### Abstract

We study the long-time behavior of symmetric solutions of the nonlinear Boltzmann equation and a closely related nonlinear Fokker-Planck equation. If the symmetry of the solutions corresponds to shear flows, the existence of stationary solutions can be ruled out because the energy is not conserved. After anisotropic rescaling both equations conserve the energy. We show that the rescaled Boltzmann equation does not admit stationary densities of Maxwellian type (exponentially decaying). For the rescaled Fokker-Planck equation we demonstrate that all solutions converge to a Maxwellian in the long-time limit, however the convergence rate is only algebraic, not exponential.

Original language | English |
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Pages (from-to) | 1321–1348 |

Number of pages | 28 |

Journal | SIAM Journal on Mathematical Analysis (SIMA) |

Volume | 51 |

Issue number | 2 |

Early online date | 18 Apr 2019 |

DOIs | |

Publication status | Published - 31 Dec 2019 |

### Keywords

- Boltzmann equation
- Fokker-Planck
- Hypocoercivity
- Objective solution

### ASJC Scopus subject areas

- Analysis
- Computational Mathematics
- Applied Mathematics

## Fingerprint Dive into the research topics of 'Rescaled Objective Solutions of Fokker-Planck and Boltzmann equations'. Together they form a unique fingerprint.

## Profiles

### Karsten Matthies

- Department of Mathematical Sciences - Senior Lecturer
- Probability Laboratory at Bath
- EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)

Person: Research & Teaching

## Cite this

Matthies, K., & Theil, F. (2019). Rescaled Objective Solutions of Fokker-Planck and Boltzmann equations.

*SIAM Journal on Mathematical Analysis (SIMA)*,*51*(2), 1321–1348. https://doi.org/10.1137/18M1202335