n = 1 in the 1 + 1-dimensional case and providing a numerical analysis for other n. The work of McCann and Seis [26] generalizes the linear analysis to higher dimensions and more general fourth-order equations by exploiting the gradient flow structure of the thin-film equation and relying on the analogous analysis for the porous medium equation [33].…”