Adrien Blanchet

Convergence of the Mass-Transport Steepest Descent Scheme for the Subcritical Patlak-Keller-Segel Model

Variational steepest descent approximation schemes for the modified Patlak-Keller-Segel equation with a logarithmic interaction kernel in any dimension are considered. We prove the convergence of the suitably interpolated in time implicit Euler scheme, defined in terms of the Euclidean Wasserstein distance, associated with this equation for subcritical masses. As a consequence, we recover the recent result about the global in time existence of weak solutions to the modified Patlak-Keller-Segel equation for the logarithmic interaction kernel in any dimension in the subcritical case. Moreover, we show how this method performs numerically in dimension one. In this particular case, this numerical scheme corresponds to a standard implicit Euler method for the pseudoinverse of the cumulative distribution function. We demonstrate its capabilities to reproduce the blow-up of solutions for supercritical masses easily without the need of mesh-refinement.

Asymptotics of the Fast Diffusion Equation via Entropy Estimates

We consider non-negative solutions of the fast diffusion equation utxa0=xa0Δum with mxa0∈ (0, 1) in the Euclidean space

Stochastic Stokes' drift, homogenized functional inequalities, and large time behavior of Brownian ratchets

{{mathbb R}^d}

On the one-dimensional parabolic obstacle problem with variable coefficients

, d ≧ 3, and study the asymptotic behavior of a natural class of solutions in the limit corresponding to t → ∞ for m ≧ mcxa0=xa0(dxa0−xa02)/d, or as t approaches the extinction time when m < mc. For a class of initial data, we prove that the solution converges with a polynomial rate to a self-similar solution, for t large enough if m ≧ mc, or close enough to the extinction time if m < mc. Such results are new in the range m ≦ mc where previous approaches fail. In the range mcxa0<xa0mxa0<xa01, we improve on known results.

Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions.

This review is dedicated to recent results on the 2d parabolic-elliptic Patlak-Keller-Segel model, and on its variant in higher dimensions where the diffusion is of critical porous medium type. Both of these models have a critical mass

Critical mass for a Patlak–Keller–Segel model with degenerate diffusion in higher dimensions

M_c

On the continuity of the time derivative of the solution to the parabolic obstacle problem with variable coefficients

such that the solutions exist globally in time if the mass is less than

Hardy-Poincaré inequalities and applications to nonlinear diffusions

M_c