Jean Dolbeault

Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions

Abstract In this paper, we find optimal constants of a special class of Gagliardo–Nirenberg type inequalities which turns out to interpolate between the classical Sobolev inequality and the Gross logarithmic Sobolev inequality. These inequalities provide an optimal decay rate (measured by entropy methods) of the intermediate asymptotics of solutions to nonlinear diffusion equations.

The optimal Euclidean Lp-Sobolev logarithmic inequality☆

We prove an optimal logarithmic Sobolev inequality in W1,p(Rd). Explicit minimizers are given. This result is connected with best constants of a special class of Gagliardo–Nirenberg-type inequalities.

A new class of transport distances between measures

We introduce a new class of distances between nonnegative Radon measures in

Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities

{\mathbb{R}^d}

Kinetic models and quantum effects: A modified Boltzmann equation for Fermi-Dirac particles

. They are modeled on the dynamical characterization of the Kantorovich-Rubinstein-Wasserstein distances proposed by Benamou and Brenier (Numer Math 84:375–393, 2000) and provide a wide family interpolating between the Wasserstein and the homogeneous

Free energy and solutions of the Vlasov-Poisson-Fokker-Planck system: external potential and confinement (Large time behavior and steady states)

{W^{-1,p}_\gamma}