### Abstract

The initial value problem for the linearized spatially-homogeneous Boltzmann equation has the form ∂f/∂t+Lf=0 with f(ξ, t=0) given. The linear operator L operates only on the ξ variable and is non-negative, but, for the soft potentials considered here, its continuous spectrum extends to the origin. Thus one cannot expect exponential decay for f, but in this paper it is shown that f decays like e^{-λ}t^{β} with β<1. This result will be used in Part II to show existence of solutions of the initial value problem for the full nonlinear, spatially dependent problem for initial data that is close to equilibrium.

Original language | English (US) |
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Pages (from-to) | 71-95 |

Number of pages | 25 |

Journal | Communications in Mathematical Physics |

Volume | 74 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1980 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*74*(1), 71-95. https://doi.org/10.1007/BF01197579

**The Boltzmann equation with a soft potential - I. Linear, spatially-homogeneous.** / Caflisch, Russel.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 74, no. 1, pp. 71-95. https://doi.org/10.1007/BF01197579

}

TY - JOUR

T1 - The Boltzmann equation with a soft potential - I. Linear, spatially-homogeneous

AU - Caflisch, Russel

PY - 1980/2

Y1 - 1980/2

N2 - The initial value problem for the linearized spatially-homogeneous Boltzmann equation has the form ∂f/∂t+Lf=0 with f(ξ, t=0) given. The linear operator L operates only on the ξ variable and is non-negative, but, for the soft potentials considered here, its continuous spectrum extends to the origin. Thus one cannot expect exponential decay for f, but in this paper it is shown that f decays like e-λtβ with β<1. This result will be used in Part II to show existence of solutions of the initial value problem for the full nonlinear, spatially dependent problem for initial data that is close to equilibrium.

AB - The initial value problem for the linearized spatially-homogeneous Boltzmann equation has the form ∂f/∂t+Lf=0 with f(ξ, t=0) given. The linear operator L operates only on the ξ variable and is non-negative, but, for the soft potentials considered here, its continuous spectrum extends to the origin. Thus one cannot expect exponential decay for f, but in this paper it is shown that f decays like e-λtβ with β<1. This result will be used in Part II to show existence of solutions of the initial value problem for the full nonlinear, spatially dependent problem for initial data that is close to equilibrium.

UR - http://www.scopus.com/inward/record.url?scp=34250258536&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34250258536&partnerID=8YFLogxK

U2 - 10.1007/BF01197579

DO - 10.1007/BF01197579

M3 - Article

VL - 74

SP - 71

EP - 95

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -