An optimal Liouville theorem for the porous medium equation

Perron’s Method for the Porous Medium Equation

This work extends Perron’s method for the porous medium equation in the slow diffusion case. The main result shows that nonnegative continuous boundary functions are resolutive in a general cylindrical domain.

Hamilton’s gradient estimates and Liouville theorems for porous medium equations

Time Splitting for the Liouville Equation in a Random Medium

We consider the Liouville equations with highly heterogeneous Hamiltonians and their numerical solution by a time splitting algorithm. Such equations model the density of particles evolving according to the corresponding Hamiltonian dynamics as well as the propagation of high frequency waves with a wavelength much smaller than the correlation length of the random Hamiltonian. Our main results a...

Probabilistic approximation for a porous medium equation

In this paper, we are interested in the one-dimensional porous medium equation when the initial condition is the distribution function of a probability measure. We associate a nonlinear martingale problem with it. After proving uniqueness for the martingale problem, we show existence owing to a propagation of chaos result for a system of weakly interacting di usion processes. The particle syste...

The Porous Medium Equation. New contractivity results

We review some lines of recent research in the theory of the porous medium equation. We then proceed to discuss the question of contractivity with respect to the Wasserstein metrics: we show contractivity in one space dimension in all distances dp, 1 ≤ p ≤ ∞, and show a negative result for the d∞ metric in several dimensions. We end with a list of problems. Mathematics Subject Classification. 3...