On the minimization problem of sub-linear convex functionals

Naoufel Ben Abdallah; Irene M. Gamba; Giuseppe Toscani

doi:10.3934/krm.2011.4.857

Article Contents

2011, Volume 4, Issue 4: 857-871. Doi: 10.3934/krm.2011.4.857

This issue Previous Article Second order averaging for the nonlinear Schrödinger equation with strongly anisotropic potential Next Article Fractional diffusion limit for collisional kinetic equations: A Hilbert expansion approach

On the minimization problem of sub-linear convex functionals

1.
Laboratoire MIP, University Paul Sabatier Toulouse, 118 route de Narbonne F-31062 Toulouse Cedex 9
2.
ICES & Department of Mathematics and ICES, University of Texas, Austin, Texas 78712
3.
University of Pavia, Department of Mathematics, Via Ferrata 1, 27100 Pavia

Received: March 31, 2011

Revised: September 30, 2011

Published: November 2011

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Abstract

The study of the convergence to equilibrium of solutions to Fokker-Planck type equations with linear diffusion and super-linear drift leads in a natural way to a minimization problem for an energy functional (entropy) which relies on a sub-linear convex function. In many cases, conditions linked both to the non-linearity of the drift and to the space dimension allow the equilibrium to have a singular part. We present here a simple proof of existence and uniqueness of the minimizer in the two physically interesting cases in which there is the constraint of mass, and the constraints of both mass and energy. The proof includes the localization in space of the (eventual) singular part. The major example is related to the Fokker-Planck equation introduced in [6, 7] to describe the evolution of both Bose-Einstein and Fermi-Dirac particles.

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Mathematics Subject Classification: Primary: 35B40; Secondary: 35K55, 39B62.

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References

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