Department of Mathematics

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Higher-order time asymptotics of fast diffusion in Euclidean space: a dynamical systems approach

by Robert McCann  (www) | University of Toronto
Time: 12:10  (Thursday, Nov. 24, 2011)
Location: FI210, Fields Institute, 222 College St.
Abstract:
With Jochen Denzler (UT Knoxville) and Herbert Koch (Bonn), we quantify the speed of convergence and higher asymptotics of fast diffusion dynamics on Euclidean space to the Barenblatt (self similar) profile. The degeneracy in the parabolicity of the equation is cured by re-expressing the dynamics on a manifold with a cylindrical end, called the cigar. The nonlinear evolution semigroup becomes differentiable with respect to Hoelder initial data on the cigar. The linearization of the dynamics is given by Laplace-Beltrami operator plus a drift term (which can be suppressed by the introduction of appropriate weights into the function space norm), plus a finite-depth potential well with a universal profile. In the limiting case of the (linear) heat equation, the depth diverges, the number of eigenstates increases without bound, and the continuous spectrum recedes to infinity. We provide a detailed study of the linear and nonlinear problems in Hoelder spaces on the cigar, including a sharp boundedness estimate for the semigroup, and use this as a tool to obtain sharp convergence results toward the Barenblatt solution. In finer convergence results (after modding out symmetries of the problem), a subtle interplay between convergence rates and tail behavior is revealed. The difficulties involved in choosing the right functional analytic spaces in which to carry out the analysis can be interpreted as genuine features of the equation rather than mere annoying technicalities.

Dates in this series

· Thursday, Jan. 07, 2010: Traveling waves for the cubic Szegö equation on the real line (Oana Pocovnicu)

· Thursday, Jan. 14, 2010: Competing Symmetries and convergence of sequences of random symmetrizations  (Almut Burchard)

· Thursday, Jan. 21, 2010: Minimal Navier-Stokes Singularities (Ian Zwiers)

· Thursday, Jan. 28, 2010: New examples on spaces of negative sectional curvature satisfying Ma-Trudinger-Wang conditions (Jiayong Li)

· Thursday, Feb. 25, 2010: The multi-marginal optimal transportation problem (Brendan Pass)

· Thursday, Mar. 04, 2010: Rectifiability of optimal transportation plans (Brendan Pass)

· Thursday, Mar. 11, 2010: Almost Sure Local Well-posedness for the Stochastic KdV-Burgers Equation (Geordie Richards)

· Thursday, Apr. 01, 2010: On the speed of the propagation of the thin film interface (Marina Chugunova)

· Thursday, Apr. 08, 2010: Ricci flow and the determinant of the Laplacian on non-compact manifolds (Frederic Rochon)

· Thursday, Apr. 15, 2010: Ricci flow and the determinant of the Laplacian on non-compact manifolds, Part 2 (Frederic Rochon)

· Thursday, Apr. 29, 2010: The probabilistic method in geometry: Bourgain's Theorem (Dominic Dotterrer)

· Wednesday, Jul. 14, 2010: Regularity of optimal transportation maps on multiple products of spheres (Young-Heon Kim)

· Friday, Jul. 23, 2010: Asymptotics of fast diffusion via dynamical systems (Jochen Denzler)

· Thursday, Sep. 16, 2010: Existence and uniquness of equilibrium in a noisy model of the marriage market (Colin Decker)

· Thursday, Sep. 30, 2010: Invariant Gibbs measures for periodic nonlinear Schrodinger equations (Geordie Richards)

· Thursday, Oct. 07, 2010: Regularity of solutions for a variational problem related to the Rochet-Chone model (Shibing Chen)

· Thursday, Oct. 21, 2010: A Generalized Euler-Lagrange Equation for the Principal-Agent Problem (Quentin Merigot)

· Thursday, Oct. 28, 2010: Global regularity of the reflector problem (Jiakun Liu)

· Thursday, Feb. 17, 2011: Sharp new conformally invariant integral inequalities (Shibing Chen)

· Thursday, Mar. 24, 2011: Aharonov--Bohm Effect in Resonances of Magnetic Schrodinger Operators with Potentials with Supports at Large Separation (Ivana Alexandrova)

· Thursday, Mar. 31, 2011: Global well-posedness for KdV in L2 (Jordan Bell)

· Thursday, Apr. 07, 2011: Structural results on optimal transportation plans (Brendan Pass)

· Thursday, Sep. 22, 2011: Rayleigh--Benard convection. A new bound on the Nusselt number (Christian Seis)

· Thursday, Sep. 29, 2011: Density functional theory and optimal transportation with Coulomb cost (Codina Cotar)

· Thursday, Oct. 06, 2011: Gradient flow methods for thin-film and related higher order equations (Ehsan Kamalinejad)

· Thursday, Oct. 13, 2011: OPTIMAL TRANSPORTATION WITH CAPACITY CONSTRAINTS (Jonathan Korman)

· Thursday, Oct. 20, 2011: Conductivity imaging from one interior measurement in the presence of perfectly conducting and insulating inclusions (Amir Moradifam)

· Thursday, Oct. 27, 2011: Loss of compactness and bubbling for complete minimal surfaces in hyperbolic space (Spryos Alexakis)

· Thursday, Nov. 03, 2011: On the coarsening rates in demixing binary viscous liquids (Christian Seis)

· Thursday, Nov. 10, 2011: Optimal transportation with infinitely many marginals (Brendan Pass)

· Thursday, Nov. 17, 2011: On uniqueness of the waiting-time type solution of the thin film equation (Marina Chugunova)

· Thursday, Nov. 24, 2011: Higher-order time asymptotics of fast diffusion in Euclidean space: a dynamical systems approach (Robert McCann)

· Thursday, Jan. 05, 2012: The Gamma-limit of the Ohta-Kawasaki energy. Derivation of the renormalized energy (Dorian Goldman)

· Thursday, Mar. 01, 2012: Well-posedness and finite-time blowup for the Zakharov system on two-dimensional torus (Nobu Kishimoto)

· Thursday, Mar. 08, 2012: A Generalized Representation Formula for Covariant Tensor Wave Equations on Curved Spacetimes (Arick Shao)

· Thursday, Mar. 29, 2012: Big frequency cascades in the cubic nonlinear Schrödinger flow on the 2-torus (James Colliander)