Seminar: PDEs as Gradient flows (WS12/13)

Time: Wednesday 14-16
Room: Endenicher Allee 60, 2.040


Schedule and Topics

10.10.12 André

  • Gradientflows in $$\mathbb{R}^n$$: Characterization of stationary points, linearization around critical points, convergence by convexity, implicit Euler time-discrete scheme and variational formulation
  • Basic Riemannian Geometry: differentiation on manifolds
  • First part of Otto theorem: convergence of (finite-dimensional) gradient flows of convex energies to stationary points on a Riemannian manifold

17.10.12 André

  • Some more Riemannian Geometry: Riemannian connection/fundamental theorem of Riemannian geometry, geodesics, length/energy of curves, geodesics as energy minimizers, formula for Hessian
  • Introduction to Fokker Planck equation: characterization of equilibrium solution, example Ornstein-Uhlenbeck process, behaviour in non-convex potential

24.10.12 Leonardo

  • Classic results porous medium equation: Self-similar solutions by rescaling, stationary solutions
  • Two gradient flow formulations for porous medium equation: $$L^2$$ and entropic gradient flow

31.10.12 Simon

07.11.12 Angelo

  • Physical derivation of PME: characterzation of unique minimizer and derivation of gradient functional (Darcy’s law, osmotic pressure)
  • Asymptotic formulation: energy functional in rescaled equation
  • Energy-entropy estimate: characterzation of unique minimizer and derivation of gradient functional
  • Convergence in induced distance

12.11.12 (8:30) Stefan

  • Construction of isometric submersion of flat manifold of diffeomorphisms on $\mathbb{R}^n$ onto manifold of densities $\mathcal{M}$
  • Pullback of curves and geodesics under the submersion

28.11.12 Simon

  • Convergence of time-discrete scheme to Fokker-Planck equation

26.11.12 (16:00) Leonardo

  • Some results from Optimal Transport: Gradient and Hessian of convex functions via Alexandrov’s Theorem
  • Identification of Wasserstein distance as induced distance
  • Computation of the Hessians: displacement convexity and lower bounds

05.12.12 Dies Academicus

12.12.12. Angelo

  • Towards rigorous results: smooth setting

19.12.12 Stefan

  • Porous medium equation on manifolds
  • Contraction in Wasserstein space by Eulerian calculus

09.01.13 Simon

  • A non-local non-linear Fokker-Planck equation
  • Example of Dynamic in special regimes
  • setting of a constraint gradient flow

14.01.13 (16:00) Stefan

  • Incompressible Euler equation and Arnolds interpretation
  • Breniers relaxation and relation to Wasserstein distance

 16.01.13 Angelo

21.01.13 (16:00) Leonardo

  • Rate of convergence for Fokker-Planck equation via transport inequalities (logarithmic Sobolev inequality, HWI-inequality)

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