Uniform estimates on the Fisher information for solutions to Boltzmann and Landau equations

Ricardo J. Alonso; Véronique Bagland; Bertrand Lods

doi:10.3934/krm.2019044

Article Contents

2019, Volume 12, Issue 5: 1163-1183. Doi: 10.3934/krm.2019044

This issue Previous Article On the blow-up criterion and global existence of a nonlinear PDE system in biological transport networks Next Article Diffusion limit for a kinetic equation with a thermostatted interface

Uniform estimates on the Fisher information for solutions to Boltzmann and Landau equations

1.
Departamento de Matemática, PUC-Rio, Rua Marquês de São Vicente 225, Rio de Janeiro, CEP 22451-900, Brazil
2.
Université Clermont Auvergne, LMBP, UMR 6620 - CNRS, Campus des Cézeaux, 3, place Vasarely, TSA 60026, CS 60026, F-63178 Aubière Cedex, France
3.
Università degli Studi di Torino & Collegio Carlo Alberto, Department ESOMAS, Corso Unione Sovietica, 218/bis, 10134 Torino, Italy

^* Corresponding author
^* Corresponding author

Received: January 31, 2019

Revised: March 31, 2019

Published: July 2019

B. L gratefully acknowledges the financial support from the Italian Ministry of Education, University and Research (MIUR), "Dipartimenti di Eccellenza" grant 2018-2022.

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Abstract

In this note we prove that, under some minimal regularity assumptions on the initial datum, solutions to the spatially homogenous Boltzmann and Landau equations for hard potentials uniformly propagate the Fisher information. The proof of such a result is based upon some explicit pointwise lower bound on solutions to Boltzmann equation and strong diffusion properties for the Landau equation. We include an application of this result related to emergence and propagation of exponential tails for the solution's gradient. These results complement estimates provided in [24,26,15,23].

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Mathematics Subject Classification: 35Q20, 82C05, 82C22, 82C40.

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References

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