# 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

### 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

### 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

### 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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