### Abstract

Original language | English |
---|---|

Pages (from-to) | 1321–1348 |

Number of pages | 28 |

Journal | SIAM Journal on Mathematical Analysis (SIMA) |

Volume | 51 |

Issue number | 2 |

DOIs | |

Publication status | Published - 18 Apr 2019 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Analysis
- Computational Mathematics
- Applied Mathematics

### Cite this

*SIAM Journal on Mathematical Analysis (SIMA)*,

*51*(2), 1321–1348. https://doi.org/10.1137/18M1202335

**Rescaled Objective Solutions of Fokker-Planck and Boltzmann equations.** / Matthies, Karsten; Theil, Florian.

Research output: Contribution to journal › Article

*SIAM Journal on Mathematical Analysis (SIMA)*, vol. 51, no. 2, pp. 1321–1348. https://doi.org/10.1137/18M1202335

}

TY - JOUR

T1 - Rescaled Objective Solutions of Fokker-Planck and Boltzmann equations

AU - Matthies, Karsten

AU - Theil, Florian

N1 - 25 pages

PY - 2019/4/18

Y1 - 2019/4/18

N2 - 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.

AB - 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.

KW - Boltzmann equation

KW - Fokker-Planck

KW - Hypocoercivity

KW - Objective solution

UR - http://www.scopus.com/inward/record.url?scp=85065502858&partnerID=8YFLogxK

U2 - 10.1137/18M1202335

DO - 10.1137/18M1202335

M3 - Article

VL - 51

SP - 1321

EP - 1348

JO - SIAM Journal on Mathematical Analysis (SIMA)

JF - SIAM Journal on Mathematical Analysis (SIMA)

SN - 0036-1410

IS - 2

ER -