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December  2011, 4(4): 857-871. doi: 10.3934/krm.2011.4.857

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, France

2. 

ICES & Department of Mathematics and ICES, University of Texas, Austin, Texas 78712, United States

3. 

University of Pavia, Department of Mathematics, Via Ferrata 1, 27100 Pavia, Italy

Received  April 2011 Revised  October 2011 Published  November 2011

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.
Citation: Naoufel Ben Abdallah, Irene M. Gamba, Giuseppe Toscani. On the minimization problem of sub-linear convex functionals. Kinetic & Related Models, 2011, 4 (4) : 857-871. doi: 10.3934/krm.2011.4.857
References:
[1]

J. A. Carrillo, A. Jüngel, P. A. Markowich, G. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities,, Monatsh. Math., 133 (2001), 1.  doi: 10.1007/s006050170032.  Google Scholar

[2]

S. Chapman and T. Cowling, "The Mathematical Theory of Non- Uniform Gases. The Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases,", Third edition, (1970).   Google Scholar

[3]

F. Demengel and R. Temam, Convex functions of a measure and applications,, Indiana Math. J., 33 (1984), 673.  doi: 10.1512/iumj.1984.33.33036.  Google Scholar

[4]

M. Escobedo, S. Mischler and M. A. Valle, Entropy maximization problem for quantum relativistic particles,, Bull. Soc. Math. France, 133 (2005), 87.   Google Scholar

[5]

R. T. Glassey and W. A. Strauss, Asymptotic stability of the relativistic Maxwellian,, Publ. Res. Inst. Math. Sciences, 29 (1993), 301.  doi: 10.2977/prims/1195167275.  Google Scholar

[6]

G. Kaniadakis and P. Quarati, Kinetic equation for classical particles obeying an exclusion principle,, Phys. Rev. E, 48 (1993), 4263.  doi: 10.1103/PhysRevE.48.4263.  Google Scholar

[7]

G. Kaniadakis and P. Quarati, Classical model of bosons and fermions,, Phys. Rev. E., 49 (1994), 5103.  doi: 10.1103/PhysRevE.49.5103.  Google Scholar

[8]

X. Lu, A modified Boltzmann equation for Bose-Einstein particles: Isotropic solutions and long-time behavior,, J. Statist. Phys., 98 (2000), 1335.  doi: 10.1023/A:1018628031233.  Google Scholar

show all references

References:
[1]

J. A. Carrillo, A. Jüngel, P. A. Markowich, G. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities,, Monatsh. Math., 133 (2001), 1.  doi: 10.1007/s006050170032.  Google Scholar

[2]

S. Chapman and T. Cowling, "The Mathematical Theory of Non- Uniform Gases. The Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases,", Third edition, (1970).   Google Scholar

[3]

F. Demengel and R. Temam, Convex functions of a measure and applications,, Indiana Math. J., 33 (1984), 673.  doi: 10.1512/iumj.1984.33.33036.  Google Scholar

[4]

M. Escobedo, S. Mischler and M. A. Valle, Entropy maximization problem for quantum relativistic particles,, Bull. Soc. Math. France, 133 (2005), 87.   Google Scholar

[5]

R. T. Glassey and W. A. Strauss, Asymptotic stability of the relativistic Maxwellian,, Publ. Res. Inst. Math. Sciences, 29 (1993), 301.  doi: 10.2977/prims/1195167275.  Google Scholar

[6]

G. Kaniadakis and P. Quarati, Kinetic equation for classical particles obeying an exclusion principle,, Phys. Rev. E, 48 (1993), 4263.  doi: 10.1103/PhysRevE.48.4263.  Google Scholar

[7]

G. Kaniadakis and P. Quarati, Classical model of bosons and fermions,, Phys. Rev. E., 49 (1994), 5103.  doi: 10.1103/PhysRevE.49.5103.  Google Scholar

[8]

X. Lu, A modified Boltzmann equation for Bose-Einstein particles: Isotropic solutions and long-time behavior,, J. Statist. Phys., 98 (2000), 1335.  doi: 10.1023/A:1018628031233.  Google Scholar

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